http://www.rssboard.org/rss-specification 720 https://escholarship.org/uc/ucdavismath_other/rss Recent eScholarship items from Other Sat, 12 Sep 2026 19:29:12 +0000 https://escholarship.org/uc/item/95d5t7zz Consider the following partial “sorting algorithm” on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size n, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this “fixed point forest” exhibits a rich structure. In this paper, we consider the fixed point forest in the limit n → ∞ and show using Stein’s method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path from a random permutation to a leaf converges to the geometric distribution with mean e − 1, and the length... https://escholarship.org/uc/item/95d5t7zz Mon, 14 May 2018 00:00:00 +0000 Johnson, Tobias Schilling, Anne https://orcid.org/0000-0002-2601-7340 Slivken, Erik https://escholarship.org/uc/item/6782j1rh Based on the results in the previous papers that the boundary value problem $y'' - y' + y = y^3, y(0) = 0, y(\infty) =1$ with the condition $y(x) > 0$ for $0 https://escholarship.org/uc/item/6782j1rh Mon, 14 May 2018 00:00:00 +0000 Wang, Chie Bing https://escholarship.org/uc/item/5d18n90g We encode the binomials belonging to the toric ideal IAassociated with an integral d×n matrix A using a short sum of rational functions as introduced by Barvinok (Math. Operations Research 19 (1994) 769) and Barvinok and Woods (J. Amer. Math. Soc. 16 (2003) 957). Under the assumption that d and n are fixed, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal IA, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate applications to enumerative combinatorics, integer programming, and statistics. © 2004 Elsevier Ltd. All rights reserved. https://escholarship.org/uc/item/5d18n90g Mon, 14 May 2018 00:00:00 +0000 De Loera, JA Haws, D Hemmecke, R Huggins, P Sturmfels, B Yoshida, R https://escholarship.org/uc/item/9xq8n2b2 We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions. We also present an extreme function that is a piecewise linear function with some irrational breakpoints, whose extremality follows from a new principle. https://escholarship.org/uc/item/9xq8n2b2 Thu, 22 Feb 2018 00:00:00 +0000 Basu, Amitabh Hildebrand, Robert Köppe, Matthias https://escholarship.org/uc/item/9xf5q08w We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs. https://escholarship.org/uc/item/9xf5q08w Thu, 22 Feb 2018 00:00:00 +0000 Ahmed, Maya Mohsin https://escholarship.org/uc/item/9wj5n6q1 We relate a universal formula for the deformation quantization of arbitrary Poisson structures proposed by Maxim Kontsevich to the Campbell-Baker-Hausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such universal formulae. https://escholarship.org/uc/item/9wj5n6q1 Thu, 22 Feb 2018 00:00:00 +0000 Kathotia, Vinay https://escholarship.org/uc/item/9vj481z1 We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embeddings in a finite sheeted covering. https://escholarship.org/uc/item/9vj481z1 Thu, 22 Feb 2018 00:00:00 +0000 Agol, Ian Long, Darren D. Reid, Alan W. https://escholarship.org/uc/item/9sq5b1zc In this paper we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the other one is conservative. We prove that if the regularizing terms have order higher than one (combined), there exists a global strong solution for arbitrarily large initial data. In the case where the regularizing terms have combined order one, we prove the global existence of solution under some size restriction for the initial data. Moreover, in the case where the conservative regularizing term vanishes, regardless of the order of the diffusion and under certain hypothesis on the initial data, we also prove the global existence of strong solution and we obtain some new entropy balances. Finally, we provide numerics suggesting that, if the order of the diffusion is $0< \alpha<1$, a finite time blow up of the solution is possible. https://escholarship.org/uc/item/9sq5b1zc Thu, 22 Feb 2018 00:00:00 +0000 Burczak, Jan Granero-Belinchón, Rafael Luli, Garving K. https://escholarship.org/uc/item/9ph0c492 We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate the gapless condition on monotone trangles into a pattern avoidance-like condition on Gog words associated. We estimate the number of gapless monotone triangles using a bijection with p-branchings. https://escholarship.org/uc/item/9ph0c492 Thu, 22 Feb 2018 00:00:00 +0000 Ayyer, Arvind Cori, Robert Gouyou-Beauchamps, Dominique https://escholarship.org/uc/item/9mz9g1th We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated subgroup without the bounded packing property. In this example the subgroup is a metabelian retract also. Thus we obtain a negative answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property. https://escholarship.org/uc/item/9mz9g1th Wed, 21 Feb 2018 00:00:00 +0000 Sardar, Pranab https://escholarship.org/uc/item/9mq6x6np This paper is being withdrawn because an error was discovered in lemma 4.3. Although the rest of the paper appears to be correct, this error invalidates the proof of theorem 3.1 and theorem 3.3. https://escholarship.org/uc/item/9mq6x6np Wed, 21 Feb 2018 00:00:00 +0000 Bowen, Lewis https://escholarship.org/uc/item/9mp2x7jm Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen, Louveaux and Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 4, 209--215], some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond... https://escholarship.org/uc/item/9mp2x7jm Wed, 21 Feb 2018 00:00:00 +0000 Basu, Amitabh Hildebrand, Robert Köppe, Matthias https://escholarship.org/uc/item/9m69x2n3 The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply laced, and some remarks made on how this restriction may be relaxed. https://escholarship.org/uc/item/9m69x2n3 Wed, 21 Feb 2018 00:00:00 +0000 Viswanath, Sankaran https://escholarship.org/uc/item/9jc9x40h For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C\log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings. https://escholarship.org/uc/item/9jc9x40h Wed, 21 Feb 2018 00:00:00 +0000 Nachtergaele, Bruno Sims, Robert https://escholarship.org/uc/item/9g5835pj Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges. https://escholarship.org/uc/item/9g5835pj Wed, 21 Feb 2018 00:00:00 +0000 De Loera, J. A. Lee, J. Malkin, P. Margulies, S. https://escholarship.org/uc/item/9fs9j6xr We describe the distribution of the number and location of the fixed points of permu- tations that avoid the pattern 321 via a bijection with rooted plane trees on n + 1 vertices. Using the local limit theorem for Galton-Watson trees, we are able to give an explicit description of the limit of this distribution. https://escholarship.org/uc/item/9fs9j6xr Wed, 21 Feb 2018 00:00:00 +0000 Hoffman, Christopher Rizzolo, Douglas Slivken, Erik https://escholarship.org/uc/item/9c85z3ch A cluster expansion is proposed, that applies to both continuous and discrete systems. The assumption for its convergence involves an extension of the neat Kotecky-Preiss criterion. Expressions and estimates for correlation functions are also presented. The results are applied to systems of interacting classical and quantum particles, and to a lattice polymer model. https://escholarship.org/uc/item/9c85z3ch Wed, 21 Feb 2018 00:00:00 +0000 Ueltschi, Daniel https://escholarship.org/uc/item/9bw5s2r0 We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon) bi-Lipschitz homeomorphism between S'_1 and S'_2 and both covers S'_i have bounded injectivity radius. 2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j: S -> M where S is a surface of bounded injectivity radius and j is a pi_1-injective local isometry onto its image. https://escholarship.org/uc/item/9bw5s2r0 Wed, 21 Feb 2018 00:00:00 +0000 Bowen, Lewis https://escholarship.org/uc/item/9bn0j439 We consider the one-particle sector of the spinless Yukawa model, which describes the interaction of a nucleon with a real field of scalar massive bosons (neutral mesons). The nucleon as well as the mesons have relativistic dispersion relations. In this model we study the dependence of the nucleon mass shell on the ultraviolet cut-off $\Lambda$. For any finite ultraviolet cut-off the nucleon one-particle states are constructed in a bounded region of the energy-momentum space. We identify the dependence of the ground state energy on $\Lambda$ and the coupling constant. More importantly, we show that the model considered here becomes essentially trivial in the limit $\Lambda\to\infty$ regardless of any (nucleon) mass and self-energy renormalization. Our results hold in the small coupling regime. https://escholarship.org/uc/item/9bn0j439 Wed, 21 Feb 2018 00:00:00 +0000 Deckert, D. -A. Pizzo, A. https://escholarship.org/uc/item/7d2127j9 In this paper, we prove the existence portion of the Bertram-Feinberg-Mukai Conjecture for an infinite family of new cases using degeneration technique. This not only leads to a substantial improvement of known results but also develops finer tools for analyzing the moduli of rank two limit linear series which should be useful for other applications to other higher-rank Brill-Noether Problems. https://escholarship.org/uc/item/7d2127j9 Wed, 21 Feb 2018 00:00:00 +0000 Zhang, Naizhen https://escholarship.org/uc/item/98z348cb Recently Swatee Naik and Theodore Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called Strong S-equivalence, and prove that two oriented, ordered links L and L' are related by a sequence of doubled-delta moves if and only if they are Strongly S-equivalent. We also show that this is equivalent to the fact that L' can be obtained from L through a sequence of Y-clasper surgeries, where each clasper leaf has total linking number zero with L. https://escholarship.org/uc/item/98z348cb Tue, 20 Feb 2018 00:00:00 +0000 Gee, Carol Gwosdz https://escholarship.org/uc/item/9826p7w4 The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dynamics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal structures which are discussed here in some detail. While the Parisi solution was proven to be correct, through recent works of F. Guerra and M. Talagrand, the reasons for the effectiveness of the Parisi ansatz still... https://escholarship.org/uc/item/9826p7w4 Tue, 20 Feb 2018 00:00:00 +0000 Aizenman, Michael Sims, Robert Starr, Shannon L. https://escholarship.org/uc/item/96n1m8zz We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic-elliptic fractional case. https://escholarship.org/uc/item/96n1m8zz Tue, 20 Feb 2018 00:00:00 +0000 Burczak, Jan Granero-Belinchón, Rafael https://escholarship.org/uc/item/947573fm Hastings recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this paper we present an expanded version of Hastings' proof. In addition to a careful elucidation of the details of the proof, we also present bounds for the minimal dimensions needed to obtain a counterexample. https://escholarship.org/uc/item/947573fm Tue, 20 Feb 2018 00:00:00 +0000 Fukuda, Motohisa King, Christopher Moser, David https://escholarship.org/uc/item/93w1t4px We study the global existence of solutions to a one-dimensional drift-diffusion equation with logistic term, generalizing the classical parabolic-elliptic Keller-Segel aggregation equation arising in mathematical biology. In particular, we prove that there exists a global weak solution, if the order of the fractional diffusion $\alpha \in (1-c_1, 2]$, where $c_1>0$ is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range $1-c_2<\alpha\leq 2$ with $0 https://escholarship.org/uc/item/93w1t4px Tue, 20 Feb 2018 00:00:00 +0000 Burczak, Jan Granero-Belinchón, Rafael https://escholarship.org/uc/item/92f7s8hn A centrally symmetric $2d$-vertex combinatorial triangulation of the product of spheres $\S^i\times\S^{d-2-i}$ is constructed for all pairs of non-negative integers $i$ and $d$ with $0\leq i \leq d-2$. For the case of $i=d-2-i$, the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order $4d$. The crux of this construction is a definition of a certain full-dimensional subcomplex, $\B(i,d)$, of the boundary complex of the $d$-dimensional cross-polytope. This complex $\B(i,d)$ is a combinatorial manifold with boundary and its boundary provides a required triangulation of $\S^i\times\S^{d-i-2}$. Enumerative characteristics of $\B(i,d)$ and its boundary, and connections to another conjecture of Sparla are also discussed. https://escholarship.org/uc/item/92f7s8hn Tue, 20 Feb 2018 00:00:00 +0000 Klee, Steven Novik, Isabella https://escholarship.org/uc/item/91m894gz We investigate the effect of a magnetic field supported at a single lattice site on the low-energy spectrum of the ferromagnetic Heisenberg XXZ chain. Such fields, caused by impurities, can modify the low-energy spectrum significantly by pinning certain excitations, such as kink and droplet states. We distinguish between different boundary conditions (or sectors), the direction and also the strength of the magnetic field. E.g., with a magnetic field in the z-direction applied at the origin and ++ boundary conditions, there is a critical field strength B_c (which depends on the anisotropy of the Hamiltonian and the spin value) with the following properties: for B < B_c there is a unique ground state with a gap, at the critical value, B_c, there are infinitely many (droplet) ground states with gapless excitations, and for B>B_c there is again a unique ground state but now belonging to... https://escholarship.org/uc/item/91m894gz Tue, 20 Feb 2018 00:00:00 +0000 Contucci, Pierluigi Nachtergaele, Bruno Spitzer, Wolfgang L. https://escholarship.org/uc/item/90w3c8h5 The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of Z≥0 occur as the set of catenary degrees of a numerical monoid (i.e., a co-finite, additive submonoid of Z≥0). In particular, we show that, with one exception, every finite subset of Z≥0 that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid. https://escholarship.org/uc/item/90w3c8h5 Tue, 20 Feb 2018 00:00:00 +0000 O'Neill, Christopher Pelayo, Roberto https://escholarship.org/uc/item/90p8d610 In this note we give a homological explanation of "pure spinors" in YM theories with minimal amount of supersymmetries. We construct A_{\infty} algebras A for every dimension D=3,4,6,10, which for D=10 coincides with homogeneous coordinate ring of pure spinors with coordinate lambda^{alpha}. These algebras are Bar-dual to Lie algebras generated by supersymmetries, written in components. The algebras have a finite number of higher multiplications. The main result of the present note is that in dimension D=3,6,10 the algebra A\otimes \Lambda[\theta^{\alpha}]\otimes Mat_n with a differential D is equivalent to Batalin-Vilkovisky algebra of minimally supersymmetric YM theory in dimension D reduced to a point. This statement can be extended to nonreduced theories. https://escholarship.org/uc/item/90p8d610 Tue, 20 Feb 2018 00:00:00 +0000 Movshev, M. https://escholarship.org/uc/item/8zh748kz We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting traveling waves resembling viscous shock profiles. https://escholarship.org/uc/item/8zh748kz Tue, 20 Feb 2018 00:00:00 +0000 Granero-Belinchón, Rafael Hunter, John K. https://escholarship.org/uc/item/8w84m2mv We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order. https://escholarship.org/uc/item/8w84m2mv Tue, 20 Feb 2018 00:00:00 +0000 Coutand, Daniel Shkoller, Steve https://escholarship.org/uc/item/8ms707mz We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors. https://escholarship.org/uc/item/8ms707mz Tue, 20 Feb 2018 00:00:00 +0000 Coward, Alexander https://escholarship.org/uc/item/8kn652jc A supermanifold P^{3|4} is a target space for twistor string theory. In this note we identify a line bundle of holomorphic volume elements BerM_gP^{3|4} defined on the moduli space of curves of genus g in P^{3|4} with a pullback of a line bundle defined on M_g(pt). We also give some generalizations of this fact. https://escholarship.org/uc/item/8kn652jc Tue, 20 Feb 2018 00:00:00 +0000 Movshev, M. https://escholarship.org/uc/item/8jq218n2 We analyze the Farey spin chain, a one dimensional spin system with effective interaction decaying like the squared inverse distance. Using a polymer model technique, we show that when the temperature is decreased below the (single) critical temperature T_c=1/2, the magnetization jumps from zero to one. https://escholarship.org/uc/item/8jq218n2 Tue, 20 Feb 2018 00:00:00 +0000 Contucci, Pierluigi Kleban, Peter Knauf, Andreas https://escholarship.org/uc/item/8jc1r85g We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erd\H{o}s-R\'enyi random graphs. Let $p_{\text{ppl}}$ and $p_{\text{puz}}$ denote the probability that an edge exists in the respective people and puzzle graphs and define $p_{\text{eff}}= p_{\text{ppl}}p_{\text{puz}}$, the effective probability. We show for constants $c_1>1$ and $c_2> \pi^2/6$ and $c_3 c_1 \log n /n$ the critical effective probability $p^c_{\text{eff}}$, satisfies $c_3 < p^c_{\text{eff}}n\log n < c_2.$ https://escholarship.org/uc/item/8jc1r85g Tue, 20 Feb 2018 00:00:00 +0000 Slivken, Erik https://escholarship.org/uc/item/8j5661bp A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly complete geometric description of smooth and rationally smooth Schubert varieties using primarily combinatorial methods. We also give some partial results for Schubert varieties in Kac-Moody flag varieties. In... https://escholarship.org/uc/item/8j5661bp Tue, 20 Feb 2018 00:00:00 +0000 Richmond, Edward Slofstra, William https://escholarship.org/uc/item/8cj6535b We consider quantum information tasks in an operator algebraic setting, where we consider normal states on von Neumann algebras. In particular, we consider subfactors N⊂M, that is, unital inclusions of von Neumann algebras with trivial center. One can ask the following question: given a normal state ω on M, how much can one learn by only doing measurements from N? We argue how the Jones index [M:N] can be used to give a quantitative answer to this, showing how the rich theory of subfactors can be used in a quantum information context. As an example we discuss how the Jones index can be used in the context of wiretap channels. Subfactors also occur naturally in physics. Here we discuss two examples: rational conformal field theories and Kitaev's toric code on the plane, a prototypical example of a topologically ordered model. There we can directly relate aspects of the general setting to physical properties such as the quantum dimension of the excitations. In the example... https://escholarship.org/uc/item/8cj6535b Tue, 20 Feb 2018 00:00:00 +0000 Naaijkens, Pieter https://escholarship.org/uc/item/8b04q4qz The quantum hyperplane section theorem is explained for nonnegative decomposable concavex bundle spaces over generalized flag manifolds. https://escholarship.org/uc/item/8b04q4qz Tue, 20 Feb 2018 00:00:00 +0000 Kim, Bumsig https://escholarship.org/uc/item/8353f5v5 We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs. https://escholarship.org/uc/item/8353f5v5 Tue, 20 Feb 2018 00:00:00 +0000 Berget, Andrew Eu, Sen-Peng Reiner, Victor https://escholarship.org/uc/item/7z16q91t It is shown that, with an appropriate scaling, the energy of low-lying excitations of the (1,1,...,1) interface in the $d$-dimensional quantum Heisenberg model are given by the spectrum of the $d-1$-dimensional Laplacian on an suitable domain. https://escholarship.org/uc/item/7z16q91t Tue, 20 Feb 2018 00:00:00 +0000 Bolina, Oscar Contucci, Pierluigi Nachtergaele, Bruno Starr, Shannon https://escholarship.org/uc/item/7wx4f0q1 The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. Entropy production and Onsager relations are relevant thermodynamic notions which are shown to emerge from the microscopic description. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy- and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable assumptions on the interaction between the reservoirs. https://escholarship.org/uc/item/7wx4f0q1 Tue, 20 Feb 2018 00:00:00 +0000 Froehlich, Juerg Merkli, Marco Ueltschi, Daniel https://escholarship.org/uc/item/7w32w7j6 We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2. https://escholarship.org/uc/item/7w32w7j6 Tue, 20 Feb 2018 00:00:00 +0000 Basu, Amitabh Hildebrand, Robert Köppe, Matthias Molinaro, Marco https://escholarship.org/uc/item/7vf9s034 This is an extended and corrected version of lecture notes originally written for a one semester course at Leibniz University Hannover. The main aim of the notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model the materials in condensed matter physics. The notes provide the necessary background material to access recent literature in the field. Some of these recent results are also discussed. The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9) Haag-Ruelle scattering... https://escholarship.org/uc/item/7vf9s034 Tue, 20 Feb 2018 00:00:00 +0000 Naaijkens, Pieter https://escholarship.org/uc/item/7vb663mn We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in $L^2(\R)$ whose generator and its translates form a Riesz basis. https://escholarship.org/uc/item/7vb663mn Tue, 20 Feb 2018 00:00:00 +0000 Tessera, Romain Wang, Haichao https://escholarship.org/uc/item/7t40z5w4 We develop the viewpoint that the opposite of the category of W*-algebras and unital normal *-homomorphisms is analogous to the category of sets and functions. For each pair of W*-algebras, we construct their free exponential, which in the context of this analogy corresponds to the collection of functions from one of these W*-algebras to the other. We also show that every unital normal completely positive map between W*-algebras arises naturally from a normal state on their free exponential. https://escholarship.org/uc/item/7t40z5w4 Tue, 20 Feb 2018 00:00:00 +0000 Kornell, Andre https://escholarship.org/uc/item/7rs6194j We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \*\sigma - o(N^{-6/11+\epsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\epsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\epsilon >0, $ one has $$ \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon}) $$ with probability going to 1 as $N \to \infty. $ https://escholarship.org/uc/item/7rs6194j Tue, 20 Feb 2018 00:00:00 +0000 Peche, Sandrine Soshnikov, Alexander https://escholarship.org/uc/item/7qj0v04m We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. We prove that for J>= 3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum. https://escholarship.org/uc/item/7qj0v04m Tue, 20 Feb 2018 00:00:00 +0000 Mulherkar, Jaideep Nachtergaele, Bruno Sims, Robert Starr, Shannon https://escholarship.org/uc/item/7qf8z2c2 Recently Hastings proved the existence of random unitary channels which violate the additivity conjecture. In this paper we use Hastings' method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of non-unital channels which violate additivity of minimal output entropy. https://escholarship.org/uc/item/7qf8z2c2 Tue, 20 Feb 2018 00:00:00 +0000 Fukuda, Motohisa King, Christopher https://escholarship.org/uc/item/7nh0z08h Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf s},{\bf D})$, i.e., a number of integer solutions to a linear system ${\bf x} \ge 0, {\bf D x} = {\bf s}$. It is shown that $W({\bf s},{\bf D})$ can be expressed through the vector Bernoulli polynomials of higher order. https://escholarship.org/uc/item/7nh0z08h Tue, 20 Feb 2018 00:00:00 +0000 Rubinstein, Boris Y. https://escholarship.org/uc/item/7n64498k In 1980 Provan and Billera defined the notion of weak $k$-decomposability for pure simplicial complexes. They showed the diameter of a weakly $k$-decomposable simplicial complex $\Delta$ is bounded above by a polynomial function of the number of $k$-faces in $\Delta$ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes. https://escholarship.org/uc/item/7n64498k Tue, 20 Feb 2018 00:00:00 +0000 De Loera, Jesus A. Klee, Steven https://escholarship.org/uc/item/7mp4q6h1 Some recent developments in the theory of quantum spin systems are reviewed. https://escholarship.org/uc/item/7mp4q6h1 Tue, 20 Feb 2018 00:00:00 +0000 Nachtergaele, Bruno Sims, Robert https://escholarship.org/uc/item/7mj738pk We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Mobius function, recovering a classical result due to Verma. https://escholarship.org/uc/item/7mj738pk Tue, 20 Feb 2018 00:00:00 +0000 Jones, Brant C. https://escholarship.org/uc/item/7kv7q58k We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R^2). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization). https://escholarship.org/uc/item/7kv7q58k Tue, 20 Feb 2018 00:00:00 +0000 Bolina, Oscar Contucci, Pierluigi Nachtergaele, Bruno Starr, Shannon https://escholarship.org/uc/item/7jt9016k A spectrally sparse signal of order $r$ is a mixture of $r$ damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of $n$ regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that $O(r^2\log^2(n))$ number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on $3$D arrays demonstrate the capability of FIHT on handling large and high-dimensional... https://escholarship.org/uc/item/7jt9016k Tue, 20 Feb 2018 00:00:00 +0000 Cai, Jian-Feng Wang, Tianming Wei, Ke https://escholarship.org/uc/item/7j87h4rb The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $HW_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a suitable choice of character, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical 0-Hecke algebra and that of the affine Hecke algebra at this specialization. https://escholarship.org/uc/item/7j87h4rb Tue, 20 Feb 2018 00:00:00 +0000 Hivert, Florent Schilling, Anne Thiéry, Nicolas M. https://escholarship.org/uc/item/7hm671sk In this paper, we study the fluctuation of linear eigenvalue statistics of Random Band Matrices defined by $M_{n}=\frac{1}{\sqrt{b_{n}}}W_{n}$, where $W_{n}$ is a $n\times n$ band Hermitian random matrix of bandwidth $b_{n}$, i.e., the diagonal elements and only first $b_{n}$ off diagonal elements are nonzero. Also variances of the matrix elmements are upto a order of constant. We study the linear eigenvalue statistics $\mathcal{N}(\phi)=\sum_{i=1}^{n}\phi(\lambda_{i})$ of such matrices, where $\lambda_{i}$ are the eigenvalues of $M_{n}$ and $\phi$ is a sufficiently smooth function. We prove that $\sqrt{\frac{b_{n}}{n}}[\mathcal{N}(\phi)-\mathbb{E} \mathcal{N}(\phi)]\stackrel{d}{\to} N(0,V(\phi))$ for $b_{n}>>\sqrt{n}$, where $V(\phi)$ is given in the Theorem 1. https://escholarship.org/uc/item/7hm671sk Tue, 20 Feb 2018 00:00:00 +0000 Jana, Indrajit Saha, Koushik Soshnikov, Alexander https://escholarship.org/uc/item/7bd6n8vr We address the question of the classification of gapped ground states in one dimension that cannot be distinguished by a local order parameter. We introduce a family of quantum spin systems on the one-dimensional chain that have a unique gapped ground state in the thermodynamic limit that is a simple product state but which on the left and right half-infinite chains, have additional zero energy edge states. The models, which we call Product Vacua with Boundary States (PVBS), form phases that depend only on two integers corresponding to the number of edge states at each boundary. They can serve as representatives of equivalence classes of such gapped ground states phases and we show how the AKLT model and its $SO(2J+1)$-invariant generalizations fit into this classification. https://escholarship.org/uc/item/7bd6n8vr Tue, 20 Feb 2018 00:00:00 +0000 Bachmann, Sven Nachtergaele, Bruno https://escholarship.org/uc/item/7b21r0np We generalize the topological recursion of Eynard-Orantin (2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle $T^*C$ of an arbitrary smooth base curve $C$. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal $\hbar$-deformation family of $D$-modules over an arbitrary projective algebraic curve $C$ of genus greater than $1$, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the $SL(2,\mathbb{C})$-character variety of the fundamental group $\pi_1(C)$. We show that the semi-classical limit through the WKB approximation of these $\hbar$-deformed $D$-modules recovers the initial family of Hitchin spectral... https://escholarship.org/uc/item/7b21r0np Tue, 20 Feb 2018 00:00:00 +0000 Dumitrescu, Olivia Mulase, Motohico https://escholarship.org/uc/item/6cz9p4m7 The hallmark of deterministic chaos is that it creates information---the rate being given by the Kolmogorov-Sinai metric entropy. Since its introduction half a century ago, the metric entropy has been used as a unitary quantity to measure a system's intrinsic unpredictability. Here, we show that it naturally decomposes into two structurally meaningful components: A portion of the created information---the ephemeral information---is forgotten and a portion---the bound information---is remembered. The bound information is a new kind of intrinsic computation that differs fundamentally from information creation: it measures the rate of active information storage. We show that it can be directly and accurately calculated via symbolic dynamics, revealing a hitherto unknown richness in how dynamical systems compute. https://escholarship.org/uc/item/6cz9p4m7 Tue, 20 Feb 2018 00:00:00 +0000 James, Ryan G. Burke, Korana Crutchfield, James P. https://escholarship.org/uc/item/77r4z3vj We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian. https://escholarship.org/uc/item/77r4z3vj Fri, 16 Feb 2018 00:00:00 +0000 Ayyer, Arvind https://escholarship.org/uc/item/73b5j3wk We give a construction of sl_2-categorifications (in the sense of Chuang-Rouquier) for representations of U_q(gl_n), for generic q and for q a root of unity. https://escholarship.org/uc/item/73b5j3wk Fri, 16 Feb 2018 00:00:00 +0000 Virk, Rahbar https://escholarship.org/uc/item/72z9p4j1 We establish global existence and uniqueness of the dynamics of classical electromagnetism with extended, rigid charges and fields which need not to be square integrable. We consider also a modified theory of electromagnetism where no self-fields occur. That theory and our results are crucial for approaching the as yet unsolved problem of the general existence of dynamics of Wheeler Feynman electromagnetism, which we shall address in the follow up paper. https://escholarship.org/uc/item/72z9p4j1 Fri, 16 Feb 2018 00:00:00 +0000 Bauer, G. Deckert, D. -A. Dürr, D. https://escholarship.org/uc/item/72v20795 In this note we give a transparent proof of a result of da Cruz and Dias da Silva on the equality of symmetrized decomposable tensors. This will be done by explaining that their result follows from the fact that the coordinate ring of a flag variety is a unique factorization domain. https://escholarship.org/uc/item/72v20795 Fri, 16 Feb 2018 00:00:00 +0000 Berget, Andrew https://escholarship.org/uc/item/71n6j5hv We extend the theory of relative trisections of smooth, compact, oriented 4-manifolds with connected boundary given by Gay and Kirby to include 4-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected 4-manifolds can be glued to one another along diffeomorphic boundary components so as to induce a trisected manifold. These two results allow us to define a category Tri whose objects are smooth, closed, oriented 3-manifolds equipped with open book decompositions, and morphisms are relatively trisected cobordisms. Additionally, we extend the Hopf stabilization of open book decompositions to a relative stabilization of relative trisections. https://escholarship.org/uc/item/71n6j5hv Fri, 16 Feb 2018 00:00:00 +0000 Castro, Nickolas A. https://escholarship.org/uc/item/70w1f61x We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together,... https://escholarship.org/uc/item/70w1f61x Fri, 16 Feb 2018 00:00:00 +0000 Gover, A. Rod Latini, Emanuele Waldron, Andrew https://escholarship.org/uc/item/703997qt In our earlier work, we constructed a specific non-compact quantum group whose quantum group structures have been constructed on a certain twisted group C*-algebra. In a sense, it may be considered as a ``quantum Heisenberg group C*-algebra''. In this paper, we will find, up to equivalence, all of its irreducible *-representations. We will point out the Kirillov type correspondence between the irreducible representations and the so-called ``dressing orbits''. By taking advantage of its comultiplication, we will then introduce and study the notion of ``inner tensor product representations''. We will show that the representation theory satisfies a ``quasitriangular'' type property, which does not appear in ordinary group representation theory. https://escholarship.org/uc/item/703997qt Fri, 16 Feb 2018 00:00:00 +0000 Kahng, Byung-Jay https://escholarship.org/uc/item/6zx6s6pk Let W -> A^2 be the universal Weierstrass family of cubic curves over C. For each N >= 2, we construct surfaces parametrizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of A^2. Since W -> A^2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0,0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S^3 with monodromy in SL_2(Z/N). https://escholarship.org/uc/item/6zx6s6pk Fri, 16 Feb 2018 00:00:00 +0000 Bernstein, Mira Tuffley, Christopher https://escholarship.org/uc/item/6zc461p4 Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy. https://escholarship.org/uc/item/6zc461p4 Fri, 16 Feb 2018 00:00:00 +0000 Coward, Alexander Hass, Joel https://escholarship.org/uc/item/6sr360s6 In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of an one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and $n$ dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation. https://escholarship.org/uc/item/6sr360s6 Thu, 15 Feb 2018 00:00:00 +0000 Bae, Hantaek Granero-Belinchón, Rafael https://escholarship.org/uc/item/6r52904p In this paper we prove instances of the cyclic sieving phenomenon for finite Grassmannians and partial flag varieties, which carry the action of various tori in the finite general linear group GL_n(F_q). The polynomials involved are sums of certain weights of the minimal length parabolic coset representatives of the symmetric group S_n, where the weight of a coset representative can be written as a product over its inversions. https://escholarship.org/uc/item/6r52904p Thu, 15 Feb 2018 00:00:00 +0000 Berget, Andrew Huang, Jia https://escholarship.org/uc/item/6pq443cm In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universal deformation formula of the Hopf algebra $\calh_1$ associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket $RC_1$ defines a noncommutative Poisson structure for an arbitrary $\calh_1$ action. https://escholarship.org/uc/item/6pq443cm Thu, 15 Feb 2018 00:00:00 +0000 Bieliavsky, Pierre Tang, Xiang Yao, Yijun https://escholarship.org/uc/item/6ph676bk We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as components in this degeneration. https://escholarship.org/uc/item/6ph676bk Thu, 15 Feb 2018 00:00:00 +0000 Dumitrescu, Olivia https://escholarship.org/uc/item/6jx7k7ns In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical constructions to take connected sums of triangulated 3-manifolds, we obtain the following result: given a minimal triangulation of a closed orientable 3-manifold M, it takes polynomial time in the number of tetrahedra to check if M is reducible or not. https://escholarship.org/uc/item/6jx7k7ns Thu, 15 Feb 2018 00:00:00 +0000 Barchechat, Alexander https://escholarship.org/uc/item/6hw9639d In this paper, some deficiencies of a method proposed by Burdet and Johnson in 1977 for solving integer programming problems are discussed. Examples where the algorithm fails to solve the IP and ways to fix these errors are given. https://escholarship.org/uc/item/6hw9639d Thu, 15 Feb 2018 00:00:00 +0000 Moazzez, Babak Cheung, Kevin https://escholarship.org/uc/item/6h7657jd We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire. In the literature, the standard description of such wires is via random matrix theory (RMT). Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work (arXiv:0912.1574) on the same topic. In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes \beta=1,2. In the \beta=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\ from TUE. In the \beta=1 class, we find a deviation from TOE. It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally... https://escholarship.org/uc/item/6h7657jd Thu, 15 Feb 2018 00:00:00 +0000 Bachmann, Sven Butz, Maximilian De Roeck, Wojciech https://escholarship.org/uc/item/6fm315cm Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac-Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac-Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac-Moody algebras. https://escholarship.org/uc/item/6fm315cm Thu, 15 Feb 2018 00:00:00 +0000 Slofstra, William https://escholarship.org/uc/item/68g7g62p In this paper we present a class of maps for which the multiplicativity of the maximal output p-norm holds when p is 2 and p is larger than or equal to 4. The class includes all positive trace-preserving maps from the matrix algebra on the three-dimensional space to that on the two-dimensional. https://escholarship.org/uc/item/68g7g62p Thu, 15 Feb 2018 00:00:00 +0000 Fukuda, Motohisa https://escholarship.org/uc/item/67m099tr We show that the leading coefficient of the Kazhdan--Lusztig polynomial $P_{x,w}(q)$ known as $\mu(x,w)$ is always either 0 or 1 when $w$ is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance by Billey--Warrington (2001) and Billey--Jones (2007). In type $A$, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's (1990) algorithm, we provide some combinatorial criteria to determine when $\mu(x,w) = 1$ for such permutations $w$. https://escholarship.org/uc/item/67m099tr Thu, 15 Feb 2018 00:00:00 +0000 Jones, Brant C. https://escholarship.org/uc/item/64m6f7ms In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a hyperbolic manifold M contains an acylindrical surface S, then Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5 V_3. https://escholarship.org/uc/item/64m6f7ms Thu, 15 Feb 2018 00:00:00 +0000 Agol, Ian https://escholarship.org/uc/item/63k2r3j2 We analyze a class of quantum spin models defined on half-spaces in the $d$-dimensional hypercubic lattice bounded by a hyperplane with inward unit normal vector $m\in\mathbb{R}^d$. The family of models was previously introduced as the single species Product Vacua with Boundary States (PVBS) model, which is a spin-$1/2$ model with a XXZ-type nearest neighbor interactions depending on parameters $\lambda_j\in (0,\infty)$, one for each coordinate direction. For any given values of the parameters, we prove an upper bound for the spectral gap above the unique ground state of these models, which vanishes for exactly one direction of the normal vector $m$. For all other choices of $m$ we derive a positive lower bound of the spectral gap, except for the case $\lambda_1 =\cdots =\lambda_d=1$, which is known to have gapless excitations in the bulk. https://escholarship.org/uc/item/63k2r3j2 Thu, 15 Feb 2018 00:00:00 +0000 Bishop, Michael Nachtergaele, Bruno Young, Amanda https://escholarship.org/uc/item/6221f965 The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals. https://escholarship.org/uc/item/6221f965 Thu, 15 Feb 2018 00:00:00 +0000 Bandlow, Jason Schilling, Anne Thiery, Nicolas M. https://escholarship.org/uc/item/5xw0k3pf We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small $H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for $H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth. https://escholarship.org/uc/item/5xw0k3pf Thu, 15 Feb 2018 00:00:00 +0000 Cheng, C. H. Arthur Granero-Belinchón, Rafael Shkoller, Steve https://escholarship.org/uc/item/5w87w4x0 A self-avoiding walk with small attractive interactions is described here. The existence of the connective constant is established, and the diffusive behavior is proved using the method of the lace expansion. https://escholarship.org/uc/item/5w87w4x0 Thu, 15 Feb 2018 00:00:00 +0000 Ueltschi, Daniel https://escholarship.org/uc/item/5rb0c81w Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a quantum curve equation, and a Schroedinger like partial differential equation. In this paper we generalize these properties to simple Hurwitz numbers with an arbitrary base curve. https://escholarship.org/uc/item/5rb0c81w Thu, 15 Feb 2018 00:00:00 +0000 Liu, Xiaojun Mulase, Motohico Sorkin, Adam https://escholarship.org/uc/item/5r66x529 The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform construction of the spectral curve and the recursion kernel from the unstable geometries of the original counting problem. We examine this construction using four concrete examples: Grothendieck's dessins d'enfants (or higher-genus analogue of the Catalan numbers), the intersection numbers of tautological cotangent classes on the moduli stack of stable pointed curves, single Hurwitz numbers, and the stationary Gromov-Witten invariants of the complex projective line. https://escholarship.org/uc/item/5r66x529 Thu, 15 Feb 2018 00:00:00 +0000 Dumitrescu, Olivia Mulase, Motohico Safnuk, Brad Sorkin, Adam https://escholarship.org/uc/item/5q9513dw Stanley conjectured in 1977 that the $h$-vector of a matroid simplicial complex is a pure $O$-sequence. We give simple constructive proofs that the conjecture is true for matroids of rank less than or equal to 3, and corank 2. We used computers to verify that Stanley's conjecture holds for all matroids on at most nine elements. https://escholarship.org/uc/item/5q9513dw Thu, 15 Feb 2018 00:00:00 +0000 DeLoera, Jesus Kemper, Yvonne Klee, Steven https://escholarship.org/uc/item/5pr6v31j Let $K$ be a hyperbolic knot in the 3-sphere. If $r$-surgery on $K$ yields a lens space, then we show that the order of the fundamental group of the lens space is at most $12g-7$, where $g$ is the genus of $K$. If we specialize to genus one case, it will be proved that no lens space can be obtained from genus one, hyperbolic knots by Dehn surgery. Therefore, together with known facts, we have that a genus one knot $K$ admits Dehn surgery yielding a lens space if and only if $K$ is the trefoil. https://escholarship.org/uc/item/5pr6v31j Thu, 15 Feb 2018 00:00:00 +0000 Goda, Hiroshi Teragaito, Masakazu https://escholarship.org/uc/item/5mm2b797 We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius $r$-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius $r$-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius. https://escholarship.org/uc/item/5mm2b797 Thu, 15 Feb 2018 00:00:00 +0000 Bowen, Lewis https://escholarship.org/uc/item/5kd92402 The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the "combinatorial... https://escholarship.org/uc/item/5kd92402 Thu, 15 Feb 2018 00:00:00 +0000 Ayyer, Arvind Romik, Dan https://escholarship.org/uc/item/5k6373p2 We extend the idea of interval pattern avoidance defined by Yong and the author for $S_n$ to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for $GL_n$, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold. https://escholarship.org/uc/item/5k6373p2 Thu, 15 Feb 2018 00:00:00 +0000 Woo, Alexander https://escholarship.org/uc/item/5j72d76s We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e., has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula... https://escholarship.org/uc/item/5j72d76s Thu, 15 Feb 2018 00:00:00 +0000 Bolina, Oscar Contucci, Pierluigi Nachtergaele, Bruno https://escholarship.org/uc/item/5g6094dh We study weakly disordered quantum wires whose width is large compared to the Fermi wavelength. It is conjectured that such wires diplay universal metallic behaviour as long as their length is shorter than the localization length (which increases with the width). The random matrix theory that accounts for this behaviour - the DMPK theory- rests on assumptions that are in general not satisfied by realistic microscopic models. Starting from the Anderson model on a strip, we show that a twofold scaling limit nevertheless allows to recover rigorously the fundaments of DMPK theory, thus opening a way to settle some conjectures on universal metallic behaviour. https://escholarship.org/uc/item/5g6094dh Thu, 15 Feb 2018 00:00:00 +0000 Bachmann, S. De Roeck, W. https://escholarship.org/uc/item/5d11z3gb We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them. https://escholarship.org/uc/item/5d11z3gb Thu, 15 Feb 2018 00:00:00 +0000 Viswanath, Sankaran https://escholarship.org/uc/item/542257wd In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane $\mathbb{R}^2$ or a bounded strip $S=\mathbb{R}\times(-\pi/2,\pi/2)$. The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in \cite{ccfgl} for the homogeneus Muskat problem with infinite depth. https://escholarship.org/uc/item/542257wd Thu, 15 Feb 2018 00:00:00 +0000 Berselli, Luigi Cordoba, Diego Granero-Belinchon, Rafael https://escholarship.org/uc/item/53t504d0 We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process. https://escholarship.org/uc/item/53t504d0 Thu, 15 Feb 2018 00:00:00 +0000 Coutand, Daniel Shkoller, Steve https://escholarship.org/uc/item/530445zg This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$. https://escholarship.org/uc/item/530445zg Thu, 15 Feb 2018 00:00:00 +0000 De Loera, Jesús A. Kim, Edward D. Onn, Shmuel Santos, Francisco https://escholarship.org/uc/item/51r1502p We show that a Tychonoff discretely star-Lindelof space can have arbitrarily big extent and note that there are consistent examples of normal discretely star-Lindelof spaces with uncountable extent. https://escholarship.org/uc/item/51r1502p Thu, 15 Feb 2018 00:00:00 +0000 Matveev, Mikhail https://escholarship.org/uc/item/5157n6r7 We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on Cayley graphs of finitely generated residually amenable groups. https://escholarship.org/uc/item/5157n6r7 Thu, 15 Feb 2018 00:00:00 +0000 Bowen, Lewis https://escholarship.org/uc/item/50x425wh We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization. https://escholarship.org/uc/item/50x425wh Thu, 15 Feb 2018 00:00:00 +0000 Aliev, Iskander De Loera, Jesus A. Oertel, Timm O'Neill, Christopher https://escholarship.org/uc/item/4zk89874 Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action $C^*$ on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos} and \cite{bkHam} for some results on the geometry and quantization of O. We construct an explicit subspace $A_{-1}\subset D(O)$ of commuting differential operators which are Euler homogeneous of degree -1. The space $A_{-1}$ is finite-dimensional, g-stable and carries the adjoint representation. $A_{-1}$ consists of (for $g \neq sp(2n,C)$) non-obvious order 4 differential operators obtained by quantizing symbols we obtained previously. These operators are "exotic" in that there is (apparently) no geometric or algebraic theory which explains them. The algebra generated by $A_{-1}$ is a maximal... https://escholarship.org/uc/item/4zk89874 Thu, 15 Feb 2018 00:00:00 +0000 Astashkevich, A. Brylinski, R.