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Recent ucdavismath_other items
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Recent eScholarship items from Other
Mon, 20 Sep 2021 20:19:35 +0000

Local limit of the fixed point forest
https://escholarship.org/uc/item/95d5t7zz
Consider the following partial “sorting algorithm” on permutations: take the first entry of the permutation in oneline 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

Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (III): Global Solution and
Asymptotics
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

Short rational functions for toric algebra and applications
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

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The
OneDimensional Case
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

Algebraic Combinatorics of Magic Squares
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

Kontsevich's Universal Formula for Deformation Quantization and the
CampbellBakerHausdorff Formula, I
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 CampbellBakerHausdorff 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

The Bianchi groups are subgroup separable on geometrically finite subgroups
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 SeifertWeber 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

On the generalized BuckleyLeverett equation
https://escholarship.org/uc/item/9sq5b1zc
In this paper we study the generalized BuckleyLeverett 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

Monotone triangles and 312 Pattern Avoidance
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 selfcomplementary 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 312avoiding
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 avoidancelike condition on Gog words associated. We
estimate the number of gapless monotone triangles using a bijection with pbranchings.
https://escholarship.org/uc/item/9ph0c492
Thu, 22 Feb 2018 00:00:00 +0000

Packing subgroups in solvable groups
https://escholarship.org/uc/item/9mz9g1th
We show that any subgroup of a (virtually) nilpotentbypolycyclic group satisfies
the bounded packing property of HruskaWise. 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 HruskaWise. 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

Sphere Packings in Hyperbolic Space: Periodicity and Continuity
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

The Triangle Closure is a Polyhedron
https://escholarship.org/uc/item/9mp2x7jm
Recently, cutting planes derived from maximal latticefree 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
latticefree 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), 233256] and Averkov [On finitely generated closures in the theory
of cutting planes, Discrete Optimization 9 (2012), no. 4, 209215], some basic questions
have remained unresolved. For example, maximal latticefree triangles are the natural
family to study beyond...
https://escholarship.org/uc/item/9mp2x7jm
Wed, 21 Feb 2018 00:00:00 +0000

On growth types of quotients of Coxeter groups by parabolic subgroups
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

A MultiDimensional LiebSchultzMattis Theorem
https://escholarship.org/uc/item/9jc9x40h
For a large class of finiterange quantum spin models with halfinteger spins, we
prove that uniqueness of the ground state implies the existence of a lowlying 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 multidimensional LiebSchultzMattis 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

Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial
Infeasibility
https://escholarship.org/uc/item/9g5835pj
Systems of polynomial equations over an algebraicallyclosed field K can be used to
concisely model many combinatorial problems. In this way, a combinatorial problem is
feasible (e.g., a graph is 3colorable, 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
largescale linearalgebra computations over K. We report on experiments based on the
problem of proving the non3colorability 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

Fixed points of 321avoiding permutations
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 GaltonWatson 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

Cluster expansions and correlation functions
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
KoteckyPreiss 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

Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture
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 locallyisometric covers S'_i of S_i (for i=1,2)
such that there is a (1+\epsilon) biLipschitz homeomorphism between S'_1 and S'_2 and both
covers S'_i have bounded injectivity radius. 2. Let M be a closed hyperbolic 3manifold.
Then there exists a map j: S > M where S is a surface of bounded injectivity radius and
j is a pi_1injective local isometry onto its image.
https://escholarship.org/uc/item/9bw5s2r0
Wed, 21 Feb 2018 00:00:00 +0000

Ultraviolet Properties of the Spinless, OneParticle Yukawa Model
https://escholarship.org/uc/item/9bn0j439
We consider the oneparticle 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 cutoff $\Lambda$. For
any finite ultraviolet cutoff the nucleon oneparticle states are constructed in a bounded
region of the energymomentum 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 selfenergy renormalization. Our results hold in the small coupling
regime.
https://escholarship.org/uc/item/9bn0j439
Wed, 21 Feb 2018 00:00:00 +0000

Towards the BertramFeinbergMukai Conjecture
https://escholarship.org/uc/item/7d2127j9
In this paper, we prove the existence portion of the BertramFeinbergMukai
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 higherrank BrillNoether Problems.
https://escholarship.org/uc/item/7d2127j9
Wed, 21 Feb 2018 00:00:00 +0000

Strong Sequivalence of ordered links
https://escholarship.org/uc/item/98z348cb
Recently Swatee Naik and Theodore Stanford proved that two Sequivalent knots are
related by a finite sequence of doubleddelta moves on their knot diagrams. We show that
classical Sequivalence is not sufficient to extend their result to ordered links. We
define a new algebraic relation on Seifert matrices, called Strong Sequivalence, and prove
that two oriented, ordered links L and L' are related by a sequence of doubleddelta moves
if and only if they are Strongly Sequivalent. We also show that this is equivalent to the
fact that L' can be obtained from L through a sequence of Yclasper 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

MeanField Spin Glass models from the CavityROSt Perspective
https://escholarship.org/uc/item/9826p7w4
The SherringtonKirkpatrick 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

On a generalized doubly parabolic KellerSegel system in one spatial dimension
https://escholarship.org/uc/item/96n1m8zz
We study a doubly parabolic KellerSegel system in one spatial dimension, with
diffusions given by fractional laplacians. We obtain several local and global
wellposedness 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 spatiotemporal 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 parabolicelliptic fractional case.
https://escholarship.org/uc/item/96n1m8zz
Tue, 20 Feb 2018 00:00:00 +0000

Comments on Hastings' Additivity Counterexamples
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

Global solutions for a supercritical driftdiffusion equation
https://escholarship.org/uc/item/93w1t4px
We study the global existence of solutions to a onedimensional driftdiffusion
equation with logistic term, generalizing the classical parabolicelliptic KellerSegel
aggregation equation arising in mathematical biology. In particular, we prove that there
exists a global weak solution, if the order of the fractional diffusion $\alpha \in (1c_1,
2]$, 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 $1c_2<\alpha\leq 2$ with $0
https://escholarship.org/uc/item/93w1t4px
Tue, 20 Feb 2018 00:00:00 +0000

Centrally symmetric manifolds with few vertices
https://escholarship.org/uc/item/92f7s8hn
A centrally symmetric $2d$vertex combinatorial triangulation of the product of
spheres $\S^i\times\S^{d2i}$ is constructed for all pairs of nonnegative integers $i$
and $d$ with $0\leq i \leq d2$. For the case of $i=d2i$, the existence of such a
triangulation was conjectured by Sparla. The constructed complex admits a vertextransitive
action by a group of order $4d$. The crux of this construction is a definition of a certain
fulldimensional subcomplex, $\B(i,d)$, of the boundary complex of the $d$dimensional
crosspolytope. This complex $\B(i,d)$ is a combinatorial manifold with boundary and its
boundary provides a required triangulation of $\S^i\times\S^{di2}$. 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

The Ferromagnetic Heisenberg XXZ chain in a pinning field
https://escholarship.org/uc/item/91m894gz
We investigate the effect of a magnetic field supported at a single lattice site on
the lowenergy spectrum of the ferromagnetic Heisenberg XXZ chain. Such fields, caused by
impurities, can modify the lowenergy 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 zdirection 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

Realizable sets of catenary degrees of numerical monoids
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 cofinite, 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

YangMills theories in dimensions 3,4,6,10 and Barduality
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 Bardual 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 BatalinVilkovisky 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

On a nonlocal analog of the KuramotoSivashinsky equation
https://escholarship.org/uc/item/8zh748kz
We study a nonlocal equation, analogous to the KuramotoSivashinsky 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

Wellposedness of the freesurface incompressible Euler equations with or without
surface tension
https://escholarship.org/uc/item/8w84m2mv
We provide a new method for treating free boundary problems in perfect fluids, and
prove localintime wellposedness in Sobolev spaces for the freesurface 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 freesurface at highestorder.
https://escholarship.org/uc/item/8w84m2mv
Tue, 20 Feb 2018 00:00:00 +0000

Algorithmically detecting the bridge number of hyperbolic knots
https://escholarship.org/uc/item/8ms707mz
We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the
3sphere. 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

On the Berezinian of a moduli space of curves in P^{nn+1}
https://escholarship.org/uc/item/8kn652jc
A supermanifold P^{34} is a target space for twistor string theory. In this note
we identify a line bundle of holomorphic volume elements BerM_gP^{34} defined on the
moduli space of curves of genus g in P^{34} 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

A Fully Magnetizing Phase Transition
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

Jigsaw Percolation on ErdosRenyi Random Graphs
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}sR\'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

BilleyPostnikov decompositions and the fibre bundle structure of Schubert
varieties
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
BilleyPostnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian
Schubert varieties, and give a new proof of Peterson's theorem that all simplylaced
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 KacMoody flag varieties. In...
https://escholarship.org/uc/item/8j5661bp
Tue, 20 Feb 2018 00:00:00 +0000

Subfactors and quantum information theory
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

Quantum Hyperplane Section Principle for Concavex Decomposable Vector Bundles
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

Constructions for cyclic sieving phenomena
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

A continuum approximation for the excitations of the (1,1,...,1) interface in the
quantum Heisenberg model
https://escholarship.org/uc/item/7z16q91t
It is shown that, with an appropriate scaling, the energy of lowlying excitations
of the (1,1,...,1) interface in the $d$dimensional quantum Heisenberg model are given by
the spectrum of the $d1$dimensional Laplacian on an suitable domain.
https://escholarship.org/uc/item/7z16q91t
Tue, 20 Feb 2018 00:00:00 +0000

Dissipative Transport: Thermal Contacts and Tunnelling Junctions
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

A (k+1)Slope Theorem for the kDimensional Infinite Group Relaxation
https://escholarship.org/uc/item/7w32w7j6
We prove that any minimal valid function for the kdimensional infinite group
relaxation that is piecewise linear with at most k+1 slopes and does not factor through a
linear map with nontrivial 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

Quantum spin systems on infinite lattices
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) LiebRobinson bounds,
(7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9)
HaagRuelle scattering...
https://escholarship.org/uc/item/7vf9s034
Tue, 20 Feb 2018 00:00:00 +0000

Uncertainty Principles in Finitely generated ShiftInvariant Spaces with additional
invariance
https://escholarship.org/uc/item/7vb663mn
We consider finitely generated shiftinvariant 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 (nontrivially) generalizes recent results obtained in the special case
of a principal shiftinvariant 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

Quantum Collections
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

On the lower bound of the spectral norm of symmetric random matrices with independent
entries
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 nonsymmetrically 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

Isolated Eigenvalues of the Ferromagnetic SpinJ XXZ Chain with Kink Boundary
Conditions
https://escholarship.org/uc/item/7qj0v04m
We investigate the lowlying 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

Entanglement of random subspaces via the Hastings bound
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 nonunital channels which violate additivity of minimal
output entropy.
https://escholarship.org/uc/item/7qf8z2c2
Tue, 20 Feb 2018 00:00:00 +0000

Extension of the Bernoulli and Eulerian Polynomials of Higher Order and Vector
Partition Function
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

Not all simplicial polytopes are weakly vertexdecomposable
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 0decomposable complexes, this bound is linear in
the number of vertices and the dimension. In this paper we exhibit the first examples of
nonweakly 0decomposable simplicial polytopes.
https://escholarship.org/uc/item/7n64498k
Tue, 20 Feb 2018 00:00:00 +0000

Recent Progress in Quantum Spin Systems
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

An explicit derivation of the Mobius function for Bruhat order
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

Finitevolume excitations of the 111 interface in the quantum XXZ model
https://escholarship.org/uc/item/7kv7q58k
We show that the ground states of the threedimensional 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

Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via LowRank
Hankel Matrix Completion
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 highdimensional...
https://escholarship.org/uc/item/7jt9016k
Tue, 20 Feb 2018 00:00:00 +0000

Hecke group algebras as quotients of affine Hecke algebras at level 0
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 0Hecke
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 0Hecke 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

Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices
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

Product vacua with boundary states and the classification of gapped phases
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 onedimensional 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
halfinfinite 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

Quantum curves for Hitchin fibrations and the EynardOrantin theory
https://escholarship.org/uc/item/7b21r0np
We generalize the topological recursion of EynardOrantin (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 semiclassical 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

Chaos Forgets and Remembers: Measuring Information Creation, Destruction, and
Storage
https://escholarship.org/uc/item/6cz9p4m7
The hallmark of deterministic chaos is that it creates informationthe rate being
given by the KolmogorovSinai 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 informationthe ephemeral
informationis forgotten and a portionthe bound informationis 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

Determinants and Perfect Matchings
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

Derived equivalences and sl_2categorifications for U_q(gl_n)
https://escholarship.org/uc/item/73b5j3wk
We give a construction of sl_2categorifications (in the sense of ChuangRouquier)
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

MaxwellLorentz Dynamics of Rigid Charges
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 selffields
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

Equality of symmetrized tensors and the coordinate ring of the flag variety
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

Trisecting Smooth 4dimensional Cobordisms
https://escholarship.org/uc/item/71n6j5hv
We extend the theory of relative trisections of smooth, compact, oriented 4manifolds with connected boundary given by Gay and Kirby to include 4manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected 4manifolds 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 3manifolds 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

PoincareEinstein Holography for Forms via Conformal Geometry in the Bulk
https://escholarship.org/uc/item/70w1f61x
We study higher form Proca equations on Einstein manifolds with boundary data along
conformal infinity. We solve these Laplacetype 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

*Representations of a Quantum Heisenberg Group Algebra
https://escholarship.org/uc/item/703997qt
In our earlier work, we constructed a specific noncompact 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 socalled
``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

Level structures on the Weierstrass family of cubics
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

Topological and physical knot theory are distinct
https://escholarship.org/uc/item/6zc461p4
Physical knots and links are onedimensional 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

Global existence for some transport equations with nonlocal velocity
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 quasigeostrophic equation and the incompressible porous media
equation, and one dimensional and $n$ dimensional models of the dissipative
quasigeostrophic equations and the dissipative incompressible porous media equation.
https://escholarship.org/uc/item/6sr360s6
Thu, 15 Feb 2018 00:00:00 +0000

Cyclic sieving of finite Grassmannians and flag varieties
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

RankinCohen brackets and formal quantization
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 WeylMoyal
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:defgpd}, 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 RankinCohen 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

Plane curves with prescribed triple points: a toric approach
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

Minimal Triangulations of Reducible 3Manifolds
https://escholarship.org/uc/item/6jx7k7ns
In this thesis, we use normal surface theory to understand certain properties of
minimal triangulations of compact orientable 3manifolds. We describe the collapsing
process of normal 2spheres and disks. Using some geometrical constructions to take
connected sums of triangulated 3manifolds, we obtain the following result: given a minimal
triangulation of a closed orientable 3manifold 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

On Burdet and Johnson's Algorithm for Integer Programming
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

Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law
https://escholarship.org/uc/item/6h7657jd
We study the electronic transport properties of the Anderson model on a strip,
modeling a quasi onedimensional 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
thickwire limit, which is the experimentally...
https://escholarship.org/uc/item/6h7657jd
Thu, 15 Feb 2018 00:00:00 +0000

A Brylinski filtration for affine KacMoody algebras
https://escholarship.org/uc/item/6fm315cm
Braverman and Finkelberg have recently proposed a conjectural analogue of the
geometric Satake isomorphism for untwisted affine KacMoody groups. As part of their model,
they conjecture that (at dominant weights) Lusztig's qanalog of weight multiplicity is
equal to the Poincare series of the principal nilpotent filtration of the weight space, as
occurs in the finitedimensional case. We show that the conjectured equality holds for all
affine KacMoody 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 qanalog of weight multiplicity. Finally,
we give some partial results for indefinite KacMoody algebras.
https://escholarship.org/uc/item/6fm315cm
Thu, 15 Feb 2018 00:00:00 +0000

An application of decomposable maps in proving multiplicativity of low dimensional
maps
https://escholarship.org/uc/item/68g7g62p
In this paper we present a class of maps for which the multiplicativity of the
maximal output pnorm holds when p is 2 and p is larger than or equal to 4. The class
includes all positive tracepreserving maps from the matrix algebra on the
threedimensional space to that on the twodimensional.
https://escholarship.org/uc/item/68g7g62p
Thu, 15 Feb 2018 00:00:00 +0000

Leading coefficients of KazhdanLusztig polynomials for Deodhar elements
https://escholarship.org/uc/item/67m099tr
We show that the leading coefficient of the KazhdanLusztig 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 BilleyWarrington (2001) and BilleyJones (2007). In type $A$, these
elements are precisely the 321hexagon 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

Lower bounds on volumes of hyperbolic Haken 3manifolds
https://escholarship.org/uc/item/64m6f7ms
In this paper, we find lower bounds for volumes of hyperbolic 3manifolds with
various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal
simplex in hyperbolic 3space. 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

Spectral Gap and Edge Excitations of $d$dimensional PVBS models on halfspaces
https://escholarship.org/uc/item/63k2r3j2
We analyze a class of quantum spin models defined on halfspaces 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
XXZtype 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

On the uniqueness of promotion operators on tensor products of type A crystals
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 KirillovReshetikhin crystals.
https://escholarship.org/uc/item/6221f965
Thu, 15 Feb 2018 00:00:00 +0000

Wellposedness of the Muskat problem with $H^2$ initial data
https://escholarship.org/uc/item/5xw0k3pf
We study the dynamics of the interface between two incompressible fluids in a
twodimensional porous medium whose flow is modeled by the Muskat equations. For the
twophase Muskat problem, we establish global wellposedness and decay to equilibrium for
small $H^2$ perturbations of the rest state. For the onephase Muskat problem, we prove
local wellposedness 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

A selfavoiding walk with attractive interactions
https://escholarship.org/uc/item/5w87w4x0
A selfavoiding 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

Quantum curves for simple Hurwitz numbers of an arbitrary base curve
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 EynardOrantin
topological recursion, an infiniteorder 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

The spectral curve of the EynardOrantin recursion via the Laplace transform
https://escholarship.org/uc/item/5r66x529
The EynardOrantin 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 highergenus 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
GromovWitten invariants of the complex projective line.
https://escholarship.org/uc/item/5r66x529
Thu, 15 Feb 2018 00:00:00 +0000

$h$vectors of small matroid complexes
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

Dehn surgeries on knots which yield lens spaces and genera of knots
https://escholarship.org/uc/item/5pr6v31j
Let $K$ be a hyperbolic knot in the 3sphere. 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
$12g7$, 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

Periodicity and Circle Packing in the Hyperbolic Plane
https://escholarship.org/uc/item/5mm2b797
We prove that given a fixed radius $r$, the set of isometryinvariant probability
measures supported on ``periodic'' radius $r$circle packings of the hyperbolic plane is
dense in the space of all isometryinvariant 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 isometryinvariant 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

New enumeration formulas for alternating sign matrices and square ice partition
functions
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
MillsRobbinsRumsey, its proof by Zeilberger, and more recent work on doublyrefined and
triplyrefined enumeration by several authors. In this paper we extend the previously known
results on this problem by deriving explicit enumeration formulas for the "topleftbottom"
(triplyrefined) and "topleftbottomright" (quadruplyrefined) 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

Interval pattern avoidance for arbitrary root systems
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

Path Integral Representation for Interface States of the Anisotropic Heisenberg
Model
https://escholarship.org/uc/item/5j72d76s
We develop a geometric representation for the ground state of the spin1/2 quantum
XXZ ferromagnetic chain in terms of suitably weighted random walks in a twodimensional
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

From the Anderson model on a strip to the DMPK equation and random matrix
theory
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

KostkaFoulkes polynomials for symmetrizable KacMoody algebras
https://escholarship.org/uc/item/5d11z3gb
We introduce a generalization of the classical HallLittlewood and KostkaFoulkes
polynomials to all symmetrizable KacMoody algebras. We prove that these KostkaFoulkes
polynomials coincide with the natural generalization of Lusztig's $t$analog of weight
multiplicities, thereby extending a theorem of Kato. For $g$ an affine KacMoody 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

Local solvability and turning for the inhomogeneous Muskat problem
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

Unique solvability of the freeboundary NavierStokes equations with surface
tension
https://escholarship.org/uc/item/53t504d0
We prove the existence and uniqueness of solutions to the timedependent
incompressible NavierStokes equations with a freeboundary governed by surface tension.
The solution is found using a topological fixedpoint theorem for a nonlinear iteration
scheme, requiring at each step, the solution of a model linear problem consisting of the
timedependent Stokes equation with linearized meancurvature 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

Graphs of Transportation Polytopes
https://escholarship.org/uc/item/530445zg
This paper discusses properties of the graphs of 2way and 3way transportation
polytopes, in particular, their possible numbers of vertices and their diameters. Our main
results include a quadratic bound on the diameter of axial 3way transportation polytopes
and a catalogue of nondegenerate 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

On spaces in countable web
https://escholarship.org/uc/item/51r1502p
We show that a Tychonoff discretely starLindelof space can have arbitrarily big
extent and note that there are consistent examples of normal discretely starLindelof
spaces with uncountable extent.
https://escholarship.org/uc/item/51r1502p
Thu, 15 Feb 2018 00:00:00 +0000

Couplings of Uniform Spanniing Forests
https://escholarship.org/uc/item/5157n6r7
We prove the existence of an automorphisminvariant 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

Sparse solutions of linear Diophantine equations
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 nonzero 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

Exotic Differential Operators on Complex Minimal Nilpotent Orbits
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 quasiaffine 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
finitedimensional, gstable and carries the adjoint representation. $A_{1}$ consists of
(for $g \neq sp(2n,C)$) nonobvious 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