The Log-Concavity of Kazhdan-Lusztig Polynomials of Uniform Matroids

Elias,et al.(2016)conjectured that the Kazhdan-Lusztig polynomial of any matroid is logconcave.Inspired by a computer proof of Moll’s log-concavity conjecture given by Kauers and Paule,the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids.

The L_(p) Shephard problem on entropy of log-concave functions

this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)f is the L_(p) projection body of a log-concave function f.Our results give a partial answer to this problem.

The Busemann-Petty problem on entropy of log-concave functions

The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we investigate the Busemann-Petty problem on entropy of log-concave functions:for even log-concave functions f and g with finite positive integrals in R^(n),if the marginal∫_(R^(n))∩H^(f(x)dx)of f is smaller than the marginal∫_(R^(n))∩H^(g(x)dx)of g for every hyperplane H passing through the origin,is the entropy Ent(f)of f bigger than the entropy Ent(g)of g?The BusemannPetty problem on entropy of log-concave functions includes the Busemann-Petty problem,and hence its answer is negative when n≥5.For 2≤n≤4,we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.

Valuations on Concave Functions and Log-Concave Functions

Recently, the theory of valuations on function spaces has been rapidly growing. It is more general than the classical theory of valuations on convex bodies. In this paper, all continuous, SL(n) and translation invariant valuations on concave functions and log-concave functions are completely classified, respectively.

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.

Unimodality of Independence Polynomials of the Cycle Cover Product of Graphs

An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.

New Results on Truncated Elliptical Distributions

Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions.Two members of this class are the multivariate truncated normal and multivariate truncated t distributions.We derive statistical properties of the truncated elliptical distributions.Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions.

Log-behavior of Two Sequences Related to the Elliptic Integrals

Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0 respectively.In this paper,we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence.Then we prove the log-convexity of{Vn^2-V(n-1)V(n+1)}n≥2 and{n!Vn}n≥1,the ratio log-concavity of{Pn}n≥0 and the sequence{An}n≥0 of Apéry numbers,and the ratio log-convexity of{Vn}n≥1.

Poincaréand Logarithmic Sobolev Inequalities for Nearly Radial Measures

Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic Sobolev inequalities.We also give various examples where the obtained bounds are quite sharp.Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.