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 ...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.展开更多
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 invaria...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.展开更多
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 alge...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.展开更多
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...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.展开更多
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 ar...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.展开更多
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 ...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.展开更多
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 tr...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.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12001291)supported by National Natural Science Foundation of China(Grant No.12071318)the Fundamental Research Funds for the Central Universities(Grant No.531118010593)。
文摘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.
基金Supported by Foundation of China Scholarship Council(201808430267)the Education Department of Hunan Province(16C0635)the Natural Science Foundation of Hunan Province(2017JJ3085)
文摘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.
基金supported by the National Natural Science Foundation of China under Grant Nos.11901431 and 12171362.
文摘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.
基金partially supported by the National Science Foundation of Xinjiang Uygur Autonomous Region(No. 2017D01C084)the National Science Foundation of China (Nos. 11771330 and 11701491)
文摘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.
基金Supported by ANR(Grant No.EFI ANR-17-CE40-0030)。
文摘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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11971206,12022105)Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province(Grant No.BK20200048)。
文摘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.
基金Conselho Nacional de Desenvolvi-mento Cientifico e Tecnologico-CNPq(Grant No.305963-2018-0).
文摘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.
