Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

On the Buchstaber Invariant of Simple Convex 3-Polytopes

In this paper the author gives a method of constructing characteristic matrices,and uses it to determine the Buchstaber invariants of all simple convex 3-polytopes,which imply that each simple convex 3-polytope admits a characteristic function.As a further application of the method,the author also gives a simple new proof of five-color theorem.     

Integer Partitions from the Polyhedral Point of View

We characterize integer partitions that are convex combinations of two partitions, which connects vertices of the partition polytopes with Sidon sets and sum-free sets. We prove that all vertices of the partition polytope can be generated from a subset of support vertices with the use of two operations of merging parts. Application of either operation results in an adjacent vertex. We present also some numerical data on vertices.     

On Domains in Which Harmonic Functions Satisfy Generalized Mean Value Properties

Potential Analysis - Assume that a bounded domain Ω??N (N ≥ 2) has the property that there exists a signed measure µ with compact support in Ω such that, for every...     

Polyhedral combinatorics of multi-index axial transportation problems

This paper studies integer points (IP) and integer vertices (IV) of the p-index axial transportation polytope (p  -ATP) of order _n1×_n2×?×_np$n_1\times n_2\times ?\times n_p$, _n1,_n2,…,_np?2$n_1\text{,}{n}_2\text{,}\dots \text{,}{n}_p?2$, p?2$p?2$, defined by integer vectors, as well as noninteger vertices of the 3-ATP. In particular, for the p  -ATP, we establish criteria for the minimum and maximum number of IPs and describe the class of polytopes for which the number of IPs coincides with the number of IVs. For the 3-ATP of order n×n×n$n\times n\times n$, we prove the theorem on the exponential growth of denominators of fractional components of the polytope vertices. Three conjectures are stated regarding the maximum number of vertices of the p-ATP, the maximum number of IVs, and the structure of the nondegenerate polytopes with the maximum number of IPs.     

Minkowski-type and Alexandrov-Type Theorems for Polyhedral Herissons

Classical H. Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H. Minkowski uniqueness theorem due to A.D. Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.     

Harnack and mean value inequalities on graphs

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.     

Growth of certain harmonic functions in an n-dimensional cone

We give the growth properties of harmonic functions at infinity in a cone, which generalize the results obtained by Siegel-Talvila.     

高阶Lagrange中值定理“中值点”的渐近性

借助Stirling数研究了高阶Lagrange微分中值定理在f(n+1)(a)=0或f(n+1)(a)不存在时的“中值点”的渐近性,并给出了渐近性估计式.     

两个微分中值定理证明中辅助函数的多种作法

在数学分析中 ,三个微分中值定理极为重要 .在证明 Lagrange中值定理和 Cauchy中值定理时 ,都少不了作辅助函数 ,各种版本的《数学分析》或《高等数学》书本中 ,都只给出了一种形式的辅助函数 .为了扩展思路 ,给出了多种形式的辅助函数 ,并得出了一般形式 .     

两个微分中值定理证明中辅助函数作法探讨

在证明拉格朗日中值定理与柯西中值定理时都要作辅助函数,这里直接依据所证明的结论,给出两种求辅助函数的方法.     

论中值定理类命题证明中的辅助函数构造

借助实例分析的方法,讨论在证明微分与积分相结合的中值定理类命题时,关于辅助函数的构造技巧及其变形思想.     

Improved Estimation of Parameter Matrices in a One-Sample and Two-Sample Problems

In this paper, the problems of estimating the covariance matrix in a Wishart distribution (refer as one-sample problem) and the scale matrix in a multi-variate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their harmonic mean is proposed. It is shown that the new estimator dominates the best linear estimator under two scale invariant loss functions.     

Ergodic theory of amenable semigroup actions

In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.     

一个微分中值命题条件的弱化

讨论一个微分中值命题条件的弱化,将条件“f′（x）g′（x）〉0”弱化为“f（a）≠f（b）”,利用介值定理和柯西中值定理给出证明,以扩大命题的适用范围,并举出实例予以说明．     

A Simplified Elementary Proof of Hadwiger's Volume Theorem

One of the most important results in geometric convexity is Hadwiger's characterization of quermassintergrals and intrinsic volumes. The importance lies in that Hadwiger's theorem provides straightforward proofs of numerous results in integral geometry such as the kinematic formulas [Santaló, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, 1976], the mean projection formulas for convex bodies [Schneider, R.: Convex Bodies: The Brunn—Minkowski Theory, Cambridge Univ. Press, 1993], and the characterization of totally invariant set functions of polynomial type [Chen, B. and Rota, G.-C.: Totally invariant set functions of polynomial type, Comm. Pure Appl. Math. 47 (1994), 187–197]. For a long time the only known proof of Hadwiger's theorem was his original one [Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche and Isoperimetrie, Springer, Berlin, 1957] (long and not available in English), until a new proof was obtained by Klain [Klain, D. A.: A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329–339., Klain, D. A. and Rota, G.-C.: Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press, 1997], using a result from spherical harmonics. The present paper provides a simplified and self-contained proof of Hadwiger's theorem.     

On the Mean Value of the Infinite Order Legendre Series

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积分第二中值定理的中间点ξ的渐近性质

讨论积分第一和第二中值定理的中间点ξ的渐近性质的一般结果,主要证明积分第二中值定理的中间点在弱条件下的渐近性质.