The goal of this paper is to study convex lattice sets by the discrete Legendre transform. The definition of the polar of convex lattice sets in ℤ^n is provided. It is worth mentioning that the polar of convex lattice sets have the self-dual property similar to that of convex bodies. Some properties of convex lattice sets are established, for instance, the inclusion relation, the union and intersection on the polar of convex lattice sets. In addition, we discuss the relationship between the cross-polytope and the discrete Mahler product. It states that a convex lattice set is the cross-polytope if and only if its discrete Mahler product is the smallest.

This paper studies the classification of origin-symmetric convex lattice sets under integer unimodular transformations. For an origin-symmetric convex lattice set, we construct an affine invariant and consider the affine invariance of convex lattice sets from different points of view. Furthermore, we provide an upper bound of coordinates of points in origin-symmetric convex lattice sets, which is accomplished by proving the upper bound of the minimum area of the square enclosing an origin-symmetric convex lattice set. Using these results we classify origin-symmetric convex lattice sets with cardinality not greater than 17.

The discrete Legendre transform is a powerful tool for analyzing the properties of convex lattice sets. In this paper, for t>0, we study a class of convex lattice sets and establish a relationship between vertices of the polar of convex lattice sets and vertices of the polar of its t−dilation. Subsequently, we show that there exists a class of convex lattice sets such that its polar is itself. In addition, we calculate upper and lower bounds for the discrete Mahler product of a class of convex lattice sets.

In this paper, the mixed Lp-surface area measures are defined and the mixed Lp Minkowski inequality is obtained consequently. Furthermore, the mixed Lp projection inequality for mixed projection bodies is established.

In this paper, we establish a Loomis-Whitney type inequality about volume normalized L-p projection body for p >= 1 with complete equality conditions for p not equal 2. Meanwhile, an estimate for the weighted L-p zonoid is given.

Let $K$ be a convex lattice set in $\mathbb{Z}^n$ containing the origin as the interior of its convex hull. In this paper, the definition of the polar of a convex lattice set $K$ is given both in $\mathbb{Q}^n$ and $\mathbb{Z}^n$. Some properties and inequalities about the convex lattice sets and their polar are established.

Based on the identification (a(11), a(21), a(22), ..., a(d1), a(d2), ..., a(dd))(T) is an element of Ed(d+1)/2 with symmetric matrix A = (a(ij)) is an element of E-d2, the relation between geometry of the cone of positive semidefinite matrices and the moment cone in four-dimensional Euclidean space E-4 is given. As for the higher dimensional case, we present part of the relation of the two types of cone.

Let 2(d) be the family of all d x d positive semidefinite matrices. We show that the form of faces of 2(d) under geometric and algebraic definitions coincides. Using this result, we get a classification of 2(d) in a geometric sense. The form of projection matrix is described more clearly, and it turns out that 2(d) is the positive hull of all projection matrices with rank 1.

In this paper, the survey about some results of the convex lattice set are given and the invariance of projection problem of convex lattice set is also obtained. And combining a famous result in the graph theory, several conjectures about the convex lattice set are presented.

随着部分高等农林院校向综合性大学发展,数学硕士学位研究生的培养在高等农林院校逐步得以开展.目前,高等农林院校数学硕士学科点的建设主要具有创建时间短、基础较弱但是教师集体凝聚力强、研究方向的互补性强、具有较多的交叉研究机会等特点;同时,面临着师资力量有待加强,研究方向分散、缺乏特色,生源数量不足、素质参差不齐,本、硕专业的关系有待理顺,学科点内外的有效学术交流有待加强等问题.为此,提出高等农林院校数学硕士学科点建设应侧重于2个关键点:一是通过引进优秀人才和优化师资结构,完善导师遴选机制、考核机制和激励机制,加强导师的职前和在岗培训,保障导师的科研工作时间等,加强导师队伍建设;二是通过加强基础数学领域相关研究方向的建设、加强与农林类相关学科的合作研究、依托所在区域的数学学科优势、在短时间内凝练出主干研究方向等,加强研究方向建设.同时,从课程建设、人才培养模式、交流合作、理顺本、硕专业的关系等方面,提出进一步加强高等农林院校数学硕士学科点建设的建议.

Can one determine a centrally symmetric lattice polygon by its projections? In 2005, Gardner et al. proposed the above discrete version of Aleksandrov's projection theorem. In this paper, we define a coordinate matrix for a centrally symmetric convex lattice set and suggest an algorithm to study this problem.

Associated with the notion of the mixed L-p-affine surface area for multiple convex bodies for all real p (p not equal n) which was introduced by Ye, et al. [D. Ye, B. Zhu, J. Zhou, arXiv, 2013 (2013), 38 pages], we define the concept of the mixed L-p-dual affine surface area for multiple star bodies for all real p (p not equal -n) and establish its monotonicity inequalities and cyclic inequalities. Besides, the Brunn-Minkowski type inequalities of the mixed L-p-dual affine surface area for multiple star bodies with two addition are also presented. (C) 2016 All rights reserved.

首先对2012年5月北京林业大学举办的第一届数学建模竞赛的概况进行了介绍,并对数学建模竞赛的B题进行了详细的点评.其次阐述了这次数学建模竞赛所展现的特点,即参赛学生具有扎实的基本功;最优选择下关系的确定是影响竞赛成绩的关键;参赛学生对试题使用了多样求解方法;部分学生误解题意,最终导致了错误的解决方案.针对这些特点,提出将数学建模纳入大学数学教学,并应增加数学问题的计算机实现环节,利用数学建模可以多角度讲述大学数学重要的知识点.

In this note, we given a version of Pick's theorem for the simple lattice polygon in two-dimensional subspace of R^3.

考虑了以数理逻辑中的等值演算为工具对一个结构较为复杂的定理的逻辑结构做了分析.这为我们常用的分析命题结构的方法如逆否命题等提供了一个新思路.

For a $m\times n$ matrix $B=(b_{ij})_{m\times n}$ with nonnegative entries $b_{ij}$ and any $k\times l-$submatrix $B_{ij}$ of $B$, let $a_{B_{ij}}$ and $g_{B_{ij}}$ denote the arithmetic mean and geometric mean of elements of $B_{ij}$ respectively. It is proved that if $k$ is an integer in $(\frac{m}{2}, m]$ and $l$ is an integer in $(\frac{n}{2}, n]$ respectively, then $$\Big(\prod_{i=k,j=l\atop B_{ij}\subset B}a_{B_{ij}}\Big)^{\frac{1}{C_m^k\cdot C_n^l}} \geq\frac{1}{C_m^k\cdot C_n^l}\Big(\sum_{i=k,j=l\atop B_{ij}\subset B}g_{B_{ij}}\Big),$$ with equality if and only if $b_{ij}$ is a constant for every $i,j$.

From different viewpoints,we discuss the function expansion of a polynomial at a given point.Through the process,the authors explore ways to train students thinking divergently.

In this paper, the section of a convex body K with volume one by any hyperplane is considered when it has isotropic surface measure. We give the lower and upper bounds of the volumes of the section of K. © 2012 Pushpa Publishing House.

Considering the similarity between quantitative increment of retail investor and infectious diseases,we establish a model for retail investor which describe the changing rule of its increment.Furthermore,the results of numerical experiments illustrate that our model shows the changing rule of retail investor in the short term.

We present a method to obtain the Hasse diagram of a partial ordering relation.