Matroids on convex geometries (cg-matroids)

We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids.     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Communicated by T.N.T. Goodman     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Partially supported by DGICYT PS93-0310 and by the EC project CHRX-CT94-0522.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Multiobjective DC programs with infinite convex constraints

New results are established for multiobjective DC programs with infinite convex constraints (MOPIC) that are defined on Banach spaces (finite or infinite dimensional) with objectives given as the difference of convex functions. This class of problems can also be called multiobjective DC semi-infinite and infinite programs, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Such problems have not been studied as yet. Necessary and sufficient optimality conditions for the weak Pareto efficiency are introduced. Further, we seek a connection between multiobjective linear infinite programs and MOPIC. Both Wolfe and Mond-Weir dual problems are presented, and corresponding weak, strong, and strict converse duality theorems are derived for these two problems respectively. We also extend above results to multiobjective fractional DC programs with infinite convex constraints. The results obtained are new in both semi-infinite and infinite frameworks.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

整循环图的秩（英文）

A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.     

Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuos homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.     

On mixed-integer sets with two integer variables

We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one on the equivalence of different definitions of split cuts presented in Cook et al. (1990) [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard (2008) [8]. Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a closed, bounded and convex set.     

Convexity of level lines of maximal space-like hypersurfaces in Minkowski space

For the maximal space-like hypersurface defined on a convex ring in R², we obtain the regularity and the strict convexity of its level lines by the continuity method, and the curvature estimate of the level lines is also derived.     

Maximum Order of Triangle‐Free Graphs with a Given Rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle‐free graphs with a given rank and characterize all such graphs achieving the maximum order.     

半模糊凸模糊映射

In this paper, a new class of fuzzy mappings called semistrictly convex fuzzy mappings is introduced and we present some properties of this kind of fuzzy mappings. In particular, we prove that a local minimum of a semistrictly convex fuzzy mapping is also a global minimum. We also discuss the relations among convexity, strict convexity and semistrict convexity of fuzzy mapping, and give several sufficient conditions for convexity and semistrict convexity.     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

Characterization of R-Evenly Quasiconvex Functions

A function defined on a locally convex space is called evenly quasiconvex if its level sets are intersections of families of open half-spaces. Furthermore, if the closures of these open halfspaces do not contain the origin, then the function is called R-evenly quasiconvex. In this note, R-evenly quasiconvex functions are characterized as those evenly-quasiconvex functions that satisfy a certain simple relation with their lower semicontinuous hulls.     

A Theorem on Strict Separability of Convex Polyhedra and Its Applications in Optimization

We propose a new approach to the strict separation of convex polyhedra. This approach is based on the construction of the set of normal vectors for the hyperplanes, such that each one strict separates the polyhedra A and B. We prove the necessary and sufficient conditions of strict separability for convex polyhedra in the Euclidean space and present its applications in optimization.     

关于半连续函数与凸函数的注记

在半连续前提下,给出凸函数和严格凸函数的不等式刻划．指出非空凸集上的半连续函数满足中间点凸性时,成为凸函数,满足中间点严格凸性时,成为严格凸函数．最后定义F—G广义凸函数和条件p1,p2等概念,列举若干满足条件p1,p2的数量函数和向量函数,并指出,对于F—G广义凸函数,在条件p1,p2及一定连续性条件下,可以得到类似结果．     

关于 Musielak-Orlicz 空间的复端点与复严格凸

Istratescu,V.I.和 Istratescu,I.在[1]中引进了复 Banach 空间中的复端点及复严格凸的概念,即下述的定义1、2。本文将以简明形式给出重要的 Musielak-Orlicz 空间关于 Luxemburg 范数的复端点和复严格凸的刻划。应当指出:对于 Musielak-Orlicz 空间关于 Luxemburg 范数的严格凸条件,Hudzik,H。已经给出,吴从炘、陈述涛则去掉了 Hudzik,H。所加的值域空间为可分这个很强的限制,并且讨论了端点的特征。     

Generalized Solution of the First Boundary Value Problem for Parabolic Monge-Ampere Equation

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).     

Linear Operators Generated by a Countable Number of Quasi-differential Expressions

The theory of linear ordinary quasi-differential operators has been considered in Lebesgue locally integrable spaces on a single interval of the real line. Such spaces are not Banach spaces but can be considered as complete, locally convex, linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete, locally convex, linear topological space but now with the topology derived as a strict inductive limit. This article extends the previous single interval results to the case when a finite or countable number of intervals of the real line is considered. Conjugate and preconjugate linear quasi-differential operators are defined and relationships between these operators are developed.     

On the Structure of the Complements of Chebyshev Sets

A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215--232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established.