关于凸性的一些探讨

现行的不少教材在叙述凸函数定义时,通常都假设函数是连续的.本文以没有连续为前提的一元凸函数的定义为基础,探讨了函数的连续性,左右导数的存在性,凸函数在区间端点的形态,最后利用左右导数,给出了判定函数为凸的一个充要条件.     

Classes of Discrete Convexity Properties

In the general first-level classification of the convexity properties for sets, discrete convexities appear in more classes. A second-level classification identifies more subclasses containing discrete convexity properties, which appear as approximations either of classical convexity or of fuzzy convexity. First, we prove that all these convexity concepts are defined by segmental methods. The type of segmental method involved in the construction of discrete convexity determines the subclass to which it belongs. The subclasses containing the convexity properties that have discrete particular cases are also presented.     

关于秩-1凸与拟凸的一个注记

证明了齐次奇次多项式或三次多项式的情形下秩-1凸与拟凸的等价性.     

Abstract Convexity,Some Relations and Applications

The aim of this article is to analyze the relationship between various notions of abstract convexity structures that we find in the literature, in connection with the problem of the existence of continuous selections and fixed points of correspondences. We focus mainly on the notion of mc -spaces, which was introduced in [J.V. LLinares (1998). Unified treatment of the problem of the existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics , 29 , 285-302], and its relationship with c -spaces [Ch.D. Horvath (1991). Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications , 156 , 341-357], simplicial convexity [R. Bielawski (1987). Simplicial convexity and its applications. Journal of Mathematical Analysis and Applications , 127 , 155-171], order convexity (used in [Ch.D. Horvath and J.V. LLinares (1996). Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics , 25 , 291-306]), B '-simplicial convexity and L -spaces [H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano and J.V. LLinares (1998). Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications , 222 , 138-150]. Moreover, in the context of mc -spaces, a characterization result of nonempty finite intersection, in the line with the Knaster-Kuratowski-Mazurkiewicz Lemma, some consequences of it and some generalizations of Browder's existence of continuous selection and fixed point theorem are presented.     

与积分有关的一类函数的凸性问题

解决了一类与积分有关的函数的凸性问题,其中的定理都是一些已知结果的加强.作为应用,它给出了二类平均和一个与Gamma函数有关的函数的凸性.     

Diameter Two Properties,Convexity and Smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space \({X}\): “For every slice \({S}\) of \({B_X}\) and every norm-one element \({x}\) in \({S}\), there is a point \({y \in S}\) in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.     

Some Geometric Properties of Julia Sets and filled-in Julia Sets of Polynomials

In this paper, some geometric properties of connected Julia sets and filled-in Julia sets of polynomials are given.     

Some Geometric Properties of Analytic Functions Involving a New Fractional Operator

In this paper, we introduce and study a new fractional operator and its implications in terms of the Ruscheweyh derivative operator, the Sălăgean operator and a certain fractional differintegral operator. Some geometric properties of the analytic functions involving this operator are derived. We also consider some applications and derive certain corollaries of our main results. Some useful consequences and relationship of certain results with known results are also pointed out.

     

NURBS曲线保凸性质的几何证明

NURBS curves are convexity preserving, i.e. once the control polygon is convex, the associated NURBS curve will also be convex. In this paper this property is proved geometrically.     

Geometric Form of the Hahn–Banach Theorem for Generalized Convexity

We investigate a class of compact sets convex with respect to a certain family of planes. For compact sets that satisfy the condition of acyclicity of sections by a certain collection of two-dimensional planes, we prove their generalized convexity.     

关于k-致凸性和k-致光滑性的几点注记

设X为Banach空间,记U(X)={x∈X:‖x‖≤1}。V.I.Istratescu引入了下面两个概念。Banach空间Z叫做k一致凸的,如果对每个ε>0,存在δ(ε)>0,当x₁,…,x_k,y₁,…,y_k为U(X)中的元素.本文证明上述k一致凸性等价于一致凸性,并且X为k一致光滑的当且仅当X为一致光滑的,因此这两个概念都不是新的概念。     

Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming

We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f ₁(y ₁), f ₂(y ₂) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ₂(y ₁,y ₂)=f ₁(y ₁)f ₂(y ₂) is convex for all (y ₁,y ₂) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions; that is, they are sufficient conditions for every f _i (y _i ), i=1,2,…,n, so as ℱ _n (y ₁,y ₂,…,y _n )=∏ _i=1 ⁿ f _i (y _i ) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f) ^κ be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis. The authors gratefully acknowledge support from the National Science Foundation.     

Convexity Properties of the Cone of Nonnegative Polynomials

We study metric properties of the cone of homogeneous nonnegative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of nonnegative polynomials and the minimum volume ellipsoid of the natural base of the cone of powers of linear forms and compute the coefficients of symmetry of the bases. The multiplication by (x₁ ² + ··· + x_n ²)^m induces an isometric embedding of the space of polynomials of degree 2k into the space of polynomials of degree 2(k+m), which allows us to compare the cone of nonnegative polynomials of degree 2k and the cone of sums of 2(k+m)-powers of linear forms. We estimate the volume ratio of the bases of the two cones and the rate at which it approaches 1 as m grows.     

Some Remarks on the Convexity Number of a Graph

Given a subgraph H in a graph G, we say H is convex if given any two vertices u and v in H, then H contains each shortest u–v path in G. As defined in [5], [8] and other places the convexity number of G, denoted con(G), is the order of a largest convex proper subgraph of G. We note that if G is not a complete graph, then the clique number of G forms a lower bound on con(G). As we shall see, in many graphs the clique number and convexity number are equal. We exploit this equality to show that computation of con is NP-complete, provide a Nordhaus-Gaddum type bound for con and produce a Ramsey type theorem for con. We shall see the latter two problems are closely related to classical Ramsey theory. In doing so, we answer a problem posed by Chartrand and Zhang in [5]. Acknowledgments.Respectfully dedicated to Professor Edgar M. Palmer, on the happy occasion of his retirement from Michigan State University. Professor Palmer's kind generosity to his students, colleagues and family is deeply appreciated by many. Salmon beware!!!     

Pseudo-uniform Convexity in Hp and Some Extremal Problems on Sobolev Spaces

We extend Newman and Keldysh theorems to the behavior of sequences of functions in H^p (μ) which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.     

Geometric Properties of Metric Spaces

Ukrainian Mathematical Journal - We study some problems of geometrization of arbitrary metric spaces. In particular, we analyze the notions of straight and flat placements of points in these...     

几何分布的几个性质

本文给出几何分布与指数分布相同及相异的几个性质．