Vector complementarity and minimal element problems

In this paper, vector complementarity problems are introduced as weak versions of vector variational inequalities in ordered Banach spaces. New dual cones are introduced and proved to be closed. In the sense of efficient point, we prove that the minimal element problem is solvable if a vector variational inequality is solvable; we also prove that any solution of a strong vector variational inequality or positive vector complementarity problem is a solution of the minimal element problem.This work was done while the author was with the Chongqing Institute of Architecture and Engineering, Chongqing, P. R. China.     

Differentiability of mappings in the geometry of Carnot manifolds

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher’s theorem) and a generalization of Stepanov’s theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis ? the nilpotent tangent cone.”     

Polynomial interpolation of minimal degree

Summary. Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing. This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view. Received June 9, 1995 / Revised version received June 26, 1996     

One-dimensional minimal fillings with negative edge weights

A.O. Ivanov and A.A. Tuzhilin proposed a particular case of Gromov??s minimal fillings problem generalized to the case of stratified manifolds using weighted graphs with a nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., the minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings.     

The functions that do not change types of minimal fillings

It is proved that a function changing distances in finite metric spaces and preserving the types of their minimal fillings has the form f (x) = kx + b. It is sufficient to assume that the types of fillings are preserved for spaces consisting of at most five points.     

Intuitionistic fuzzy metric spaces

Using the idea of intuitionistic fuzzy set due to Atanassov [Intuitionistic fuzzy sets. in: V. Sgurev (Ed.), VII ITKR's Session, Sofia June, 1983; Fuzzy Sets Syst. 20 (1986) 87], we define the notion of intuitionistic fuzzy metric spaces as a natural generalization of fuzzy metric spaces due to George and Veeramani [Fuzzy Sets Syst. 64 (1994) 395] and prove some known results of metric spaces including Baire's theorem and the Uniform limit theorem for intuitionistic fuzzy metric spaces.     

Topological minimal sets and existence results

In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets.     

A generalization of the Bochner integral to locally convex spaces

We present a generalization of the Bochner integral to locally convex spaces. This generalization preserves the nuclearity of the mapping of the space of continuous functions on a compactum represented by the Bochner integral. We introduce locally convex spaces in which the study of a broad class of vector measures with values in these spaces reduces to the study of measures with values in a normed space. The results obtained are used to describe Fréchet spaces possessing the RN property.Translated from Matematicheskie Zametki, Vol. 18, No. 4, pp. 577–588, October, 1975.     

Fine properties of basis vector fields on Carnot-Carathéodory spaces under minimal assumptions on smoothness

We deduce some fine geometric properties of basis vector fields on Carnot-Carathéodory spaces under minimal assumptions on smoothness. These lead to the estimates of the approximation of a Carnot-Carathéodory space by local homogeneous groups.     

Fuzzy拓扑线性空间的Fuzzy赋准范化

引进（ＱＬ）型Ｆｕｚｚｙ拓扑线笥空间的优基坯概念,在此基础上得到了分离的Ｆｕｚｚ６ｙ拓扑线性空间可Ｆｕｚｚｙ赋准范化的一个充分必要条件,进上步阐明了Ｆｕｚｚｙ准范线性空间与（ＱＬ）型Ｆｕｚｚｙ拓扑线笥空间的关系。     

Fuzzy Modal Logics

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 201–230, 2003.     

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ₁ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.     

On some closed sets in ideal minimal spaces

The purpose of this paper is to introduce ideal minimal spaces and to investigate the relationships between minimal spaces and ideal minimal spaces. We define some closed sets in these spaces to establish their relationships. Basic properties and characterizations related to these sets are given.     

Some properties of minimal surfaces in singular spaces

This paper involves the generalization of minimal surface theory to spaces with singularities. Let be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in .

     

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.     

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics??a generalization of a game called Tug of War that has been used in Peres et?al. (J Am Math Soc 22(1):167?C210, 2009) to study real-valued AMLEs.     

A New Form of Fuzzy Compact Spaces and Related Topics via Fuzzy Idealization

Fuzzy ideals and the notion of fuzzy local function were introduced and studied by Sarkar[12] and by Mahmoud in [9]. The purpose of this paper deals with a fuzzy compactness modulo a fuzzy ideal. Many new sorts of weak and strong fuzzy compactness have been introduced to fuzzy topological spaces in the last twenty years but not have been studied using fuzzy ideals so,the main aim of our work in this paper is to define and study some new various types of fuzzy compactness with respect to fuzzy ideals namely fuzzy L-compact and L*-compact spaces. Also fuzzy compactness with respect to ideal is useful as unification and generalization of several others widely studied concepts. Possible application to superstrings and E∞ space-time are touched upon.     

On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.

     

On the classification of fuzzy projective planes of fuzzy 3-dimensional projective space

In this work, the classifications of fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector space and fuzzy projective planes of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space are given.