Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ ^N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.      

Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg–Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Appropriate algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient first to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and R_δ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.     

Nonlinear impulsive systems on infinite dimensional spaces

In this paper we consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, non-monotone continuous operators with respect to suitable Banach spaces. We present here a new result on compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.     

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

On extension of continuity and application of countability axioms in topological tensor products

In this paper an attempt has been made to prove an analogue of the result¹ on the extension of linear and continuous mappings in topological vector spaces to the context of topological tensor products. We also prove that the topological tensor-product of two separable locally convex spaces is separable, and the projective tensor product of any two locally-convex 1st countable (2nd countable) spaces is 1st countable (2nd countable). For the terminology, and basic notations, we refer to Treves¹ and Jha.^2,3     

Prevalence of Multifractal Functions in S<Superscript>v</Superscript> Spaces

Spaces called S^v were introduced by Jaffard [16] as spaces of functions characterized by the number ≃ 2^ν(α)j of their wavelet coefficients having a size ≳ 2^−αj at scale j . They are Polish vector spaces for a natural distance. In those spaces we show that multifractal functions are prevalent (an infinite-dimensional “almost-every”). Their spectrum of singularities can be computed from ν, which justifies a new multifractal formalism, not limited to concave spectra.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

On the generalized vector equilibrium problems

In this paper, generalized vector equilibrium problems are studied and some existence theorems of solutions for these problems in the setting of topological vector spaces are proved. Sufficient conditions for the set of solutions to be compact and convex are given. Our results improve some recent results in this field.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

Renormalized solutions of some transport equations with partially <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript> velocities and applications

We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not W^1,1 with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.     

Connectedness of Cone-Efficient Solution Set for Cone-Quasiconvex Multiobjective Programming in Locally Convex Spaces

This paper deals with the connectedness of the cone-efficient solution set for vector optimization inlocally convex Hausdorff topological vector spaces.The connectedness of the cone-efficient solution set is provedfor multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set ofalternatives.The generalized saddle theorem plays a key role in the proof.     

Topological hypercovers and <Superscript>1</Superscript>-realizations

We show that if U_* is a hypercover of a topological space X then the natural map hocolim U_* X is a weak equivalence. This fact is used to construct topological realization functors for the ¹-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.Mathematics Subject Classification (2000):55U35, 14F20, 14F42The second author was supported by an NSF Postdoctoral Research Fellowship     

拓扑向量空间中锥拟凸多目标规划锥有效解集的连通性(英文)

本文研究局部凸的拓扑向量空间中锥拟凸多目标规划锥有效解集的连通性问题.证明了定义在紧凸集上目标映射为一对一的锥拟凸多目标规划的锥有效解集是连通的.在证明中,广义鞍点定理起着关键的作用.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153     

Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations

We consider piecewise monotone (not necessarily, strictly) piecewise C ² maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy h _top(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.