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In this paper, we present a unified treatment on the linear, differentiable and topological equivalences for time-invariant linear control systems governed by ordinary differential equations (ODEs). Especially, we give a new solution to the topological classification problem for linear control systems by means of an elementary analytic approach, which differs considerably from J.C. Willems’ algebraic method developed in (Willems, 1980) [17] . More precisely, based on P. Brunovsky’s results in (Brunovsky, 1970) [2] , we determine all topological invariants for the linear control system. As an interesting by-product, we show that there exist only finitely many topological equivalence classes for any stabilizable system with given numbers of state and control variables, which might be useful for some engineering applications.

We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T ⊕ M g is universal, where M g is multiplication by a generating element of a compact topological group. We use this result to characterize R + -supercyclic operators and to show that whenever T is a supercyclic operator and z 1 , … , z n are pairwise different non-zero complex numbers, then the operator z 1 T ⊕ ⋯ ⊕ z n T is cyclic. The latter answers affirmatively a question of Bayart and Matheron.

In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure C defined on a topological space has the weak selection property then C satisfies H 0 -condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property.

A considerable number of non-normed topological *-algebras admit a C*-enveloping algebra. Various characterizations of such algebras are given in terms of diverse properties of decisive features of a given topological *-algebra, like the sets of its continuous *-representations, its continuous positive functionals, the hermitian spectrum, etc. Interconnections of these sorts of algebras with hermiticity and C*-spectrality are also discussed. An analogy in structure, between (not necessarily involutive) commutative Q-algebras and topological *-algebras having a C*-enveloping algebra, leads to the study of some automatic continuity results. Furthermore, different examples are presented concerning the question: When does a dense Frechet *-subalgebra of a C*-algebra have a C*-enveloping algebra? Some questions and conjectures are stated that arise from the whole study.

In this paper, we introduce generalized R-KKM mapping and discuss some new generalized R-KKM theorem under the nonconvexity setting of topological space. As applications, some new minimax inequalities, saddle point theorem are proved in topological space. Our theorems unified and extend many known results in recent literature.

A series of basic summability results are established for matrices of linear and some nonlinear operators between topological vector spaces.

Let K be the scalar field of all real or complex numbers, let (Ω, A, μ) be a σ-finite measure space, and let L(μ, K) be the algebra of the μ-equivalence classes of all K-valued μ-measurable functions defined on (Ω, A, μ). L(μ, K) is a topological algebra over K when endowed with the topology of convergence locally in measure; topological modules over this topological algebra L(μ, K) (briefly, L(μ, K)-topological modules) are an extensive class of topological modules, which arise naturally in the course of the study of the theory of probabilistic normed spaces. The purpose of this paper is to show that an arbitrary regular L(μ, K)-topological module admits enough canonical module homomorphisms if and only if all of its quasi-free submodules of finite rank are complemented in the sense of topological modules.

Given a multifunctionAfrom a topological spaceXinto the topological dualE* of a real topological vector spaceE, we deal with the problem of findingx ∈ Xsuch that 0E* ∈ A(x). Some existence results for the last inclusion are proved. In particular, we extend to multi-valued operators the main result of 4 .

A fuzzy topological analog of the R1 separation axiom of topology is introduced and its appropriateness is established.

Functional analysis techniques, mainly pertaining to the theory of intinitedimensional representations, have been recently applied, mostly by Soviet mathematicians, to compute Hermitian K-groups related to the classification of quadratic forms with coefftcients in group rings (see, for instance, the relevant survey articles by MiSEenko [59, 801. In this concern, it has been pointed out that without loss of generality, one may profitably consider instead as “coefficient ring” an (abstract) C*-algebra [59, p. 931. In a similar framework, namely, that of (vector) bundles with coefftcients in C*algebras, topological invariants of elliptic operators related to the classical Atiyah-Singer formula have also been studied by MiiEenko in [60] (see also [61 I), whereas, Kasparov has given in [42] a similar study by still applying C*-algebra techniques (cf. the comments at the end of this paper). Besides, vector bundles over Banach algebras and, in particular C*algebras, commutative or not, have already been applied by Kandelaki [37, 381, and independently by Fujii [27], one of their main results being a generalization of the Serre-Swan theorem [41] for complex vector bundles (cf. Theorem 4.2 in this paper). Furthermore, connections with the theory of “extensions of C*-algebras” ([ 12, “BDF-Theory”]) have also been considered by Kandelaki [39, 401. On the other hand, topological (in particular, locally m-convex) *-algebras (necessarily not normed ones) have been recently used in connection, for instance, with applications to relativistic quantum theory [18, 191. Thus, the extension of the preceding considerations to vector bundles with coefftcients in topological algebras, more general than normed ones and, in particular, in locally m-convex ones [55, 581 is naturally revealed. Now, this task is further feasible as a consequence of the generalization, within the topological algebras theory framework, of previous relevant work concerning applications of algebraic topology and complex analysis to Banach algebras theory [24, 53-55, 621. 452 0022-247X/83$3.00

In this paper duality theory for infinite dimensional linear programs is discussed in a topological vector space setting. Infinite dimensional linear programs occur in many different areas and the duality theory of such problems has been discussed by a number of authors. However, the results are scattered in the literature and are proved in a variety of different settings. The purpose of this paper is to bring together the main results on this subject and to present them in a unified setting and notation with some new simpler proofs.

In this paper we extend the notions of quasi-uniformities on topological spaces to fuzzy topological space. We prove theorems corresponding to many of the usual theorems. In particular we show every fuzzy topological space is quasi-uniformizable and characterize uniformizability in terms of a type of complete regularity. We also construct a natural uniformity on the fuzzy unit interval.

The concept of induced fuzzy topological spaces was introduced by Weiss [ 11. Subsequently, Lowen [2, 31 defined two mappings o and I which established the relationship between fuzzy topological spaces and topological spaces, and proved some deeper results on induced fuzzy topological spaces. Since then several other authors continued the investigation of such spaces. In this paper we will use the decomposition theorem for fuzzy sets to simplify the proofs of some theorems due to Lowen, and discuss the closure and interior operators, separability, Ci, C,, and separation properties of induced fuzzy topological spaces.

The purpose of this paper is to determine necessary and sufficient conditions under which certain families of linear mappings can be represented by a given bilinear mapping. More precisely, let E, F be topological vector spaces and suppose B is a continuous bilinear mapping of E x F into a topological vector space G. If dp is a collection of continuous linear mappings of F into G, under what conditions is each member of 2 represented by B(x, .) for a unique x E E ? In case E is a Hilbert space and B is a sesquilinear form on E x E the following is well-known

Considering complete Boolean algebras as sets of truth values the structure of a fuzzy topological group, fuzzy topological ring, etc., is specified. The probabilistic completion of ordinary topological algebraic structures shows the applicability of these concepts to the theory of stochastic processes, e.g., a new definition of the stochastic integral is presented in Section 5.

In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679–690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.

L -fuzzy sets and, subsequently, L -fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L -fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.

In recent years, there has been a systematic development of the theory of A-bundles, that is, vector bundles where the field of coefficients is replaced by a topological algebra A, usually an involutive one. The above extension is justified by its applications in pure mathematics (differential topology [15]) as well as in relativistic quantum mechanics, via, e.g., the Serre-Swan theorem [22]. Bundles with coefficients in a commutative unital Banach algebra, or yet a C*-algebra, were considered by T. K. Kandelaki [6, 71, K. Fujii [S], and A. S. MiSEenko and Yu. P. Solov’ev [ 15, 161. Furthermore, A. Mallios studied A-bundles, with A being a locally m-convex (*-)algebra with unit [l&12]. Note also that certain results are still valid for more general topological algebras [ 13, Sect. 53, or even topological rings [21]. The results already obtained are mainly within the framework of topological A-bundles. The basic difficulty in the development of a theory of differential A-bundles is the lack of a suitable differential calculus in the libres, the latter being topological modules over A. One might, of course, use the existing methods of differentiation in the underlying topological vector spaces (cf. for instance [18]). However, besides the intrinsic weaknesses of these methods (regarding the higher order chain rule, continuity of differentiable maps, differentiability of the composition map and evaluation map, which are necessary for differential geometric considerations on bundles), the additional algebraic operations (scalar multiplication by elements of A, involution) should be taken into account, because they are essentially involved in the structure of A-bundles (cf. [ 11, 183). For the same reason (incompatibility with involution), the 255 0022-247X/92 55.00