Convex Topology Research Articles

Order topology in minkowski space and applications

In this article, we consider and study the order topology in Minkowski spacetime of the special theory of relativity , i.e. the finest locally convex topology τ on spacetime for which every order bounded subset of spacetime is τ -bounded. This order topology that is introduced into spacetime as an ordered vector space proves to be Hausdorff and differs from Zeeman's order topology. Applying the order topology we obtain new results by applying and extending previous results on the mean ergodic theorem and functional differential evolution equations in the Minkowski space .

Non-Linear Inner Structure of Topological Vector Spaces

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. On the other hand, it was proved in the inner structure literature that isomorphisms of vector spaces and translations preserve the sets of inner points and outer points. In this manuscript, we show that in general, affine maps and convex maps do not preserve inner points. Finally, by making use of the inner structure, we find a simple proof of the fact that a convex and absorbing set is a neighborhood of 0 in the finest locally convex vector topology. In fact, we show that in a convex set with internal points, the subset of its inner points coincides with the subset of its internal points, which also coincides with its interior with respect to the finest locally convex vector topology.

Positive Miyadera–Voigt perturbations of bi-continuous semigroups

We discuss positive Miyadera–Voigt type perturbations for bi-continuous semigroups on \(\mathrm {AL}\)-spaces with an additional locally convex topology generated by additive seminorms. The main example of such spaces is the space of bounded Borel measures (on a Polish space).

Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

Representing Systems of Exponents in Weight Subspaces H (D)

In this paper, weight subspaces of the space of analytic functions on a bounded convex domain of the complex plane are considered. Descriptions of spaces that are strongly dual to the inductive and projective limits of uniformly weight spaces of analytic functions in a bounded convex domain D ⊂ ℂ are obtained in terms of the Fourier–Laplace transform. For each normed, uniformly weight space H(D, u), we construct the minimal vector space ℋi(D, u) containing H(D, u) and invariant under differentiation and the maximal vector space ℋp(D, u) contained in H(D, u) and invariant under differentiation. We introduce natural locally convex topologies on these spaces and describe strongly dual spaces in terms of the Fourier–Laplace transform. The existence of representing exponential systems in the space ℋi(D, u) is proved.

Distinguished $$ C_{p}(X) $$ spaces

We continue our initial study of $$C_{p}(X) $$ spaces that are distinguished, equiv., are large subspaces of $$\mathbb {R}^{X}$$ , equiv., whose strong duals $$L_{\beta }( X) $$ carry the strongest locally convex topology. Many are distinguished, many are not. All $$L_{\beta }(X) $$ spaces are, as are all metrizable $$C_{p}(X) $$ and $$ C_{k} ( X) $$ spaces. To prove a space $$C_{p}(X) $$ is not distinguished, we typically compare the character of $$L_{\beta }(X) $$ with |X|. A certain covering for X we call a scant cover is used to find distinguished $$C_{p} ( X) $$ spaces. Two of the main results are: (i) $$C_{p}(X) $$ is distinguished if and only if its bidual E coincides with $$\mathbb {R}^{X}$$ , and (ii) for a Corson compact space X, the space $$ C_{p}(X) $$ is distinguished if and only if X is scattered and Eberlein compact.

On Relatively Solid Convex Cones in Real Linear Spaces

Having a convex cone K in an infinite-dimensional real linear space X, Adan and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex topology $$\tau _{c}$$ on X, while the reverse implication is not true if K is not $$\tau _{c}$$ -closed. However, in the main result of this paper, we prove that the latter implication is true if the algebraic interior of the positive dual cone of K is nonempty; the general case remains an open problem. As a by-product, a result about separation of cones is obtained that improves Theorem 2.2 of the work mentioned above.

Surface Nanostructure-Wettability Coupling Leads to Unique Topological Evolution Dictating Water Transport over Nanometer Scales.

Surface nanostructure, either designed or generated as an artifact of the fabrication procedure, is known to influence interfacial phenomena intriguingly. While surface roughness-wettability coupling over nanometer scales has been addressed to some extent, the explicit interplay of hydrodynamics and confinement toward dictating the underlying characteristics for practically relevant material interfaces remains unexplored. Here, we bring out unique roles of surface nanostructures toward altering flow of water in a copper nanochannel, by capturing an exclusive interplay of confinement, roughness, wettability and flow dynamics. Toward this, non-equilibrium molecular dynamics (NEMD) simulations are performed to examine the effect of nanoscale triangular roughness. The width and height of the triangular microgroove are varied along with different driving forces at the channel inlet, and the results are compared with those corresponding to smooth-walled nanochannels. We also unveil the nontrivial characteristics of the interfacial topology as a consequence of spontaneous phase separation at the fluid-solid interface. For a constant driving force, we show that the interface may exhibit concave or convex topology, depending on the nanogroove geometry. Our results provide new vistas on how designed nanoscale roughness structures can be harnessed toward controlling the transport of water in a practically engineered nanosystem, as demanded by the specific application on hand.

Principal Submodules in the Schwartz Module

In this paper we consider the Schwartz module of entire functions of exponential type having polynomial growth along the real axis. This module is equipped with the non-metrizable locally convex topology. We establish that any principal submodule is the set of limits of converging sequences which members are polynomials multiplied by the generator of the submodule. We also obtain a weight weak localizability criterion for principal submodules and some results concerning the notion «synthesizable sequence» recently introduced by A. Baranov and Yu. Belov.

A Note on the Topological Transversality Theorem for Weakly Upper Semicontinuous, Weakly Compact Maps on Locally Convex Topological Vector Spaces

A simple theorem is presented that automatically generates the topological transversality theorem and Leray–Schauder alternatives for weakly upper semicontinuous, weakly compact maps. An application is given to illustrate our results.

Pore network model of evaporation in porous media with continuous and discontinuous corner films

During evaporation in porous media, two types of corner films are distinguished. A continuous corner film is connected to the bulk liquid, while a discontinuous one is not. To disclose their effects on evaporation in porous media, a pore network model with both continuous and discontinuous corner films is developed, which considers the capillary and viscous forces as well as the effects of corner films on the threshold pressures of pores. The capillary valve effect induced by the sudden geometrical expansion between the small and large pores is also taken into account in the model. The developed pore network model agrees well with the evaporation experiment with a quasi 2D micro model porous medium, in terms of not only the variation of the liquid saturation in each pore but also the variation of the total evaporation rate. The pore network models that neglect the corner films or the liquid viscosity are also compared with the experiment so as to shed light on the roles of the corner films. The continuous corner films, which contribute to sustain the high evaporation rate, can be interrupted to be the discontinuous ones not only by the gas invasion into pores but also by the capillary scissors effect due to the local convex topology of the solid matrix.

A locally Convex Topology on the Beurling Algebras

A locally Convex Topology on the Beurling Algebras

A Solution to the Faceless Problem

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

Relative interior and closure of the set of inner points

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of vector spaces, which, among other things, allows to characterize the linear dimension of a vector space and also serves to provide an intrinsic characterization of linear manifolds that was not possible by using internal points. Inner points of convex sets can be seen as the affine internal points, that is, the internal points with respect to the affine hull. In this manuscript we continue this research line in the scope of topological vector spaces to study the topological structure of the inner points. First, we prove that every infinite dimensional vector space has a convex subset free of inner points which is dense in every vector topology the vector space is endowed with. Also, we find the existence of convex sets in which the set of inner points is not open in the relative topology. Following this line, we also characterize the closed convex sets for which the set of inner points is not empty and open in the relative topology. Finally, we find an example of a convex set whose set of inner points is not contained in the set of inner points of its closure.

A convergent star product on the Poincaré disc

On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra.

On (L,M)-fuzzy convex structures

This paper defines a new class of L-fuzzy sets called r-L-fuzzy biconvex sets in (L,M)-fuzzy convex structures (X,C), where C is an (L,M)-fuzzy convexity on X, and some of their properties were studied. In addition, weintroduce (L,M)-fuzzy topological convexity space and study some of its properties. Finally, we introduce locally (L,M)-fuzzy topology (L,M)-fuzzy convexity space and study some of its properties.

M-FUZZIFYING TOPOLOGICAL CONVEX SPACES

The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtained from $M$-fuzzifying topological convex spaces.

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

AbstractIn the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand–Shilov spaces.

Inner structure in real vector spaces

Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.

K-spaces and duals of non-archimedean metrizable locally convex spaces

Abstract The main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field 𝕂 {\mathbb{K}} is locally compact, but fails for any non-locally compact 𝕂 {\mathbb{K}} . Here we deal with a topological subcategory, the one formed by the k 0 {k_{0}} -spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also k 0 {k_{0}} -spaces) whose products are not k 0 {k_{0}} -spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two k 0 {k_{0}} -spaces whose product is not a k 0 {k_{0}} -space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.