Theory Of Vector Spaces Research Articles

CHARACTERISATION OF EQUIVALENT NORMS ON A LINEAR SPACE USING EXPONENTIAL VECTOR SPACE

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (‘evs’ in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace C(X) comprising all nonempty compact subsets of a Hausdorff topological vector space X. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.

On integration with respect to filter

This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.

Ideals, bands and direct sum decompositions in mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections.

Bilinear spaces over a fixed field are simple unstable

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for ω-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are ω-categorical.

OLEG GEORGIYEVICH SMOLYANOV

The article is dedicated to the outstanding mathematician and pedagogue, Professor of Lomonosov Moscow State University, Oleg Georgiyevich Smolyanov (9 Feb. 1938 – 16 Dec. 2021), whose full life path and multifaceted activities were aimed at science and education development in the USSR and Russia. Oleg Georgiyevich, being a talented pedagogue and organiser, was a devotee of the scientific school in the field of the theory of topological vector spaces, infinite-dimensional analysis, beneficial for the formation of a huge number of scientists and teachers of higher educational institutions.

Study of Duality of Locally Convex Space

In this paper, we studied about the duality of locally convex space. The key to most of the results in topological vector space theory is to exploit duality - the relationship between on l.c.s. X and its dual X^*. The results of this section, particularly, show how this works. We will need to work with a variety of topologies on an l.c.s. X and it dual X^*. The results of this section, particularly, show how this works.

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. 
 In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. 
 In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. 
 The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra.
 By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

Axiomatic Approach in the Analytic Theory of Singular Perturbations

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

Induced L-bornological vector spaces and L-Mackey convergence1

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

ON MODULATED TOPOLOGICAL VECTOR SPACES AND APPLICATIONS

We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.

Regular prehomogeneous vector spaces for valued Dynkin quivers

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $(\Gamma, \boldsymbol{v})$ having a fixed oriented modulation $(𝔐, \Omega)$ over the ground field $K$. Here $K$ is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces. Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting $\Lambda$-module, where $\Lambda$ is the tensor $K$-algebra of $(𝔐, \Omega)$. We count the number of isomorphism classes of hom-orthogonal partial tilting $\Lambda$-modules of type $\mathbf{B}_n$, $\mathbf{C}_n$, $\mathbf{F}_4$ and $\mathbf{G}_2$. As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.

Extreme points of a continuum

We import into continuum theory the notion of extreme point of a convex set from the theory of topological vector spaces. We explore how extreme points relate to other established types of “edge point” of a continuum; for example we prove that extreme points are always shore points, and that any extreme point is also non-block if the continuum is either decomposable or irreducible (in particular, metrizable).In addition we discuss some continuum-theoretic analogues of the celebrated Krein-Milman theorem.

An Application of Mathematics to Computer Programming: Connecting Translation Vectors, the Minkowski Difference, and Collision Detection

Translation by a vector in the coordinate plane is first introduced in precalculus and connects to the basic theory of vector spaces in linear algebra. In this article, we explore the topic of collision detection in which the idea of a translation vector plays a significant role. Because collision detection has various applications in video games, virtual simulations, and robotics (Garcia-Alonso, Serrano, and Flaquer 1994; Rodrigue 2012), using it as a motivator in the study of translation vectors can be helpful. For example, students might be interested in the question, “How does the computer recognize when a player's character gets hit by a fireball?” Computer science provides a rich context for real-life applications of mathematics-programmers use mathematics for coding an algorithm in which the computer recognizes two objects nearing each other or colliding. The Minkowski difference, named after the nineteenth century German mathematician Hermann Minkowski, is used to solve collision detection problems (Ericson 2004). Applying the Minkowski difference to collision detection is based on translation vectors, and programmers use the algorithm as a method for detecting collision in video games.

Optimal aggregate sustainability assessment of total and selected factors of industrial processes

Vector space theory has been applied successfully to sustainability assessment of industrial processes. Processes sharing similar factors of production are represented by vectors in the space generated by the factors. The length of the vectors measures their sustainability index. This procedure leads to an ordered inequality among the set of processes. The present work shows first that each process vector is an optimal solution based on the projection theorem of optimization theory. The best option is usually the process with the smallest index in this hierarchy. It may occur though that this process is not the most sustainable in terms of particular environmental factors. For instance, the least index may be mainly due to smaller resource factors needed for production and yet has larger emissions of greenhouse gases and toxic materials. The decision-maker may choose a more environmentally sustainable process whose overall aggregate index of total factors is not the least index in the hierarchy. The set of critical factors define a subspace within the whole factor space. Another aim of the present work consists also in finding out the optimal process vectors on the selected subspaces by employing the same procedure based on the projection theorem. The decision-maker may then compare the resultant subspace hierarchies to the one from all factors. The method is then applied to automobile fender designs and food supply chain, whose factors in both cases are obtained from life cycle assessments.

A functional equation and degenerate principal series

A functional equation between the $\zeta $ distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.

Invariance and robustness of the ordered inequality of aggregate sustainability indices by vector space theory

Sustainability assessment of manufacturing sector has been described by a single aggregate index for each process. A set of production processes give rise to an inequality ordering of such indices. This ordered hierarchy must be representative of the physical system otherwise it may yield misleading results. A recent paper has treated processes as vectors whose components are the manufacturing factors in a Euclidean space generated by a set of orthonormal vectors. Aggregate indices are defined by the Euclidean magnitude of the process vectors. A subsequent paper considered an enlarged space where the Euclidean space is a particular case. In this increased space, the robustness of the Euclidean indices was analyzed. The present work in this broader space focuses on the robustness of the ordered inequality. The central aim consists in determining the conditions by which the Euclidean indices belong to an invariant hierarchy in this space. The formalism was confirmed by providing consistent results when applied to real production processes.

A novel framework for analyzing the green value of food supply chain based on life cycle assessment

The environmental impact is one of the major pillars of concerns when addressing the sustainability of food supply chain. In life cycle assessment of food supply chain, a particular environmental impact category from farming to distribution is aggregated. These values are widely used in determining which supply chain is performing more environmentally sound on the basis of a particular impact category. However, when different categories of environmental impacts are represented in one matrix, different impact categories can exhibit increasing and decreasing trends simultaneously for the same product. Moreover, different impact categories have different units of measurement. All of these factors make it difficult to clearly prefer one supply chain to the other. Therefore, a quantifiable indicator is desirable for each individual product which can represent all impact categories for a single product through a single index and can be easily comparable during decision making. This paper presents a new framework to compute a single index based on Life cycle assessment using vector space theory. This index can be utilized to compare the environmental sustainability among various food supply chains. The index is named as green value of a supply chain. The use of this proposed green value is explained through three case studies with different goals, scopes, system boundaries, and functional units. It is also shown that how the order of green value varies by changing the functional unit of the same product. The contribution of green value is the better interpretation of Life cycle assessment result for the purpose of decision making and is easily understandable to customers, manufacturers, policy makers, and other stakeholders.

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a difficult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.

Scalable and robust channel allocation for densely-deployed urban wireless stations

This paper presents a scalable and robust scheduling algorithm for a set of wireless stations. Recently, road-side access points for vehicular networks and outdoor WiFi stations are deployed in wide urban areas and compete with each other for limited wireless resources. We focus on pursuing the best balance among (i) optimality of resource utilization, (ii) robustness to new station installation and traffic demand, and (iii) scalability to the population of stations and area size. The algorithm is designed based on a vector space theory with correctness proof. Compared with our previous work that presents the basic algorithm and theory, we extend the algorithm to a multi-layer (i.e. hierarchical) version to handle more realistic situations. Additionally, we have conducted extensive simulations with a variety of scenarios. Finally, the applicability of approach has been testified by a case study on a scheduling problem for roadside access points of vehicular networks in cooperation with a manufacturing corporation.