Theory Of Vector Spaces Research Articles

Combining different theoretical perspectives for analyzing students’ difficulties in vector spaces theory

This paper originates from our doctorate research project, in which we investigated graduate and undergraduate students’ errors and difficulties in vector space theory (VST). After a brief account of the whole study, we will focus on the analyses carried out through the lenses of two different theoretical frameworks: Fischbein’s Theory of Tacit Models and Sfard’s process/object duality theory. We also propose a retrospective reflection concerning the reasons why we chose to carry out our analysis according to more than one theoretical framework and why we chose those specific theoretical frameworks as well. Hence this contribution has a double focus: on the one hand it reports on the results drawn from our study on students’ difficulties in VST; on the other hand it presents a personal retrospective reflection on the use of theoretical frames within that study.

FUSION OVER A VECTOR SPACE

Let T1 and T2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T1 ∪ T2 has a strongly minimal completion.

Fusion over sublanguages

AbstractGeneralising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theoriesT1.T2intersecting in a totally categorical sub-theoryT0. We show that if. e.g.,T0is the theory of infinite vector spaces over a finite field then the fusion theoryTω, exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis ofTω, proving that if bothT1,T2are 1-based then.Tω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied ifT0is the theory of infinite vector spaces over a finite field.

An application of vector space theory in data transmission

This work illustrates an application of vector spaces to data transmission theory. We show how Hamming code error detection and error correction are done through the tool of various theories in vector space. It is hoped that this article will explain the importance of abstract mathematics, such as vector space and basis, in the application of data transmission, which enlightens mathematics and computer science majors.

LINEARLY INDEPENDENT ELEMENTS IN N-GROUPS WITH FINITE GOLDIE DIMENSION

The concepts linearly independent elements and u-linearly independent elements in an N-group G where N is a near-ring, were introduced and studied. A few important results in the theory of vector spaces were generalized to N-groups.

Approximation of vector valued smooth functions

AbstractA real locally convex space is said to be convenient if it is separated, bornological and Mackey‐complete. These spaces serve as underlying objects for a whole theory of differentiation and integration (see [4]) upon which infinite dimensional differential geometry is based (cf. [8]). We investigate the question of density of the subspaces C∞(E) ⊗ F and 𝒫f (E) ⊗ F of smooth (polynomial) decomposable functions in the space C∞(E, F) of smooth functions between convenient vector spaces E, F with respect to various natural structures. A characterization is given for density with respect to the c∞‐topology and also some classical locally convex topologies on C∞(E, F). It is shown furthermore, that for the space ℝ(ℕ) the convenient analogon of the Schwartz kernel theorem for C∞‐functions holds. Spaces of C∞‐functions on both separable and non‐separable manifolds are considered and an example of a non‐separable manifold is given failing the above property of approximability by decomposable functions. Those notions and features of the theory of convenient vector spaces which are essential for the results of this paper are explained in the introductory section below and where needed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Classroom note: A teaching method for the vector triple product identity

In texts on vector analysis one finds many different methods of proving the vector triple product identity. Most of these rely on complicated geometrical constructions or theorems drawn from vector space theory. The component method is generally regarded as being merely long and tedious. Yet, viewed in the correct way, the component method is actually the simplest. With correct guidance students can easily construct the proof for themselves by this method.

On the space of quadruples of quinary alternating forms

We discuss some aspects of the invariant theory and arithmetic of the prehomogeneous vector space of quadruples of quinary alternating forms. In particular, we complete the explicit construction of all prehomogeneous covariants of this space and give the rational classification of maximal singular orbits.

On the convergence of MoM and mode matching solutions for infinite array and waveguide problems

It is shown that the method of moment (MoM) fails to analyze an infinite array of strip discontinuities if the number of basis functions and the number of expansion modes are set equal, however large they individually may be. Using spectral analysis and linear vector space theory it is found that for a given number of basis functions, there exists a minimum number of expansion modes required for a reasonable accuracy. Subsequently, for a given number of expansion modes, there is an upper limit for the number of basis functions. The mode matching (MM) analysis is viewed as a special case of the MoM analysis and the accuracy of the analysis is discussed with respect to the number of modes in the waveguides.

Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements:

Nonregular 2-Simple Prehomogeneous Vector Spaces of Type I and Their Relative Invariants

After the fundamental theorem for nonregular reductive prehomogeneous vector spaces was proved by A. Gyoja (Tsukuba J. Math.14 (1990), 437–457), the construction of the theory of nonregular prehomogeneous vector spaces became an interesting problem. However, only (Spn × GL2, Λ1 ⊗ 2Λ1, M(2n, 3)) is known as an interesting example of nonregular reductive prehomogeneous vector spaces. In this paper, we give some other interesting examples by constructing all relative invariants of nonregular 2-simple prehomogeneous vector spaces of type I. We consider everything over the complex number field C.

Generalization of inductive topologies to L-topological vector spaces

In this paper, using the theory of L-topological vector spaces, we study some properties of inductive topologies determined by the family of L-fuzzy linear order-homomorphisms defined by Fang Jin-xuan (J. Fuzzy Math. 4(1) (1996) 93), and give a characterization of inductive topologies determined by a single L-fuzzy linear order-homomorphism. As an example, we point out the quotient L-topology of L-tvs is an inductive topology.

WEAK REFLECTION IN CONCRETE CATEGORIES

In this art cle I give the de .n t on of the strong and weak re .ex v ty and the com- pat bility relation of objects in a categorical language.These concepts will generalize the corresponding concepts n the theory of topolog cal vector spaces. The main theorem makes clear the connection between these mportant concepts and we can show a lot of objects in our category,which are not strongly re .exive,but compatible with strongly re .ex ve objects.We also consider a lot of interesting examples and try to throw new light upon the ex stence of such Abelian groups which satisfy Pontryagin duality but do not respect compactness.We w ll prove also that the weak topolog cal vector space- structures are not re .ex ve in general.

Time-varying threshold integral pulse frequency modulation.

Several methods have been proposed so far for the analysis of the integral pulse frequency modulation (IPFM) model and detecting its corresponding physiological information. Most of these methods rely on the low-pass filtering method to extract the modulating signal of the model. In this paper, we present an entirely new approach based on vector space theory. The new method is developed for a more comprehensive form of the IPFM model, namely the time-varying threshold integral pulse frequency modulation (TVTIPFM) model. The new method decomposes the driving signals of the TVTIPFM model into a series of orthogonal basis functions and constructs a matrix identity through which the input signals can be obtained by a parametric solution. As a particular case, we apply this method to R-R intervals of the SA node to discriminate between its autonomic nervous modulation and the stretch induced effect.

Some properties of Banach‐valued sequence spaces ℓp[X

We discuss some properties of the Banach‐valued sequence space ℓp[X](1 ≤ p < ∞), the space of weakly p‐summable sequences on a Banach space X. For example, we characterize the reflexivity of ℓp[X], convergent sequences on ℓp[X], and compact subsets of ℓp[X].

Information theoretic similarity measures for content based image retrieval

Content-based image retrieval is based on the idea of extracting visual features from image and using them to index images in a database. The comparisons that determine similarity between images depend on the representations of the features and the definition of appropriate distance function. Most of the research literature uses vectors as the predominate representation given the rich theory of vector spaces. While vectors are an extremely useful representation, their use in large databases may be prohibitive given their usually large dimensions and similarity functions. In this paper, we propose similarity measures and an indexing algorithm based on information theory that permits an image to be represented as a single number. When use in conjunction with vectors, our method displays improved efficiency when querying large databases.

Information theoretic similarity measures for content based image retrieval

AbstractContent‐based image retrieval is based on the idea of extracting visual features from image and using them to index images in a database. The comparisons that determine similarity between images depend on the representations of the features and the definition of appropriate distance function. Most of the research literature uses vectors as the predominate representation given the rich theory of vector spaces. While vectors are an extremely useful representation, their use in large databases may be prohibitive given their usually large dimensions and similarity functions. In this paper, we propose similarity measures and an indexing algorithm based on information theory that permits an image to be represented as a single number. When use in conjunction with vectors, our method displays improved efficiency when querying large databases.

On Mackey topology for groups

The present paper is a contribution to fill in a gap existing between the theory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Functional Analysis. It is natural to expect that some important and elegant theorems about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably the lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilenkin [26] and the work of Banaszczyk [1] have paved the way to obtain theorems of this nature. We study here the group topologies compatible with a given duality. We have obtained, among others, the following result: for a complete metrizable topological abelian group, there always exists a finest locally quasi-convex topology with the same set of continuous characters as the original topology. We also give a description of this topology as an G-topology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.

The role of formalism in the teaching of the theory of vector spaces

In the French tradition of Bourbaki, the theory of vector spaces is usually presented in a very formal setting, which causes severe difficulties to many students. The aim of this paper is to analyze the underlying reasons of these difficulties and to suggest some ways to make the first teaching of the theory of vector spaces less ineffective for many students. We do not reject the necessity for formalism. On the contrary, on the basis of a historical analysis we can explain the specific meaning it has in the theory. From this mathematical analysis with a historical perspective, we analyze the teaching and the apprehension of vector space theory in a new approach. For instance, we will show that mistakes made by many students can be interpreted as a result of a lack of connection between the new formal concepts and their conceptions previously acquired in more restricted, but more intuitively based areas. Our conclusions will not plead for avoiding formalism but for a better positioning of the formal concepts with regard to previous knowledge of the students as well as special care to be given in making the role and the meaning of formalism in linear algebra explicit to the students.

A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms

About twenty-five years ago T. Shintani defined and studied four Dirichlet series whose coefficients are class numbers of integral binary cubic forms by using the theory of prehomogeneous vector spaces. In this paper, we give a conjecture that two of these Dirichlet series are essentially the same as the remaining two series. The conjecture is based on our calculation of the first two hundred nonzero coefficients of the four Dirichlet series, and is also consistent with the known functional equation and residues at poles. If the conjecture is true, we get simpler symmetric functional equations than previously known. Namely, if we take a certain linear combination of Shintani's Dirichlet series, that a single function is invariant under the variable change of s into 1 - s .