scholarly journals M-Hazy Vector Spaces over M-Hazy Field

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1118
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Binary Operation ◽  
Vector Spaces ◽  
Fundamental Properties ◽  
Classical Algebra ◽  
Important Development ◽  
Fuzzy Algebra ◽  
Convex Spaces

The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called M-hazy vector spaces over M-hazy field. Some fundamental properties of M-hazy field, M-hazy vector spaces, and M-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of M-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that M-fuzzifying convex spaces are induced by an M-hazy subspace of M-hazy vector space.

scholarly journals ωb-Topological Vector Spaces

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Vector Spaces ◽  
Closed Set ◽  
Topological Vector Spaces ◽  
New Class ◽  
Fundamental Properties ◽  
General Properties

The purpose of the present paper is to introduce the new class of ω b - topological vector spaces. We study several basic and fundamental properties of ω b - topological and investigate their relationships with certain existing spaces. Along with other results, we prove that transformation of an open (resp. closed) set in aω b - topological vector space is ω b - open (resp. closed). In addition, some important and useful characterizations of ω b - topological vector spaces are established. We also introduce the notion of almost ω b - topological vector spaces and present several general properties of almost ω b - topological vector spaces.

On the Nonstandard Duality Theory of Locally Convex Spaces

1980 ◽  
Vol 32 (2) ◽  
pp. 460-479 ◽  
Author(s):  
Arthur D. Grainger
Keyword(s):  
Vector Space ◽  
Duality Theory ◽  
Dual System ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Nonstandard Model ◽  
Locally Convex ◽  
External Property ◽  
Convex Spaces

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

scholarly journals MATRIKS BAKU UNTUK TRANSFORMASI LINIER PADA RUANG VEKTOR DIMENSI TIGA

2019 ◽  
Vol 15 (2) ◽  
pp. 88-93
Author(s):  
Khasnah Aris Friantika ◽  
Harina O. L. Monim ◽  
Rium Hilum
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Finite Dimension ◽  
Linear Operators ◽  
Vector Spaces ◽  
Linear Transformations ◽  
Dimension Vector ◽  
Compression And Expansion ◽  
University Level ◽  
The University

The linear transformation is a function relating the vector   ke . If , then the transformation is called a linear operator. Several examples of linear operators have been introduced since SMA such as reflexive, rotation, compression and expansion and shear. Apart from being introduced in SMA, these linear operators were also introduced to the linear algebra course. Linear transformations studied at the university level include linear transformation in finite dimension vector spaces . The discussion includes how to determine the standard matrix for reflexive linear transformations, rotation, compression and expansion and given shear. Through the column vectors of reflexive, rotation, compression and expansion and shear, a standard matrix of 2x2 size is formed for the corresponding linear transformation. however, in this study, the authors studied linear transformations in dimensioned vector spaces . The results of this study are if known  is a vector space with finite and  the standard matrix for reflexivity, rotation, expansion, compression and shear is obtained. Each of these linear transformations is performed on x-axis, y-axis and z-axis on  to get column vectors. The column vectors as a result of the linear transformation at form the standard matrix for the corresponding linear transformation in the vector space. The standard matrix for linear transformations in the vector space  is obtained by determining reflexivity, rotation, expansion, compression and shear. The process of obtaining a standard matrix for linear transformation is carried out by rewriting the standard basis, determining the column vectors, and rearranging them as the standard matrix for each linear transformation in the vector space

Totally Variant Sets in Finite Groups and Vector Spaces

1968 ◽  
Vol 20 ◽  
pp. 701-710 ◽  
Author(s):  
Frederick Hoffman ◽  
Lloyd R. Welch
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Finite Groups ◽  
Dimensional Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Quaternion Group ◽  
Finite Dimensional ◽  
Identity Automorphism ◽  
Space Case

We are concerned here with the question of which finite groups and vector spaces possess subsets which are moved by every non-identity automorphism (in the vector-space case—non-singular linear transformation). We find that this is the case for all but four finite-dimensional vector spaces (2-, 3-, and 4-dimensional space over Z2, 2-dimensional space over Z3), and for all finite groups except for those corresponding to the vector-space exceptions, and the quaternion group of order eight. The question was first posed to the authors, in the vector-space case, by Morris Marx.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

scholarly journals FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

2016 ◽  
Vol 101 (2) ◽  
pp. 277-287
Author(s):  
AARON TIKUISIS
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Space Structure ◽  
Vector Spaces ◽  
Ordered Vector Space ◽  
Finite Dimensional ◽  
Ordered Vector Spaces

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$.

scholarly journals Cross-Ratio in Real Vector Space

2019 ◽  
Vol 27 (1) ◽  
pp. 47-60
Author(s):  
Roland Coghetto
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Cross Ratio ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Complex Vector ◽  
The Real ◽  
Mizar Mathematical Library ◽  
The Cross

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

scholarly journals On Dentability in Locally Convex Vector Spaces

2017 ◽  
Vol 10 ◽  
pp. 14-19
Author(s):  
Oleg Reinov ◽  
Asfand Fahad
Keyword(s):  
Vector Space ◽  
Wide Class ◽  
Positive Answer ◽  
Vector Spaces ◽  
Bounded Subset ◽  
Locally Convex

The notions of V-dentability, V-s-dentability and V-f-dentability are introduced. It is shown, in particular, that if B is a bounded sequentially complete convex metrizable subset of a locally convex vector space E and V is a neighborhood of zero in E, then the following are equivalent: 1). B is subset V-dentable; 2). B is subset V-s-dentable; 3). B is subset V-f-dentable. It follows from this that for a wide class of locally convex vector spaces E, which strictly contains the class of (BM) spaces (introduced by Elias Saab in 1978), the following is true: every closed bounded subset of E is dentable if and only if every closed bounded subset of E is f-dentable. Also, we get a positive answer to the Saab's question (1978) of whether the subset dentability and the subset s-dentability are the same forthe bounded complete convex metrizable subsets of any l.c.v. space.

scholarly journals M – STAR – Irresolute Topological Vector Spaces

2021 ◽  
Vol 7 ◽  
pp. 20-36
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Vector Space ◽  
Extreme Point ◽  
Topological Vector Space ◽  
Convex Subset ◽  
Hausdorff Space ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
New Class

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

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