scholarly journals Normed Interval Space and Its Topological Structure

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 983
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Vector Space ◽  
Topological Structure ◽  
Zero Element ◽  
Scalar Multiplication ◽  
Closed Intervals ◽  
Vector Addition ◽  
Natural Vector ◽  
Different Types ◽  
Interval Space ◽  
Null Set

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.

A r-maximal vector space not contained in any maximal vector space

1978 ◽  
Vol 43 (3) ◽  
pp. 430-441 ◽  
Author(s):  
J. Remmel
Keyword(s):  
Vector Space ◽  
Quotient Space ◽  
Independent Set ◽  
Scalar Multiplication ◽  
Effective Procedure ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Vector Addition ◽  
Zero Vector

In [4], Metakides and Nerode define a recursively presented vector space V∞. over a (finite or infinite) recursive field F to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Given a subspace W of V∞, we write dim(W) for the dimension of W. Given subspaces V and W of V∞, V + W will denote the weak sum of V and W and if V ∩ W = {0) (where 0 is the zero vector of V∞), we write V ⊕ W instead of V + W. If W ⊇ V, we write W mod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is a r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case that A is extendible.

scholarly journals A New Concept of Fixed Point in Metric and Normed Interval Spaces

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 219 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Closed Intervals ◽  
Interval Space ◽  
Inverse Element ◽  
Null Set

The main aim of this paper is to propose the concept of so-called near fixed point and establish many types of near fixed point theorems in the set of all bounded and closed intervals in R . The concept of null set will be proposed in order to interpret the additive inverse element in the set of all bounded closed intervals. Based on the null set, the concepts of metric interval space and normed interval space are proposed, which are not the conventional metric and normed spaces. The concept of near fixed point is also defined based on the null set. In this case, we shall establish many types of near fixed point theorems in the metric and normed interval spaces.

scholarly journals Informal Norm in Hyperspace and Its Topological Structure

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 945
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Vector Space ◽  
Topological Structure ◽  
Different Types

The hyperspace consists of all subsets of a vector space. Owing to a lack of additive inverse elements, the hyperspace cannot form a vector space. In this paper, we shall consider a so-called informal norm to the hyperspace in which the axioms regarding the informal norm are almost the same as the axioms of the conventional norm. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets. In this case, the topologies generated by these different concepts of open sets are investigated.

scholarly journals Isomorphisms on countable vector spaces with recursive operations

1974 ◽  
Vol 18 (2) ◽  
pp. 230-235 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Vector Space ◽  
Independent Set ◽  
Vector Spaces ◽  
Recursive Structure ◽  
Recursively Enumerable ◽  
Linearly Independent ◽  
Vector Addition

Terminology and notation may be found in Dekker [1] and [2]. Briefly, we fix a recursively enumerable (r.e.) field F with recursive structure, and let Ū be the vector space over F consisting of ultimately vanishing countable sequences of elements of F with the usual definitions of vector addition and multiplication by a scalar. A subspace V of Ū is called an α-space if V has a basis B which is contained in some r.e. linearly independent set S.

scholarly journals A SURVEY OF THE DIFFERENT TYPES OF VECTOR SPACE PARTITIONS

2012 ◽  
Vol 04 (01) ◽  
pp. 1250001 ◽  
Author(s):  
OLOF HEDEN
Keyword(s):  
Vector Space ◽  
Design Theory ◽  
Projective Planes ◽  
Error Correcting Codes ◽  
Necessary Condition ◽  
Space Partition ◽  
Different Types ◽  
Vector Space Partitions ◽  
Historical Remarks ◽  
Zero Vector

A vector space partition is here a collection [Formula: see text] of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of [Formula: see text]. Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.

scholarly journals Basic Concepts of Structural Design for Architecture Students

2021 ◽  
Author(s):  
Anahita Khodadadi ◽  
Keyword(s):  
Material Properties ◽  
Structural Design ◽  
Structural Systems ◽  
Structural Elements ◽  
Math Problem Solving ◽  
Vector Addition ◽  
Rules Of Thumb ◽  
Different Types ◽  
Internal Forces ◽  
Basic Concepts

This book aims to narrate fundamental concepts of structural design to architecture students such that they have minimum involvement with math problem-solving. Within this book, students learn about different types of loads, forces and vector addition, the concept of equilibrium, internal forces, geometrical and material properties of structural elements, and rules of thumb for estimating the proportion of some structural systems such as catenary cables and arches, trusses, and frame structures.

scholarly journals Performance Evaluation of Scalar Multiplication in Elliptic Curve Cryptography Implementation using Different Multipliers Over Binary Field GF (2233)

2020 ◽  
Vol 26 (9) ◽  
pp. 45-64
Author(s):  
Alaa Mohammed Abdul-Hadi ◽  
Yousraa Abdul-sahib Saif-aldeen ◽  
Firas Ghanim Tawfeeq
Keyword(s):  
Elliptic Curve Cryptography ◽  
Performance Index ◽  
High Frequency ◽  
Low Frequency ◽  
Scalar Multiplication ◽  
Maximum Frequency ◽  
Binary Field ◽  
Point Multiplication ◽  
Additional Layer ◽  
Different Types

This paper presents a point multiplication processor over the binary field GF (2233) with internal registers integrated within the point-addition architecture to enhance the Performance Index (PI) of scalar multiplication. The proposed design uses one of two types of finite field multipliers, either the Montgomery multiplier or the interleaved multiplier supported by the additional layer of internal registers. Lopez Dahab coordinates are used for the computation of point multiplication on Koblitz Curve (K-233bit). In contrast, the metric used for comparison of the implementations of the design on different types of FPGA platforms is the Performance Index. The first approach attains a performance index of approximately 0.217610202 when its realization is over Virtex-6 (6vlx130tff1156-3). It uses an interleaved multiplier with 3077 register slices, 4064 lookup tables (LUTs), 2837 flip-flops (FFs) at a maximum frequency of 221.6Mhz. This makes it more suitable for high-frequency applications. The second approach, which uses the Montgomery multiplier, produces a PI of approximately 0.2228157 when its implementation is on Virtex-4 (6vlx130tff1156-3). This approach utilizes 3543 slices, 2985 LUTs, 3691 FFs at a maximum frequency of 190.47MHz. Thus, it is found that the implementation of the second approach on Virtex-4 is more suitable for applications with a low frequency of about 86.4Mhz and a total number of slices of about 12305.

scholarly journals A few remarks on bounded operators on topological vector spaces

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 763-772
Author(s):  
Omid Zabeti ◽  
Ljubisa Kocinac
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Operators ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Topological Vector Spaces ◽  
Different Types ◽  
Locally Convex

We give a few observations on different types of bounded operators on a topological vector space X and their relations with compact operators on X. In particular, we investigate when these bounded operators coincide with compact operators. We also consider similar types of bounded bilinear mappings between topological vector spaces. Some properties of tensor product operators between locally convex spaces are established. In the last part of the paper we deal with operators on topological Riesz spaces.

Research on Methodology of Correlation Analysis of Sci-Tech Literature Based on Deep Learning Technology in the Big Data

2018 ◽  
Vol 29 (3) ◽  
pp. 67-88 ◽  
Author(s):  
Wen Zeng ◽  
Hongjiao Xu ◽  
Hui Li ◽  
Xiang Li
Keyword(s):  
Big Data ◽  
Deep Learning ◽  
Correlation Analysis ◽  
Vector Space ◽  
Input Word ◽  
Learning Technology ◽  
Depth Analysis ◽  
Different Types ◽  
High Level ◽  
Deep Learning Model

In the big data era, it is a great challenge to identify high-level abstract features out of a flood of sci-tech literature to achieve in-depth analysis of data. The deep learning technology has developed rapidly and achieved applications in many fields, but has rarely been utilized in the research of sci-tech literature data. This article introduced the presentation method of vector space of terminologies in sci-tech literature based on the deep learning model. It explored and adopted a deep AE model to reduce the dimensionality of input word vector feature. Also put forward is the methodology of correlation analysis of sci-tech literature based on deep learning technology. The experimental results showed that the processing of sci-tech literature data could be simplified into the computation of vectors in the multi-dimensional vector space, and the similarity in vector space could be used to represent similarity in text semantics. The correlation analysis of subject contents between sci-tech literatures of the same or different types can be made using this method.

Quadratic Forms of Type F4

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
Scalar Multiplication ◽  
Quadratic Space

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

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