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 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.

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 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.

Controlling the dependence degree of a recursively enumerable vector space

1978 ◽  
Vol 43 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Vector Space ◽  
Topological Structure ◽  
Independent Set ◽  
Original Data ◽  
Recursion Theory ◽  
Vector Spaces ◽  
Additional Structure ◽  
Recursively Enumerable ◽  
Added Benefit ◽  
And Algebra

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

scholarly journals On the topological structure of affinely connected manifolds

1984 ◽  
Vol 96 ◽  
pp. 41-60
Author(s):  
Tatsuo Higa
Keyword(s):  
Riemannian Manifold ◽  
Vector Space ◽  
Betti Number ◽  
Topological Structure ◽  
Geometric Objects ◽  
The Relationship

The purpose of the present paper is to investigate the relationship between the topological structure and differential geometric objects for affinely connected manifolds.Let M be a compact, connected and oriented Riemannian manifold, Pr(M) the vector space of all parallel r-forms on M and br(M) the r-th Betti number of M.

scholarly journals Evaluation of Vector Transformations for Russian Word2Vec and FastText Embeddings

2020 ◽  
pp. paper18-1-paper18-12
Author(s):  
Olga Korogodina ◽  
Olesya Karpik ◽  
Eduard Klyshinsky
Keyword(s):  
Vector Space ◽  
Word Pair ◽  
Proper Names ◽  
Semantic Space ◽  
Russian Language ◽  
Different Types ◽  
Analogy Problem ◽  
The Russian Language

Authors of Word2Vec claimed that their technology could solve the word analogy problem using the vector transformation in the introduced vector space. However, the practice demonstrates that it is not always true. In this paper, we investigate several Word2Vec and FastText model trained for the Russian language and find out reasons of such inconsistency. We found out that different types of words are demonstrating different behavior in the semantic space. FastText vectors are tending to find phonological analogies, while Word2Vec vectors are better in finding relations in geographical proper names. However, we found out that just four out of fifteen selected domains are demonstrating accuracy more that 0.8. We also draw a conclusion that in a common case, the task of word analogies could not be solved using a random word pair taken from two investigated categories. Our experiments have demonstrated that in some cases the length of the vectors could differ more than twice. Calculation of an average vector leads to a better solution here since it closer to more vectors.

Amplitude Spectra of Fitness Landscapes

1998 ◽  
Vol 01 (01) ◽  
pp. 39-66 ◽  
Author(s):  
Wim Hordijk ◽  
Peter F. Stadler
Keyword(s):  
Fourier Transform ◽  
Topological Structure ◽  
Empirical Studies ◽  
Amplitude Spectrum ◽  
Fitness Landscapes ◽  
Configuration Spaces ◽  
Amplitude Spectra ◽  
Different Types ◽  
Reliable Procedure ◽  
The Fourier Transform

Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying configuration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used for classification and comparison purposes. We consider here three very different types of landscapes using both mutation and recombination to define the topological structure of the configuration spaces. A reliable procedure for estimating the amplitude spectra is presented. The method is based on certain correlation functions that are easily obtained from empirical studies of the landscapes.

scholarly journals Fixed Points of Contractive Mappings inb-Metric-Like Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Nawab Hussain ◽  
Jamal Rezaei Roshan ◽  
Vahid Parvaneh ◽  
Zoran Kadelburg
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Periodic Point ◽  
Topological Structure ◽  
Fundamental Lemma ◽  
Contractive Mappings ◽  
Different Types ◽  
Convergence Of Sequences

We discuss topological structure ofb-metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences. As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results inb-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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