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Applied Sciences
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Real, Complex and Hypercomplex Number Systems in Data Processing and...
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Real, Complex and Hypercomplex Number Systems in Data Processing and Representation

A special issue of Applied Sciences (ISSN 2076-3417). This special issue belongs to the section "Computing and Artificial Intelligence".

Deadline for manuscript submissions: closed (20 January 2023) | Viewed by 12761

Special Issue Editors

Prof. Dr. Aleksandr Cariow

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Guest Editor
Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin Żołnierska 52, 71-210 Szczecin, Poland
Interests: algorithms; numbers; computations; parallel processing; embedded systems; parallel and distributed computing; signal, image, and video processing; cryptography; deep learning
Prof. Dr. Oleg Finko

E-Mail Website
Guest Editor
Institute of Mathematics and Information Technologies (named after Prof. Nikolay Chervyakov), North-Caucasus Federal University, Postal address: 1, Pushkin Street, 355017 Stavropol, Russia
Interests: data representation; IT security; polynomials; computer arithmetic; logic, data protection; coding, information security; data security; information privacy; data encryption; cryptography; security; computer security; computer algebra

Special Issue Information

Dear Colleagues,

The permanent development of the theory and practice of data processing, as well as the need to solve increasingly complex problems of computational intelligence, inspire the use of complex and advanced mathematical methods and formalisms for representing and processing big data sets. Various number systems are used in the synthesis, description, and implementation of advanced algorithms and structures of specialized processing units. This Special Issue is aimed at a wide coverage of various aspects of data processing and presentation using both familiar number systems and non-traditional ones, including exotic ones.

First of all, this concerns the use of real (rational, integer, natural), complex (elliptic, parabolic, hyperbolic), and hypercomplex (quaternions, octonions, sedenions, etc.) numbers in the implementation of computations in intelligent systems.

Secondly, issues related to the use of, among others, the following number systems are welcome:

  • Residue number system;
  • Multiple-base number systems;
  • Logarithmic number system;
  • Redundant binary number system;
  • Decimal floating point number system;
  • Fuzzy number system.

Topics include but are not limited to:

  • Numbers in data processing and representation;
  • Numerical algorithms for digital signal and image processing using various number systems;
  • Various number systems in pattern recognition and representation;
  • Circuits and systems for processing 1D, 2D, and 3D arrays, using data processing algorithms in various number systems;
  • Real-, complex- and hypercomplex-valued deep learning techniques.

We invite researchers and investigators to contribute their original research or review articles to this Special Issue.

Prof. Dr. Aleksandr Cariow
Prof. Dr. Oleg Finko
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Applied Sciences is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (7 papers)

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Editorial

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2 pages, 156 KiB  
Editorial
Special Issue: Real, Complex and Hypercomplex Number Systems in Data Processing and Representation
by Aleksandr Cariow and Oleg Finko
Appl. Sci. 2023, 13(11), 6563; https://doi.org/10.3390/app13116563 - 28 May 2023
Viewed by 814
Abstract
The evolution of human society is inevitably associated with the widespread development of computer technologies and methods, and the constant evolution of the theory and practice of data processing, as well as the need to solve increasingly complex problems in computational intelligence, have [...] Read more.
The evolution of human society is inevitably associated with the widespread development of computer technologies and methods, and the constant evolution of the theory and practice of data processing, as well as the need to solve increasingly complex problems in computational intelligence, have inspired the use of complex and advanced mathematical methods and formalisms for representing and processing big data sets [...] Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)

Research

Jump to: Editorial

12 pages, 1316 KiB  
Article
The Lossless Adaptive Binomial Data Compression Method
by Oleksiy Borysenko, Svitlana Matsenko, Toms Salgals, Sandis Spolitis and Vjaceslavs Bobrovs
Appl. Sci. 2022, 12(19), 9676; https://doi.org/10.3390/app12199676 - 26 Sep 2022
Cited by 2 | Viewed by 1605
Abstract
In this paper, we propose a new method for the binomial adaptive compression of binary sequences of finite length without loss of information. The advantage of the proposed binomial adaptive compression method compared with the binomial compression method previously developed by the authors [...] Read more.
In this paper, we propose a new method for the binomial adaptive compression of binary sequences of finite length without loss of information. The advantage of the proposed binomial adaptive compression method compared with the binomial compression method previously developed by the authors is an increase in the compression rate. This speed is accompanied in the method by the appearance of a new quality—noise immunity of compression. The novelty of the proposed method, which makes it possible to achieve these positive results, is manifested in the adaptation of the compression ratio of compressible sequences to the required time, which is carried out by dividing the initial set of binary sequences into compressible and incompressible sequences. The method is based on the theorem proved by the authors on the decomposition of a stationary Bernoulli source of information into the combinatorial and probabilistic source. The last of them is the source of the number of units. It acquires an entropy close to zero and practically does not affect the compression ratio at considerable lengths of binary sequences. Therefore, for the proposed compression method, a combinatorial source generating equiprobable sequences is paramount since it does not require a set of statistical data and is implemented by numerical coding methods. As one of these methods, we choose a technique that uses binomial numbers based on the developed binomial number system. The corresponding compression procedure consists of three steps. The first is the transformation of the compressible sequence into an equilibrium combination, the second is its transformation into a binomial number, and the third is the transformation of a binomial number into a binary number. The restoration of the compressed sequence occurs in reverse order. In terms of the degree of compression and universalization, the method is similar to statistical methods of compression. The proposed method is convenient for hardware implementation using noise-immune binomial circuits. It also enables a potential opportunity to build effective systems for protecting information from unauthorized access. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
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21 pages, 1238 KiB  
Article
On Hyperbolic Complex Numbers
by Wolf-Dieter Richter
Appl. Sci. 2022, 12(12), 5844; https://doi.org/10.3390/app12125844 - 08 Jun 2022
Cited by 5 | Viewed by 1668
Abstract
For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify [...] Read more.
For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
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22 pages, 2722 KiB  
Article
On the Octonion Cross Wigner Distribution of 3-D Signals
by Łukasz Błaszczyk and Kajetana Snopek
Appl. Sci. 2022, 12(11), 5358; https://doi.org/10.3390/app12115358 - 25 May 2022
Cited by 1 | Viewed by 1323
Abstract
This paper introduces definitions of the octonion cross Wigner distribution (OWD) and the octonion ambiguity function, forming a pair of octonion Fourier transforms. The main part is devoted to the study of the basic properties of the OWD. Among them are the properties [...] Read more.
This paper introduces definitions of the octonion cross Wigner distribution (OWD) and the octonion ambiguity function, forming a pair of octonion Fourier transforms. The main part is devoted to the study of the basic properties of the OWD. Among them are the properties concerning its nature (nonlinearity, parity, space support conservation, marginals) and some “geometric” transformations (space shift, space scaling) similar to the case of the complex Wigner distribution. This paper also presents specific forms of the modulation property and an extended discussion about the validity of Moyal’s formula and the uncertainty principle, accompanied by new theorems and examples. The paper is illustrated with examples of 3-D separable Gaussian and Gabor signals. The concept of the application of the OWD for the analysis of multidimensional analytic signals is also proposed. The theoretical results presented in the papers are summarized, and the possibility of further research is discussed. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
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21 pages, 671 KiB  
Article
Some FFT Algorithms for Small-Length Real-Valued Sequences
by Dorota Majorkowska-Mech and Aleksandr Cariow
Appl. Sci. 2022, 12(9), 4700; https://doi.org/10.3390/app12094700 - 07 May 2022
Cited by 3 | Viewed by 1798
Abstract
This paper proposes fast algorithms for computing the discrete Fourier transform for real-valued sequences of lengths from 3 to 9. Since calculating the real-valued DFT using the complex-valued FFT is redundant regarding the number of needed operations, the developed algorithms do not operate [...] Read more.
This paper proposes fast algorithms for computing the discrete Fourier transform for real-valued sequences of lengths from 3 to 9. Since calculating the real-valued DFT using the complex-valued FFT is redundant regarding the number of needed operations, the developed algorithms do not operate on complex numbers. The algorithms are described in matrix–vector notation and their data flow diagrams are shown. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
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20 pages, 341 KiB  
Article
Error-Correction Coding Using Polynomial Residue Number System
by Igor Anatolyevich Kalmykov, Vladimir Petrovich Pashintsev, Kamil Talyatovich Tyncherov, Aleksandr Anatolyevich Olenev and Nikita Konstantinovich Chistousov
Appl. Sci. 2022, 12(7), 3365; https://doi.org/10.3390/app12073365 - 25 Mar 2022
Cited by 7 | Viewed by 2048
Abstract
There has been a tendency to use the theory of finite Galois fields, or GF(2n), in cryptographic ciphers (AES, Kuznyechik) and digital signal processing (DSP) systems. It is advisable to use modular codes of the polynomial residue number system (PRNS). Modular codes of [...] Read more.
There has been a tendency to use the theory of finite Galois fields, or GF(2n), in cryptographic ciphers (AES, Kuznyechik) and digital signal processing (DSP) systems. It is advisable to use modular codes of the polynomial residue number system (PRNS). Modular codes of PRNS are arithmetic codes in which addition, subtraction and multiplication operations are performed in parallel on the bases of the code, which are irreducible polynomials. In this case, the operands are small-bit residues. However, the independence of calculations on the bases of the code and the lack of data exchange between the residues can serve as the basis for constructing codes of PRNS capable of detecting and correcting errors that occur during calculations. The article will consider the principles of constructing redundant codes of the polynomial residue number system. The results of the study of codes of PRNS with minimal redundancy are presented. It is shown that these codes are only able to detect an error in the code combination of PRNS. It is proposed to use two control bases, the use of which allows us to correct an error in any residue of the code combination, in order to increase the error-correction abilities of the code of the polynomial residue number system. Therefore, the development of an algorithm for detecting and correcting errors in the code of the polynomial residue number system, which allows for performing this procedure based on modular operations that are effectively implemented in codes of PRNS, is an urgent task. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
11 pages, 258 KiB  
Article
Binomial Number System
by Oleksiy Borysenko, Svitlana Matsenko and Vjaceslavs Bobrovs
Appl. Sci. 2021, 11(23), 11110; https://doi.org/10.3390/app112311110 - 23 Nov 2021
Cited by 5 | Viewed by 1704
Abstract
This paper presents and first scientifically substantiates the generalized theory of binomial number systems (BNS) and the method of their formation for reliable digital signal processing (DSP), transmission, and data storage. The method is obtained based on the general theory of positional number [...] Read more.
This paper presents and first scientifically substantiates the generalized theory of binomial number systems (BNS) and the method of their formation for reliable digital signal processing (DSP), transmission, and data storage. The method is obtained based on the general theory of positional number systems (PNS) with conditions and number functions for converting BNS with a binary alphabet, also allowing to generate matrix BNS, linear-cyclic, and multivalued number systems. Generated by BNS, binomial numbers possess the error detection property. A characteristic property of binomial numbers is the ability, on their basis, to form various combinatorial configurations based on the binomial coefficients, e.g., compositions or constant-weight (CW) codes. The theory of positional binary BNS construction and generation of binary binomial numbers are proposed. The basic properties and possible areas of application of BNS researched, particularly for the formation and numbering of combinatorial objects, are indicated. The CW binomial code is designed based on binary binomial numbers with variable code lengths. BNS is efficiently used to develop error detection digital devices and has the property of compressing information. Full article
(This article belongs to the Special Issue Real, Complex and Hypercomplex Number Systems in Data Processing and Representation)
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