Editor(s)
Dr. Manuel Alberto M. Ferreira
Professor, Department of Mathematics, ISTA-School of Technology and Architecture,  Lisbon University,  Portugal.

ISBN 978-93-5547-727-9 (Print)
ISBN 978-93-5547-728-6 (eBook)
DOI: 10.9734/bpi/nramcs/v4

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Chapters


On Integrability of Mathematical Physics Equations

L. I. Petrova

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 1-7
https://doi.org/10.9734/bpi/nramcs/v4/2161A

A study of differential equations has shown that the integrability of mathematical physics equations depends on the consistency of the derivatives of described functions. The study of the consistency of derivatives and equations that form the equations of mathematical physics showed that on the original coordinate space the differential equation turns out to be non-integrable. That is, the solution is not a function. However, if there are any degrees of freedom, then integrable structures, on which the derivatives of a differential equation form a differential, are realized. That is, discrete functions are the solution to the equations of mathematical physics on integrable structures. This indicates to the differential equation integrability. Double solutions of the equations of mathematical physics describes the emergence of various formations such as waves, vortex,  and so on.  

Performance Optimization on GPGPU & Multicore CPU using Roofline Model: A Recent Study

Noor Mowafeq Al layla, Shefa A. Dawwd

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 8-16
https://doi.org/10.9734/bpi/nramcs/v4/16040D

In this chapter, the roofline model is used to determine the optimum optimized platform for training a neural network that recognizes handwritten digits in a multicore CPU and general purpose GPU (GPGPU) hardware environment. For the MNIST dataset, the pattern parallel training technique is used. The training of MNIST's parallel network utilizing several data layouts on multicore CPU and GPGPU is demonstrated. The roofline model has been used to explain several bottlenecks.  As this roofline model is so simple, it can be implemented quickly. The best platform is chosen based on layouts and constraints, such as memory or compute limits. All rooflines' computational intensity is shifted to the right, and subsequently performance is improved. The most appropriate hardware platform is chosen as a result of optimization and the diversity of available data size, core number, and operational strength.

This article investigates whether there is a connection between the square root of ten, which is one of the irrational numbers, and genetic sequences. First, the square root digits of the number ten after the comma are subjected to an individual addition operation. Secondly, the result of the addition corresponds to the nucleotide bases. Thirdly, the results obtained in this way are expressed as nucleotide bases (A, T, C and G) [(A) Adenine, (T) Thymine, (C) Cytosine and (G) Guanine]. From this point of view, approximately when the first four hundred digits of the square root of the number ten after the comma are calculated [1], the resulting gene sequencing is as follows: [ATAAGTCATAAGTGTATTAGTTTAAAACTG]. Fourthly, at the same time some repetitions exactly like this were detected.: as”AGT” and “ATA”. Fifthly, after searching this sequence in NCBI (National Biotechnology Information Center), the search result was similar to bony fish, especially Danio Aesculapii [2]. Finally, the species Danio Aesculapii [3] is closely related to the Zebrafish. In summary, With these results, not only the square root of ten in mathematics, but many more irrational numbers (as explained by the similar QUANTUM PERSPECTIVE MODEL in previous articles; See Table-2) to give a common perspective on these different sciences; the link between genetic codes in biochemistry and irrational numbers in mathematics has shown to be significant and has revealed very valuable results. In other words, with this original research, a new window has been opened that can lead to new interdisciplinary discoveries.

Objectives: This investigation present an analytical study on a two-dimensional steady hydro-magnetic natural convective couette flow with mass transfer of a viscous incompressible and electrically conducting Newtonian fluid between two infinite vertical parallel plates.  A magnetic field of uniform strength is assumed to be applied transversely to the direction of the flow taking into account the induced magnetic field. The plates are subjected to constant suction and injection.

Methods: The non-linear coupled differential equations are solved analytically by regular perturbation technique with Eckert number Ec (<1) as perturbation parameter. The expressions for velocity, temperature, concentration fields and the coefficients of skin frictions at the walls, the rate of heat transfer (in terms of Nusselt numbers) from the walls to the fluid, the rate of mass transfer (in terms of Sherwood numbers) at the walls are obtained in non-dimensional form and the effects of different physical parameters involved in the problem on these fields are discussed graphically and the results are interpreted physically.

Findings: It is observed that concentration decreases for chemical reaction and thermal diffusion whereas it is increased due to high rate of molar diffusivity and low viscosity. Also fluid temperature falls due to thermal diffusion but it rises for chemical reaction.

Applications: The problem setup and subsequent findings are relevant to applications in chemical engineering and manufacturing industries.

Twin Edge Coloring of Some Path and Cycle Related Graphs: A Recent Study

J. Naveen

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 57-75
https://doi.org/10.9734/bpi/nramcs/v4/1649A

A twin edge K-coloring of a graph  is a proper edge K-coloring  of G with the elements of Zk so that the induced vertexK-coloring, in which the color of a vertex V inis the sum in Zof the colors of the edges incident with v is a proper vertexK-coloring. The minimum K for which  G  has a twin edge K-coloring  is called the twin chromatic index of G .Twin chromatic index of the splitting graph of path and cycle, middle graph of path and cycle and shadow graph of path and cycle are determined. Twin chromatic index  of Cm X Pis also determined, where Xdenotes the direct product of Cand Pr are, respectively, the cycle and the path on r vertices each.

Tripolar Fuzzy Weak Bi-ideals of a Near Ring: A Recent Study

Rakshita Deshmukh, P. Narasimha Swamy, T. Srinivas

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 76-81
https://doi.org/10.9734/bpi/nramcs/v4/16235D

We want to introduce the concept of a tripolar fuzzy weak bi-ideal of a near ring and look at some of its aspects in this work. The tripolar fuzzy set is very useful in discriminating relevant elements, irrelevant elements and contrary elements.

Tripolar Fuzzy Ideals of a Near Algebra

P. Narasimha Swamy, Rakshita Deshmukh , M. Murali Krishna Rao, B. Jyothi

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 82-88
https://doi.org/10.9734/bpi/nramcs/v4/2668B

We present the idea of tripolar fuzzy ideals of a near algebra based on the theory of tripolar fuzzy sets. It is a generalisation of fuzzy set, bipolar fuzzy set, intuitionistic fuzzy set, fuzzy ideals. We also discuss some of its properties.

J-Closed Functions via J-Closed Sets in Topological Spaces

P. L. Meenakshi, R. Sudha

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 89-98
https://doi.org/10.9734/bpi/nramcs/v4/6372F

In this article, J-closed functions using the concept of J-closed sets and J-open functions using the notion of J-open sets are initiated. The interrelationships of these newly introduced functions with various existing functions are examined and its properties are analysed.The composition of two J-closed function need not be a J-closed function which is proved by Counter Example.The J-closed functions are used to define homeomorphisms using J-closed sets in topological spaces.Homeomorphisms are the isomorphisms in the category of topological spaces-that is, they are the mappings that preserve all the topological properties of a given space.

The study of J-closed functions has been done by the following methods:

  • Analytical method of comparing J-closed functions with other existing closed functions.
  • Obtaining counter examples wherever necessary to substantiate the result.
  • Interpreting the results as diagrams.
  • Analysis of preservation of topological properties by J-closed functions.

Wedderburn Decomposition and Structure of a Semi-Simple Dihedral Group Algebra

Yordan Epitropov, Ivanka Gradeva

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 99-107
https://doi.org/10.9734/bpi/nramcs/v4/15764D

Let \(K\) be an arbitrary field, whose characteristic does not divide the order of the dihedral group \(D_{2 m}\) of order \(2 m\), where \(m\) is odd, and \(K D_{2 m}\) be the group algebra of \(D_{2 m}\) over the field \(K\). The structure of the semisimple dihedral group algebra \(K D_{2 m}\) is examined in this work. We find a complete system of minimal central orthogonal idempotents of the group algebra for this purpose. We define the simple components of \(K D_{2 m}\) and its Wedderburn decomposition through it. The results are as general as possible, i.e. they do not require a finite field.

Methods for Increasing Accuracy in the Process of Information Exchange and Processing

A. M. Mehdiyeva, I. Z. Sardarova, S. V. Quliyeva

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 108-122
https://doi.org/10.9734/bpi/nramcs/v4/6408F

In the research work discusses the problem of eliminating errors that arise at the processing stage, taking into account the growing need for digital signal processing of measurement results in the oil and gas industry. At the research work addresses the problem of eliminating errors that occur during the processing stage, taking into account the increased demand for digital signal processing of measurement results in the oil and gas industry. We investigated the problems associated with corrective filtering and discrete averaging in digital signal processing. The proposed approach to the creation of information-measuring systems of the parameters under consideration consists of a cumulative analysis of measurement processes and corrective filtration with the aim of achieving balanced metrological, structural, algorithmic and functional indicators of the effectiveness of the developed tools. Thus, the proposed approach to improve the accuracy of measurement objectives is a comprehensive analysis of measurement processes and a corrective filter to provide a balanced assessment of the effectiveness, structural and functional efficiency of the developed algorithmic tools. In the research work discusses the problem of eliminating errors that arise at the processing stage, taking into account the growing need for digital signal processing of measurement results in the oil and gas industry. And investigated the problems with equalizing filtering and discrete averaging in digital signal processing. Therefore, it can be said that the numerical mean operator applied for both types of error correction is intended for any error. This has been proven by modeling in program Matlab. A mathematical model is proposed for evaluating the noise immunity characteristics of a coherent modem reception in a message transmission system that functions under the influence of unintended interference sources. The developed mathematical model takes into account the demodulator synthesis algorithm, decoding methods, effective M-PSK modulations and Reed-Solomon codes in the modem. Integrated expressions are obtained that evaluate the noise immunity characteristics of a modem receive, taking into account the energy performance of the receiver.

The objective of this chapter is to introduce two new kinds of fuzzy spaces, namely \(T_{1 / 2}^{*}\)-fuzzy space and \(T_{1 / 2}^{* *}\) fuzzy space in fuzzy biclosure spaces. The concept of \(\partial\)-fuzzy closed sets has been used to define these two types of spaces. The study the various properties of these fuzzy spaces in fuzzy biclosure spaces has also been done.

The ILL-Posedness of Derivative and Regularized Interpolation for Non-band limited Functions

Weidong Chen

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 131-143
https://doi.org/10.9734/bpi/nramcs/v4/6158F

In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The regularized derivative interpolation's convergence is investigated. The numerical results are presented, and the Tikhonov regularization method is used to compare them to derivative interpolation. In this study, the regularized derivative interpolation is more accurate in computing. The regularized derivative interpolation by sampling can be applied. The convergence property is proved and tested by some examples. The numerical results are better than Tikhonov regularization method.

Design of an Algorithm Regarding Analytic Hierarchy Process

Miron Pavlus, Rostislav Tomeš

Novel Research Aspects in Mathematical and Computer Science Vol. 4, 28 May 2022, Page 144-158
https://doi.org/10.9734/bpi/nramcs/v4/2403B

People make three general types of judgments to express importance, preference, or likelihood and use them to choose the best among alternatives in the presence of environmental, social, political, and other influences. They base these judgments on knowledge in memory or from analyzing benefits, costs, and risks. From past knowledge, we sometimes can develop standards of excellence and poorness and use them to rate the alternatives one at a time. This is useful in such repetitive situations as student admissions and salary raises that must conform with established norms. Without norms one compares alternatives instead of rating them. Comparisons must fall in an admissible range of consistency. The analytic hierarchy process (AHP) includes both the rating and comparison methods. Rationality requires developing a reliable hierarchic structure or feedback network that includes criteria of various types of influence, stakeholders, and decision alternatives to determine the best choice [8].

for well-known statement that the maximal eigenvalue \(\lambda_{\max }\) is equal to \(n\) for the eigenvector problem \(\mathrm{A} w=\lambda w\) where \(A\) is, so called, the consistent matrix of pairwise comparisons of type \(\mathrm{n} \times \mathrm{n}(\mathrm{n} \geq 2)\) with the solution vector \(w\) that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution \(A w=\mathrm{n} w\) as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix \(A\) can be simply programmed and used.