Editor(s)

Dr. Xingting Wang
Assistant Professor,
Department of Mathematics, Howard University, Washington, USA.

ISBN 978-93-91473-89-1 (Print)
ISBN 978-93-91473-06-8 (eBook)
DOI: 10.9734/bpi/ctmcs/v6

This book covers key areas of mathematics and computer science. The contributions by the authors include dust diffusion, atmospheric diffusion equation, longitudinal diffusion, wind speed, unsteady state, mathematics of computing, mathematical analysis, numerical analysis, computations on matrices, bulk service, working vacation, infinitesimal matrix, direct truncation method, cayley Hamilton, complex variable theory, rotlet, stokeslet, stresslet, stokes flows, entropy, fuzzy entropy, codeword length, decipherable code, crisp set, holder’s inequality, cloud computing, cloudsim, CAT(0) space, nonexpansive multivalued mapping, strong convergence, one?step, iterative scheme, round function, multi-term fractional differential equations, Riemann-Liouville fractional integral, Caputo derivative, Adomian polynomials, vectors, matrices, vector differential operators, vector algebraic operators, matrix multiplication, coordinate transformation, autoregressive process, laplace distribution, moving average process, random coefficient models. This book contains various materials suitable for students, researchers and academicians in the field of mathematics and computer science.

 

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Chapters


The mathematical model for the diffusion of dust particles emitted from a fixed source in the presence of the longitudinal diffusion and absence of latitudinal and vertical diffusions, is investigated. The diffusion of dust particles in the atmosphere is governed by the atmospheric diffusion equation. In the previous paper [1], the general case of the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time, was solved. The calculations showed that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. In the previous paper, we examined the model in the presence of vertical diffusion and absence of other diffusions to show that for small times, the dust spreads with a front that travels with the speed of the wind. In this study, we shall assume that the flow is a non-zero constant.  In the current paper, the vertical and latitudinal diffusions are absent while the longitudinal diffusion is present. It is found that the solution depends on the source of time dependence. To study the nature of the solution, two special cases of the source are specified. In both cases, it is found that there is no discontinuity front, and the dust particles spread slowly into the direction of the wind.

An Algorithm for Generating N-Dimensional Rotation Matrix

Ognyan Ivanov Zhelezov

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 15-28
https://doi.org/10.9734/bpi/ctmcs/v6/3301F

This paper describes a new algorithm for generating an N-dimensional rotation matrix M that rotates a given N-dimensional vector X in the direction of a given N-dimensional vector Y of the same dimension. The N-dimensional Rotation Matrix Generation Algorithm (NRMG) uses two-dimensional rotations to rotate provided vectors X and Y in the direction of coordinate axis x1. Matrix M is created by multiplying matrix MX by the inverse of matrix MY, which rotates the given vectors in the direction of axis x1. RMG algorithm does not determine how the Mx and My matrices are calculated. An algorithm for their calculation using rotations in the coordinate planes is proposed, but they can also be obtained by the Householder transformation, which will be more efficient for "dense" vectors. The prospect of performing parallel calculations of two-dimensional rotations is also investigated.

Determination of a Special Case of Symmetric Matrices and Their Applications

Ognyan Ivanov Zhelezov

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 29-45
https://doi.org/10.9734/bpi/ctmcs/v6/3302F

This article presents a special case of symmetric matrices, matrices of transpositions (Tr matrices) that are created from the elements of given n-dimensional vector XÎRn, n=2m, mÎN. Has been proved that Tr matrices are symmetric and persymmetric. Has been proposed algorithm for obtaining matrices of transpositions with mutually orthogonal rows (Trs matrices) of dimensions 2, 4, and 8 as Hadamard product of Tr matrix and matrix of Hadamard and has been investigated their application for QR decomposition and n-dimensional rotation matrix generation. Tests and analysis of the algorithm show that obtaining an orthogonal Trs matrix of sizes 4 and 8 that rotates a given vector to the direction of one of the coordinate axes requires less processing time than obtaining a Housholder matrix of the same size. This makes the Tr and Trs matrices useful in matrix calculations.

A Computational Approach for Evaluating the Transient Behaviour of M/M (a, b)/1 Bulk Service Queueing System with Working Vacation

S. Shanthi, A. Muthu Ganapathi Subramanian, Gopal Sekar

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 46-61
https://doi.org/10.9734/bpi/ctmcs/v6/3211F

A new computational technique is used to evaluate the Transient behaviour of Single Server Bulk Service Queueing System with Working Vacation with arrival rate \(\lambda\) which follows a Poisson process and the service will be in bulk. In this model the server provides two types of services namely normal service and lower service. The normal service time follows an exponential distribution with parameter \(\mu\)1. The lower service rate follows an exponential distribution with parameter \(\mu\)2. The vacation time follows an exponential distribution with parameter \(\alpha\). According to Neuts, the server begins service only when a minimum of ‘a’ customers in the waiting room and a maximum service capacity is ‘b’. An infinitesimal generator matrix is formed for all transitions. Time dependent solutions and Steady state solutions are acquired by using Cayley Hamilton theorem. Numerical studies have been done for Time dependent average number of customers in the queue, Transient probabilities of server in vacation and server busy for several values of t, \(\lambda\), µ1, \(\mu\)2, \(\alpha\), a and b. In this model we have provided transient probability distribution of number of customers in the queue at time t and also time dependent system measures.

Study on Singularity Induced Interior Stokes Flows

N. Akhtar, G. A. H. Chowdhury

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 62-80
https://doi.org/10.9734/bpi/ctmcs/v6/3305F

Three complex variable circle theorems for studying the two-dimensional Stokes flows interior to a circular cylinder are presented. These theorems are formulated in terms of the complex velocities of the fundamental singularities in an unbounded incompressible viscous fluid. Illustrative examples are given to demonstrate their usefulness.

A Study on the Generalized Fuzzy Mean Codeword Lengths

D. S. Hooda, Divya Jain

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 81-95
https://doi.org/10.9734/bpi/ctmcs/v6/3167F

Information Theory has ideas which are widely applicable to the situations remote from its original inspiration. Although, the applicability of ideas is not always exact; yet these are very useful. One of the best applications of information measure is noiseless coding theorem which provides the bounds for suitable encoding of entropies and fuzzy information measures.

In present chapter, mean code word and fuzzy mean codeword lengths are defined, and some generalizations of mean codeword length are also described. A generalized fuzzy mean codeword length of degree \(\beta\) is defined and its bounds in the term of a generalized fuzzy information measure are studied. Further, the fuzzy mean codeword length of type (\(\alpha\), \(\beta\)) is introduced and its bounds are studied. Monotonic behavior of these fuzzy mean codeword lengths is illustrated graphically by taking some empirical data.

Study on Proposed Resource Allocation Management for Cloud Computing Using Tabu Search Algorithm

Md Imran Alam, Manjusha Pandey, Siddharth S. Rautaray

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 96-103
https://doi.org/10.9734/bpi/ctmcs/v6/11081D

Cloud computing is an emerging technology that provides computing resources depending on user demand. Some policies can be used to schedule this demand or supply of resources. The management of resource allocation should supply resources in a shorter time and at a lower cost. The services provided by cloud computing are software as service, platform as service and infrastructure as service. A dynamic optimised resource allocation management algorithm is designed in this paper based on three factors: optimum solution, deadline constraint, and cost constraint. The algorithm employs the Tabu Search Algorithm, which is then followed by prioritisation and task grouping.

The study of metric spaces without linear structure has played a vital role in various branches of pure and applied sciences. In this paper, we introduced one-step iterative process for approximating of common fixed points of two multivalued nonexpansive mappings in CAT (0) spaces and established strong convergence theroems for proposed process under some conditions. Our results extend important results.

Determination of Range of Outputs Precise of Digits Rounding in SPSS and MS Excel

Zaher Saif

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 111-119
https://doi.org/10.9734/bpi/ctmcs/v6/3347F

The concept of this research is calculates rounding and truncate in SPSS and round function in MS Excel for 10 random digits numbers and comparing the results in the two programs with the same digits numbers. Compare the rounding outputs in both programs in four decimal places. The statistical operations done by many specialist programs, by them can do these operations fluently and precisely. There are many functions in such programs can calculate in SAS, STAT and the analytical program SPSS. There is Microsoft Excel program that is calculate like these functions. The level of some programs may be different than others within these functions they are calculating. From these functions are Sum, Average, Maximum and Minimum. Round function is also from these functions that can measure it's accuracy through this research. In this research i chose ten digits numbers and I also chose three criteria under, equal and more than 5. According to that the rounding operations are done based on if wanted decimal place is less, equal or higher than 5. Rounding applied on ten digits using SPSS and MS Excel programs. The outputs findings are the same except that Microsoft Excel is truncating the last zeroes of the digit after the decimal point. Wherever the decimal place specified in the digit is want it will truncate after the decimal point. The SPSS is more precise than MS Excel based on the decimal place in the digit number wanted statistically and analytically.

One Solution of Multi-term Fractional Differential Equations by Adomian Decomposition Method: Scientific Explanation

Abdollah Sadeghinia, Prabhat Kumar

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 120-130
https://doi.org/10.9734/bpi/ctmcs/v6/11542D

The Adomian decomposition method (ADM) is applied to solve the of nonlinear multi-term fractional differential equations of the Form \(D_{*}^\alpha\)Y(X) = \(\textstyle \sum_{i=0}^n\)\(a_i\)(x)\(\textstyle D_{*}^{\beta_i}\) Y(X) + \(a_0\) (x)y(x) + N ((x, y(x), \(D_{*}^{\beta_1}\)Y(X), ..., \(D_{*}^{\beta_n}\)Y(X)) + g (x) under the initial conditions \(y^{(i)}(0)=c_i\) (0 \(\le\) i \(\le\) m -1) where N is nonlinear function of x,y(x), \(D_{*}^{\beta_1}\)Y(X), ... ..., \(D_{*}^{\beta_n}\)Y(X) and g(x) and ai(x) are functions of x. Also \(\alpha\) > \(\beta_n\)> …… \(\beta_1\) > 0, (m -1 <\(\alpha\) \(\le m,\ and \ m\in N)\) . Some examples of the solution are also presented for better comprehension.

Study on Vector Operations Transform into Matrix Operations

Feng Cheng Chang

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 131-149
https://doi.org/10.9734/bpi/ctmcs/v6/3452F

In this paper a useful technique, "Vector operation Transforms into Matrix operation" (VTM), is developed to simplify the manipulation of vector algebraic and differential operations. An efficient technique is developed to simplify the computations in the field of vector analysis. The evaluation of vector algebraic and differential operations becomes more simple and straightforward by simply transforming the vector operations into matrix operations. The matrix operations are especially useful when there are mixed coordinate basis involved in the vector operations.  

Study on Some Theorems of Random Coefficient Models with Laplace Marginals

Bindu Krishnan

Current Topics on Mathematics and Computer Science Vol. 6, 21 July 2021, Page 150-159
https://doi.org/10.9734/bpi/ctmcs/v6/3423F

In this article, a first order random coefficient autoregressive model with Laplace distribution as marginal is developed. A random coefficient moving average model of order one with Laplace as marginal distribution is introduced and its properties are studied. By combining the two models, a first order random coefficient autoregressive moving average model with Laplace marginal is developed and discussed its properties. Various theorems based on the new developed models are shown. The simulated sample path is generated from first order autoregressive Laplace process from a set of observations. A first order random coefficient moving average process with generalized Laplace innovations is also obtained.