Editor(s)
Dr. Luigi Giacomo Rodino
Professor,
Department of Mathematics, University of Turin, Italy.

ISBN 978-93-5547-324-0 (Print)
ISBN 978-93-5547-325-7 (eBook)
DOI: 10.9734/bpi/ramrcs/v5

This book covers key areas of Mathematical Research and Computer Science. The contributions by the authors include Bayes’ theorem, multiplication rule, probabilities, Integrodifferential equations, nonlinear evolution equation, nonlocal initial condition, fixed point theorem for a sum of operators, pseudoblocks, modules, endomorphism algebra, Marker set of a graph, M-set distance matrix, M-set distance Laplacian, characteristic polynomial, M-set distance eigenvalues, univalent functions, Bernardi operator, fractional derivative, linear operator, q-derivative, Measure of entropy, measure of cross-entropy, primal problem, dual problem, continuous variate distribution, concave function, convex function, discrete variate probability distribution, flow graph, matrix algebra, linear equations, cascade graph, graph order, path ordinal, path-set, quantum communications, neuronal networks models, multiple interferometry, primality testing, Riemann zeta function, factorization, platonic world, savant syndrome, distributed computer networks, information security, security analysis, Supercomputer, performance, benchmark, IBMPower8, HPC, Parallel Computing, cognitive radio, correct detection(hit), signal detection, degree of separability, Information technology, cloud computing. This book contains various materials suitable for students, researchers and academicians in the field of Mathematical Research and Computer Science.

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### Chapters

##### Problem Solving Using Bayes Theorem: A Mathematical Approach

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 1-12
https://doi.org/10.9734/bpi/ramrcs/v5/5258F

The study on understanding of Bayes’ Theorem and to use that knowledge to investigate practical problems in various professional fields. Provides a means for making probability calculations after revising probabilities when obtaining new information in an important phase of probability analysis. When given P(A) and P(A B), one can calculate P(B/A) by manipulating the information in the Multiplication Rule. However, one could not calculate P(A/B). Similarly, when given P(B) and P(A B), one can calculate P(A/B) by manipulating the information in the Multiplication Rule. There is where one can now apply Bayes’ Theorem. The derivation for Bayes’ theorem is straightforward, not everyone is the future, you are using it to make inferences about the past. People who think in terms of causality have trouble with this.

##### Study on Integrodifferential Evolution Systems with Nonlocal Initial Conditions

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 13-27
https://doi.org/10.9734/bpi/ramrcs/v5/3535F

The work is concerned with systems of abstract integrodifferential equations with general nonlocal initial conditions. To allow the nonlinear terms of the equations to behave as independently as possible, we employ a vector approach based on matrices, vector-valued norms, and a vector version of Krasnoselskii's fixed point theorem for a sum of two operators. The assumptions take into account the system's hybridity and the support for nonlocal initial conditions. To demonstrate the principle, two examples are given.

##### The Concept of Pseudoblocks of Endomorphism Algebra

Ahmed A. Khammash

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 28-32
https://doi.org/10.9734/bpi/ramrcs/v5/11635D

We introduce the concept of pseudoblocks for the endomorphism rings of modules over finite dimensional algebras. The concept is defined in terms of the homomorphism space between direct summands and the idea was originated from studying the impact of the Brauer-Fitting correspondence on block theory.

##### Describing the Properties of Characteristic Polynomial of Marker Set and Its Laplacian

Medha Itagi Huilgol

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 33-46
https://doi.org/10.9734/bpi/ramrcs/v5/14192D

The study of marker set distance matrices, associated with their eigenvalues, characteristic polynomials find a lot of applications. Also the Laplacian associated with marker set distances need to be studied in detail to get range of applications. In recent years these concepts have been studied well. In this chapter, we consider different properties related to characteristic polynomials of M-set distance matrix and its Laplacian.

##### Review on Fractional Q-calculus with Varying Arguments

N. Ravi Kumar

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 47-59
https://doi.org/10.9734/bpi/ramrcs/v5/15106D

In this Chapter we introduce a new subclasses of functions which are analytic defined by using fractional q - calculus integral operators. We also introduce q-Bernadi integral operator for analytic function using the concept of q-calculus. We discuss coefficient estimates, growth, distortion theorems and many more properties for the function f belonging to the classes V(A, B, q, $$\delta$$) and K(A, B, q, $$\delta$$) .

##### Entropy Optimization Problems for Modified Verma Measures in Primal and Dual Spaces

Rohit Kumar Verma

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 60-68
https://doi.org/10.9734/bpi/ramrcs/v5/15105D

In the present paper we formulate the dual problem for a primal problem concerned with the maximization of the modified version of Verma measure of entropy [1-2] or minimization of corresponding measure of cross-entropy. In this case, it is shown that the maximum value of that entropy (or minimum value of the corresponding cross-entropy, which is motivated by Kullback [3]-Leibler [4] is equal to the minimum (or maximum) value of the objective function of the dual problem. The recovery of the primal problem, when the dual problem is given, is also discussed.

##### Solving Higher-Order Graphs Method: A Linear Analytical Treatment

Eusebio Bernabeu, Hala Kamal, Alicia Larena

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 69-82
https://doi.org/10.9734/bpi/ramrcs/v5/14560D

Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can be enumerated and represented by ordinals. The method allows one to extract partial information and gives an alternative to classical graph approach. An analysis of higher-order graphs as potent tool to generate algorithms for SoS, IoT, quantum communications, neuronal networks models, multiple interferometric devices is also reported.

##### Mathematical Machine for Primality Testing of Numbers: An Advanced Research

Takaaki Musha

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 83-93
https://doi.org/10.9734/bpi/ramrcs/v5/15133D

Like the optical prism to break white light up into its constituent spectral colors, the machine to show a prime as a single spectrum is proposed. From the theoretical analysis, it can be shown that the machine to recognize the prime number as a single spectrum can be realized by using the Fourier transform of the correlation function of Riemann zeta function. Moreover, this method can be used for a factorization of the integer composed of two primes. Integer factorization is the decomposition of a composite number into a product of smaller integers, for which there is not known efficient algorithm. From the theoretical analysis, we can see that prime factorization for the integer composed of two different primes can be conducted within a polynomial time.

##### Confidence Analysis of a Solo Sign-On Device for Distributed Computer Networks: A Modeling Approach

C. M. Sumanth, Suresha D.

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 94-104
https://doi.org/10.9734/bpi/ramrcs/v5/1548B

Solo sign-on (SSO) is a new authentication mechanism that allows a legal user to be authenticated by numerous service providers in a distributed computer network using a single credential. A SSO technique recently suggested and claimed security by presenting well-organized security reasons. However, their technique is insecure since it violates credential privacy and authentication soundness. We describe two impersonation attacks in particular: credential recovery attacks and impersonation attacks without credentials. As a result, we present a stronger authentication technique that uses efficient verifiable encryption of RSA signatures to overcome these attacks and flaws. As one open problem, we support the formal study of the soundness of authentication.

##### Study on Dense Matrix Multiplication Algorithms and Performance Evaluation of HPCC in 81 Nodes IBM Power 8 Architecture

Eduardo Patricio Estévez Ruiz, Giovanny Eduardo Caluña Chicaiza, Fabian Rodolfo Jiménez Patiño, Joaquín Cayetano López Lago, Saravana Prakash Thirumuruganandham

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 105-125
https://doi.org/10.9734/bpi/ramrcs/v5/14371D

Optimizing HPC systems based on performance factors and bottlenecks is essential for designing an HPC infrastructure with the best characteristics and at a reasonable cost. Such insight can only be achieved through a detailed analysis of existing HPC systems and the execution of their workloads. The "Quinde I" is the only and most powerful supercomputer in Ecuador and is currently listed third on the South America. It was built with the IBM Power 8 servers. In this work, we measured its performance using different parameters from High-Performance Computing (HPC) to compare it with theoretical values and values obtained from tests on similar models. To measure its performance, we compiled and ran different benchmarks with the specific optimization flags for Power 8 to get the maximum performance with the current configuration in the hardware installed by the vendor. In computer science, benchmarking is a technique to measure the performance of a system or one of its components.  The inputs of the benchmarks were varied to analyze their impact on the system performance. In addition, we compile and compare the performance of two algorithms for dense matrix multiplication SRUMMA and DGEMM.

##### Suggesting a New Approach on Identifying Degree of Separability in Signal Detection, for Using in Channel Estimation

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 126-131
https://doi.org/10.9734/bpi/ramrcs/v5/5097F

Signal Detection Noise removal, is a very important issue in channel estimation, and increasing performance of signal transformation in cognitive networks. Therefore it is necessary to have a criterion for evaluating the degree of correctness and reliability of the signals. Nowadays neural networks has very important role in calculations and if it combined with statistical methods they will produce perfect results in separability detection.   In this paper, we used the separability degree as a criterion for separating and identifying noise from the main signal. We use statistical Hypotheses and declare some statistical  thresholds  for signal validity to get the signal more suitable by increasing noise detection quality. This method supposes two states for our signal that are false detection of weak signal, and correct detection of the main signal. All these will be done by statistical_neural methods.

##### A Study on Optimization of Truck-Drone Parcel Delivery Using Metaheuristics

Sarab AlMuhaideb, Taghreed Alhussan, Sara Alamri, Yara Altwaijry, Lujain Aljarbou, Haifa Alrayes

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 132-153
https://doi.org/10.9734/bpi/ramrcs/v5/1641A

This research addresses a variant of the traveling salesman problem in drone-based delivery systems known as the TSP-D. The TSP-D is a combinatorial optimization problem in which a truck and a drone collaborate to deliver parcels to customers, with the objective of minimizing the total delivery time. Determining the optimal solution is NP-hard; thus, the size of the problems that can be solved optimally is limited. Therefore, metaheuristics are used to solve the problem. Metaheuristics are adaptive and intelligent algorithms that have proved their success in many similar problems. This study proposes a hybrid meta-heuristic solution to the TSP-D problem, where the solution is initialized from the optimal TSP solution reached by the Concorde TSP solver. Next, the delivery routes for the truck and the drone are obtained using the greedy, randomized adaptive search procedure (GRASP) with two local search alternatives. The main contribution of this work is the application of self-adaptive selection when searching the neighbourhood in GRASP. The proposed approach was tested on 200 instances with different properties from the publicly available ”Instances of TSP with Drone” benchmark. Results were evaluated against state-of-the-art algorithms. Nonparametric statistical tests concluded that the proposed approach has comparable performance to the rival algorithms (p = 0:074) in terms of tour duration. The proposed approach has better or similar performance in instances where the drone and truck have the same speed ($$\alpha$$ = 1).

##### Study on Implementation of Cloud Computing in Education: A Revolution

Saju Mathew

Recent Advances in Mathematical Research and Computer Science Vol. 5, 22 November 2021, Page 154-160
https://doi.org/10.9734/bpi/ramrcs/v5/14014D

Innovation is necessary to ride the inevitable tide of change and one such hot recent area of research in Information Technology (IT) is cloud computing. Cloud computing is a distributed computing technology offering required software and hardware through Internet. It also provides storage, computational platform and infrastructure which are demanded by the user according to their requirement. Due to the growing need of infrastructure educational institutes, organizations have to spend a large amount on their infrastructure to fulfill the needs and demands of the users. Cloud computing is a next generation platform that allows institutions and organizations with a dynamic pools of resource and to reduce cost through improved utilization. In the present scenario, many education institutions are facing the problems with the growing need of IT and infrastructure. Cloud computing which is an emerging technology and which relies on existing technology such as Internet, virtualization, grid computing etc. can be a solution to such problems by providing required infrastructure, software and storage. Developing a cloud architecture for education can be distinct according to the purpose and infrastructure of the institution and can be challenging. In this paper a basic research has been carried out to show how cloud computing can be introduced in the education to improve teaching, agility and have a cost-effective infrastructure which can bring a revolution in the field of education. It also tries to bring out its benefits and limitations.