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

ISBN 978-93-5547-214-4 (Print)
ISBN 978-93-5547-219-9 (eBook)
DOI: 10.9734/bpi/ramrcs/v4

This book covers key areas of Mathematical Research and Computer Science. The contributions by the authors include model assistance, model calibration, semiparametric model, penalized splines, Gambler’s ruin, random walks, Brownian motions, reserves, pensions fund, Two dimensional wave equation, one dimensional heat equation, unsteady motion, Navier Stoke’s equation, incompressible flow, Auxiliary attribute, mean square error, Fixed effects, random effects, coefficient of determination, panel data and Hausman test, space-action, Dual bi-directional heuristic, heuristic, global optimum solution, bi-directional heuristic, optimum solution, multi-start heuristic, Gaussian distribution, channel capacity, directed divergence, information theory, uniform distribution, geometric algebra, limit cycle, measure, Gagliardo-Nirenberg, Prekopa-Leindler, Debreu, Brouwer, Lusin, transcendental equation, lebesgue integral, finite time blowup, NP-complete problem, polynomial algorithm, independent and identically distributed random variables, Optimal control, singular extremals, strengthened legendre-clebsch condition, Network reliability, probability-ready expression, multi-state system, multiple-valued logic, symbolic expression, multi-state delivery network. 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

##### Model Calibration at Different Stages in Cluster Sampling; Use of Penalized Splines in Semiparametric Estimation

Pius Nderitu Kihara

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 1-14
https://doi.org/10.9734/bpi/ramrcs/v4/14456D

Estimation of finite population total using internal calibration and model assistance on semiparametric models based on kernel methods have been considered by several authors. In this book chapter we extend this so as to consider model calibration based on penalized splines in two stage sampling where the auxiliary information is available both at the element level and at the cluster level. Specifically, we derive estimators of population total that incorporates model calibration in estimating the cluster totals and in estimation population total. We show that the proposed estimators are robust in the face of misspecified models, are asymptotic design unbiased, have reduced model bias, are design consistent and asymptotic normal. We have shown that estimators based on penalized splines perform better than corresponding kernel based estimators and model calibrated estimators perform better than internally calibrated estimators do.

##### Addressing Reserves and Pension Funds through Gambler’s Ruin and Generalized Brownian Motion Process

Manuel Alberto M. Ferreira, José António Filipe

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 15-24
https://doi.org/10.9734/bpi/ramrcs/v4/14551D

We used the random walk to model the problem of reserves. The classic case of a stochastic process is the example of random walks, which are used to study a set of phenomena and, particularly, as in this article, to study models of reserves evolution. Random walks also allow the construction of significant complex systems and are used as an instrument of analysis, being used in this study for giving a theoretical characteristic to specific types of systems. Our goal is mainly to study reserves to see how to ensure that pension funds are sustainable. This paper, by covering a classic approach to the study of pension funds, makes possible to draw interesting conclusions about the problem of reserves. We also consider the Brownian motion to model the pensions fund assets and liability management politics. In this context, it was possible to obtain expressions for the expected value of the pensions fund perpetual maintenance cost present value, also for the expected value of the maintenance cost up to time t, indicators of a fund maintenance policy expenditures.

##### Derivation of Second Order Partial Differential Equation Indicating Wave and Heat Equation through the Use of the Navier Stokes Equation for Unsteady and Incompressible Flow: Scientific Explanation

N. Akhtar, G. A. H. Chowdhury

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 25-30
https://doi.org/10.9734/bpi/ramrcs/v4/1937C

In this paper, we have tried to approach the concepts of two dimensional wave equation and one dimensional heat equation through the means of the Navier Stoke’s equation for unsteady and incompressible flow. Our pursuit to do so has been supported with ample justifications and analytic discussions. The strong relation shared by the fluid dynamics, wave mechanics and heat flow has been brought to light through our attempts. Garnering sufficient insights on such relations can expand the knowledge levels and bring forth significant implications of the Navier Stokes equation.

##### Study on Generalized Exponential Type Estimators for Ratio of Two Population Means Using Auxiliary Attribute in the Presence of Non-response

Kamlesh Kumar, Sunit Kumar, Anupam Kumar

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 31-41
https://doi.org/10.9734/bpi/ramrcs/v4/4664F

In this paper, generalized exponential type estimators for the ratio of two population means using auxiliary attribute in the presence of non-response, have been proposed. The expressions for the mean square error of the proposed estimators are obtained up to the first order of approximation. Some particular cases of the proposed estimators are also given. The conditions, under which the proposed estimators do better in comparison to the relevant estimators, are obtained.  An empirical study using a real data set is also given to judge the performance of the proposed estimators.  The aim of this paper is to find the estimators which are more efficient than the conventional estimator and exponential type estimators for ratio of two population means in the presence of non-response.

##### Study on Fixed Effect versus Random Effects Modeling in a Panel Data Analysis; a Consideration of Economic and Political Indicators in Six African Countries

M. T. Nwakuya, M. A. Ijomah

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 42-49
https://doi.org/10.9734/bpi/ramrcs/v4/11245D

Individual heterogeneity can be controlled in panel data analysis to avoid bias in the final estimates. Panel data can be balanced or unbalanced, short or long panel. To identify the appropriate model, the fixed effects and random effects modelling approaches were applied to an economic data set, "Africa" in the R Amelia package. Both models were fitted to the data, taking into account the assumptions of the two models. The Breusch-Pagan Lagrange Multiplier test on the random model estimates revealed that the random model was acceptable for the data, however it had a poor coefficient of determination,    of 0.48697. The fixed effect was then estimated using four different approaches (Pooled, LSDV, Within-Group and First differencing) and testing each against the random effect model using Hausman test. Our findings revealed that the random effect was inconsistent in all tests, indicating that the fixed effect was more appropriate for the data. with an  of  , the LSDV was found to have the best match among the fixed effects models.

##### A Phenomenological Relation for Decay Times: A Quantum Mechanical Approach

Frans De Medts, T. U. Eindhoven

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 50-59
https://doi.org/10.9734/bpi/ramrcs/v4/14004D

In 1931, Georges Lemaitre was invited to London to attend a British Association symposium on the relationship between the physical Universe and spirituality. He postulated that the Universe expanded from a starting point, which he dubbed the "Primeval Atom" and developed in a report published in Nature titled ‘The beginning of the World from the Point of View of Quantum Theory’ [1]. This article explains how the way the universe is claimed to have come into being governs (and hence can be the deeper cause of) all quantum mechanical phenomena. A ‘smallest space-action' (or ‘smallest limited quantity of uncertainty') is constructed on the basis of hidden symmetries, which offers a connection between all that is and establishes the boundaries within which all variation is possible; unity in diversity then.

##### Study on Dual Bi-Directional Heuristic Development Framework

Stanley Murairwa

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 60-69
https://doi.org/10.9734/bpi/ramrcs/v4/14224D

The study creates a multi-start heuristic framework. The heuristic development approach accelerates heuristic convergence to high-quality optimum solutions. The Dual Bi-Directional (DBD) heuristic is a multi-start heuristic that searches for the global optimum solution (GOS) in four concurrent directions using a pair search starting from both the beginning and ending nodes. When the four optimum tours connect to form an optimum round tour of all the search space nodes, the search ends. The DBD heuristic then begins to improve the discovered optimum round tour in a unidirectional approach by employing a global search metaheuristic. By introducing the unidirectional heuristic to improve the multi-start heuristic constructed optimum round tour, the multi-start heuristic framework reduces the non-convergence of the bi-directional approach. The approach of the multi-start heuristic framework will allow the convergence of the bi-directional heuristic.

##### Channel Capacity Enhancement on Maximization of Modified Verma and Bose-Einstein Entropies by Gaussian Distribution

Rohit Kumar Verma

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 70-78
https://doi.org/10.9734/bpi/ramrcs/v4/12064D

This chapter discussed about the channel capacity enhancement on maximization of modified Verma and Bose-Einstein entropies by Gaussian distribution [5] and Kullback-Leibler’s [3] information measure of directed divergence.

##### On the 4th Clay Millennium Problem for the Periodic Navier Stokes Equations

Terry E. Moschandreou

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 79-92
https://doi.org/10.9734/bpi/ramrcs/v4/14379D

A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R3/Z3 is hereby shown using Geometric Algebra and subsequently the Gagliardo-Nirenberg and Pr kopa-Leindler inequalities are used to prove that the integrand of the integral form of the solution obtained can be set to zero everywhere in space and time, as well as results on the velocity-pressure distribution using Debreu's, Brouwer's, Lusin's and a final theorem proving no blowup on [0,$$\infty$$]. A complex equation's values for positive and t tending to large values indicates with the help of five theorems that as  approaches positive infinity,  approaches infinity.

##### A Counterexample of the Statement P = NP: Scientific Explanation

Peter Kopanov

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 93-98
https://doi.org/10.9734/bpi/ramrcs/v4/4968F

In this paper is shown that the classes P and NP do not coincide, by constructing a relatively simple example of unsolvability in polynomial time in a particular well-known NP-complete problem. For a particular NP-complete problem, a random process is constructed that generates solution with random numbers. Here will be shown that the assumption that a concrete NP-complete problem belongs to the P class will contradict the random and independent choice of a concrete solution to the problem. Such a solution cannot be found with a polynomial algorithm but it can be verified with one if known. In this way, the impossibility of equality P = NP is shown.

##### Determining the Sufficiency in Optimal Control without the Strengthened Condition of Legendre

Gerardo Sanchez Licea

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 99-112
https://doi.org/10.9734/bpi/ramrcs/v4/2119E

In this study, we construct a sufficiency theorem for an unconstrained Lagrange fixed-endpoint problem that offers sufficient conditions for processes that do not satisfy the traditional nonsingularity assumption, i.e., the new sufficiency theorem does not impose the strengthened Legendre condition. In contrast to various generalisations of conjugate points, solutions of certain matrix Riccati equations, invariant integrals, or the Hamiltonian-Jacobi theory, the latter leverages explicitly the positivity of the second variation in the demonstration of sufficiency.

##### Reliability Analysis of a Multi-State Delivery Network through the Symbolic Derivation of a Probability-Ready Expression

Ali Muhammad Ali Rushdi, Motaz Hussain Amashah

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 113-132
https://doi.org/10.9734/bpi/ramrcs/v4/2897E

This chapter deals with the reliability of a multi-state delivery network (MSDN) with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. The main contribution of this chapter is to provide a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is of paramount importance since it allows a direct transformation of a random logical expression, on a  one-to-one one, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). The running example used for demonstrating PRE derivation in this chapter is admittedly long and tedious. The statistical expectation of the expression is its probability of being equal to 1, and is simply called network reliability. The proposed method is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier by an alternative means. This chapter is a part of an ongoing activity to develop pedagogical material for various candidate techniques for assessing multi-state reliability.

##### The Vertices of a Graph and its Dimension

Volker W. Thürey

Recent Advances in Mathematical Research and Computer Science Vol. 4, 12 November 2021, Page 133-139
https://doi.org/10.9734/bpi/ramrcs/v4/13796D

We show that the dimension of a graph is less or equal the cardinality of the set of its vertices.