METHODOLOGY OF THE TEACHING OF FINITE MATHEMATICS AT THE PEDAGOGICAL UNIVERSITIES DUE TO THE NEW TRENDS IN THE EVOLUTION OF THE MATHEMATICAL SCIENCES

There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the newly revised textbooks and corresponding syllabi for the Mathematics Courses taught at the Junior two years Colleges and Pedagogical Universities.These newly developed problems are reflection of the Development of Mathematical and Engineering Sciences to offer great amount of learning conclusion/sequel to those who pursue a bachelor’s degree at the universities of the pedagogical orientation. The problems presented in the article here are developed and restructured in terms of the newly developed techniques to solve the problem in Finite Mathematics and Engineering sciences. Moreover, the techniques offered in the article here are more likely to get utilized in Advanced Engineering Sciences, too, within the content of the problems, which require to obtain finite numerical solutions to the Real Phenomena Natural Problems in Engineering Sciences and Applied Problems in Mathematical Physics.The aim of this present publication is to offer new advanced techniques and instructional strategies to discuss methodology and instructional strategies of the mathematical training of the students at the Pedagogical Universities. Moreover, these new teaching techniques and strategies introduced may be extended to the engineering sciences at the technical universities, too.The results and scientific novelty of the introduced methodology and learning conclusions and sequel of the new knowledge the students at the Pedagogical universities may be benefited from are in the following list of the learning conclusions, presented in the article here. The students of the pedagogical orientation may attain the mastery skills in the following sections of the combinatorics in Finite Mathematics subject:- The n! Combination of n different terms.- Evaluate the expressions with factorials.- Identify that there are -!!( )!nrnr various of combinations of r identical terms in n variations.- Identify and evaluate the combinatorial coefficients from the Binomial Theorem.- Identify and able to build the Pascal’s triangle of the binomial coefficients.- Utilize the Binomial Theorem to expand the binomial formula for any natural powers.- Utilize the Binomial Theorem to obtain the general formula for the n-th term of binomial expansion.- Utilize the Sigma Symbols in the Binomial Theorem for the n-th terms of the binomial expansion. 19- Generate the expansion for the power of the ex, where e is the base of natural logarithmic function y = f(x) = x.Practical significance: the methods of teaching and new teaching strategies offered here in the article alongside with the application of the new trends in the development of mathematical and mathematics education sciences can be useful for prospective and currently practicing teachers of mathematics. Moreover, the materials presented here in this article can be useful for the educational professionals in their professional development plans to improve the quality in education

A Combinatorial Reliability Analysis of Generic Service Function Chains in Data Center Networks

In data center networks, the reliability of Service Function Chain (SFC)—an end-to-end service presented by a chain of virtual network functions (VNFs)—is a complex and specific function of placement, configuration, and application requirements, both in hardware and software. Existing approaches to reliability analysis do not jointly consider multiple features of system components, including, (i) heterogeneity, (ii) disjointness, (iii) sharing, (iv) redundancy, and (v) failure interdependency. To this end, we develop a novel analysis of service reliability of the so-called generic SFC, consisting of n = k + r sub-SFCs, whereby k ≥ 1 and r ≥ 0 are the numbers of arbitrary placed primary and backup (redundant) sub-SFCs, respectively. Our analysis is based on combinatorics and a reduced binomial theorem—resulting in a simple approach, which, however, can be utilized to analyze rather complex SFC configurations. The analysis is practically applicable to various VNF placement strategies in arbitrary data center configurations, and topologies and can be effectively used for evaluation and optimization of reliable SFC placements.

Symmetric Key and Polynomial Based Key Generation Mechanism for Secured Data Communications in 5G Networks

Fifth Generation (5G) networks provide data communications through various latest technologies including Software Defined Network (SDN), Artificial Intelligence, Machine Learning and Cloud Computing. In 5G, secure data communication is a challenging issue due to the presence of enormous volume of users including malicious users communicating with latest technologies and also based their own requirements. In such a scenario, fuzzy rules and cryptographic techniques can play a major role in providing security to the data which are either communicated through the network or stored in network based databases including distributed databases and cloud databases with cloud networks. Therefore, new and efficient mechanisms for generation and exchange of keys are necessary since they are the most important component of cryptographic methods. Since most of the existing key generation techniques are focusing on 3G and 4G networks, new key generation methods that can be generalized to n-th order polynomials are necessary to suit the security requirements of 5G networks which is smart by using rules from Artificial Intelligence. This paper proposes a new key generation and encryption/decryption mechanism which is based on both symmetric key cryptography and polynomial operations for providing effective security on data communication in 5G networks. In this work, we introduce the usage of fuzzy rules and Binomial Theorem (Pascal triangle) technique for performing the data encryption process more efficiently since it is not used in any of the existing cryptographic algorithms. Moreover, two different polynomial equations, one of degree three and another of degree two are used in the proposed work for effective key generation. Here, we have applied differential calculus for finding the second-degree polynomial. In the decryption part of the proposed mechanism, nth root operation is applied which is able to reduce the number of steps used in a single mode operation. The experimental results of the proposed work proved that the proposed security model with fuzzy rule-based approach is better than other related systems that are available in the literature in terms of reduction in computational complexity and increase in security.

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

A Note on Generalized q-Difference Equations and Their Applications Involving q-Hypergeometric Functions

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators T(a,b,c,d,e,yDx) and E(a,b,c,d,e,yθx) to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.