Research and Applications Towards Mathematics and Computer Science Vol. 9

In this chapter, the new concept SP Mean difference labeling has been introduced and a formula for SP mean labeling has been established. A function x is called a SP Mean difference labeling of a graph G with q edges, if the vertices of G to the set \(\{\) 1,2,.....,p\(\}\) such that when each edge \(\upsilon\)v is assigned the label $$x(uv) = {x(u + v) - x(u - v)\over 2},$$then the resulting edges labels are distinct numbers. In this chapter, some families of graphs which admits SP Mean difference labeling are investigted.

Since that fateful and chilly dawn of April 15, 1912, the world has witnessed the construction of larger ships, some already dismantled or lying, solitary, in the darkness of the bottom of the oceans and others still in circulation. However, no other ship has become as famous and significant for popular naval and imaginary history as the Royal Mail Ship Titanic. The RMS Titanic joined the imagination of the navy, literature, and cinema. It fed the dreams and nightmares of generations, from the one from 1912 who was perplexed to receive the news of the disaster to the present generation that has it in the ambivalence of an engineering feat of its time, as well as a fruit of the arrogance of its creators.

Its history is known to all and its data is used in many studies. It should be mentioned that these data are composed of records of various variables and various natures. In addition, they are easily generalizable to several other situations.

In this study, the researchers will make use of the regression models. This model is one of the most important statistical tools in data analysis when the objective is to study relationships between variables, or more particularly, to analyze the influence that one or more variables (explanatory variables) may have on a variable of interest (response variable).

The purpose of this study is to describe in detail the construction of this type of model using a dataset on the Titanic tragedy. Not forgetting the limitations. Understanding these limitations is crucial for making informed decisions about when and how to use linear regression and when alternative methods might be more appropriate.

Time is used in mathematical modeling which measure physical changes. We have demonstrated in two different ways that it was an invention of thought. Indeed, archeology has put in light the first trace of time which started to be used in Mesopotamia (4800-4500 years ago) [6]. Physics makes appear that time has no physical properties; it means that it’s a concept; instead, it owns interesting mathematical properties [12]. Strangely enough, time is still and wrongly considered a phenomenon; obviously, its role needs to be clarified through some contrasting examples in diverse disciplines.

Computing an exact solution for the differential equations, with differences, system that allows the determination of the M|M|m|m system transient probabilities is a hard task. The respective complexity grows with m. The computations are extremely fastidious and the length and the fact that the expressions obtained are often approximate, and not exact, will not allow the transient probabilities behavior as time functions characterization. To overcome these problems, in this work it is analyzed how that system can supply approximate values to the M|M|m|m queue system, through matrix calculations. It is also presented an asymptotic method to solve the system that becomes possible in many cases to obtain simple approximated expressions for those probabilities using the M|M| \(\propto\) transient probabilities, very well-known and very much easier to study.

Let G(V,E) be a simple undirected graph. The Inverse sum indeg index of a graph G is defined as $$ISI(G) = \sum_{r,s{\in}E(G)} {drds \over dr + ds}$$ where ds is the degree of the vertex s in G. The new idea of proper colourings in the Inverse sum indeg index graph has been proposed in this chapter. Some families of graphs such as cycle, generalized Petersen graph and complete graph which satisfies the condition of r–regular Inverse sum indeg index. The inequalities for the chromatic number related to Inverse sum indeg index are developed.

The pendulum itself is one of the mechanical systems that have been studied the most in the field of mechanics and has innumerable applications in very different scopes. The inverted spring pendulum is analyzed. We derive the equations of motion using Lagrange mechanics approach. Several ways to solve it are provided. The pendulum is linearized near the equlibrium position to obtain a system that may be solved using Mathieu functions. We solve it by means of a simple ansatz method and the Krýlov-Bogoliúbov-Mitropólsky method. The results are illustrated through several examples.

In this paper we apply several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. We permorm Melnikov analysis in order to detect chaotic beahvior.

Geographic Information Systems (GIS) are programs that connect math, science, and social science concepts with many resources for teachers to provide and integrate activities with math and science. This paper reflects on proposals to merge Mathematics and Geography when teaching undergraduates from inter- and trans-disciplinary spatial models using mathematical modeling to enable spatial analysis and visualization through GIS—Geographic Information Systems. Mathematics Education incorporates facts from real-world reality with models that use new geotechnologies. GIS brings forward an approach to Geography with new technologies integrating it with Statistics and Computer Science. Thus, it describes some practical cases that are examples of geographical problems solved mathematically. As an example, a case study about COVID-19 infection in a city is presented. The results show that GIS can be an educational tool for teaching both Geography and Mathematics based on real problems. Integrating this tool enables active and constant participation of the students, which can make them realize the applicability of mathematics in solving real problems related to geographical space, is required by any school committed to education.

In the present paper we consider the polynomials of the type q_n (x) = (x+1)ⁿ P (x)+xⁿ Q (x), where P (x) and Q(x) are nonconstant polynomials with real coefficients of degree m such that \(\lim\limits_{x\to\pm\infty}\frac{P(x)}{Q(x)}\) is a finite positive number. We investigate the number of real zeros of q_n(x) when \(n\to\infty\).

This chapter is dedicated to the development and application of an extractive summarization procedure and is a detailed account of the set of experiments performed to study the utility of applying a combined structural property of a text’s sentences and term expansion using WordNet along with a local thesaurus all combined and used in the selection of the most appropriate extractive text summarization in a particular document.  Sentences were tagged and normalized. Next, they were subjected to the well-known Longest Common Subsequence (LCS) algorithm for the selection of the most similar subset of sentences. Calculated similarity was based on LCS of pairs of sentences making up the document. A normalized score was calculated and used to rank the sentences. A selected top subset of the most similar sentences was then tokenized to produce a set of important keywords or terms. The produced terms were further expanded into two subsets using 1) WorldNet; and 2) a local electronic dictionary/thesaurus.  The three sets obtained (the original and the expanded two) were then re-cycled to further refine and expand the list of selected sentences from the original document. The process was repeated a number of times in order to find the best representative set of sentences. A final set of the top (best) sentences was selected as candidate sentences for summarization.  In order to verify the utility of the procedure, a number of experiments were conducted using an email corpus. The results were compared to those produced by human annotators as well as to those results produced using basic sentences similarity calculation method. Produced results were very encouraging and compared well to those of human annotators and Jacquard sentences similarity.

New closed form solutions of SU(2) Yang-Mills (YM) gauge field equations found by the Carmeli-Charash-Kaye-Kh. Huleihil null-tetrad formalism. Unlike earlier reported solutions with special functions of elliptic integrals, this new solutions are with complex valued circular functions and with two arbitrary integration constants. Electric parts obtained as zero and having only nonzero magnetic parts and so concluded as new magnetic monopole solutions of SU(2) YM gauge filed equations.