Research Updates in Mathematics and Computer Science Vol. 5

Linear discriminant analysis (LDA) is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. Discriminant analysis is one method used in the data analysis that involves many variables. The purpose of the use of discriminant analysis is to classify an individual or observations into groups where there is a free and thorough number of independent variables. It is one of the methods offered in multivariable analysis that can help in land classification. This research resulted in functions and values discriminant that could be used for classification of sub-district land in Banda Aceh by using Fisher discriminant analysis. The functions and values discriminant that is obtained had been implemented on vacant land data in the sub-district of Baiturrahman. The results of this research are the sub-district of Baiturrahman has 340 vacant lands, by which 282 vacant lands are used as protected areas and 58 others are used as cultural areas. The vacant land area in the sub-district of Baiturrahman is 40.54 Ha, where the land area that can be used as a protected area is 32.216 Ha and land area that can be used as cultural areas is 8.324 Ha. Furthermore, the discriminant function that is obtained has an accuracy of classifying land of 62.82%. This research is expected to be a recommendation to classify vacant land in Banda Aceh city, vacant land use for the sub-district of Baiturrahman can be a consideration to the government, so that further research can be done to classify vacant land directly into 11 land uses in Banda Aceh city. The percentage of test accuracy is quite high, so the function can be used to classify lands blank.

In this chapter, we focus, on the one hand, on the estimation of the linear mixed model parameters and, on the other hand, on the solution of the generalized estimating equations. The most frequently used models for statistical data analysis are regression models. In linear regression, the objective is to study the relationship between a response variable (explained variable) and one or more explanatory variables, based on linear models (LM). The estimation of the considered parameters model is one of the most crucial processes in the statistical data analysis process. The greatest likelihood and least squares are the two most used estimation techniques. There is no preference for one approach over the other when the data are gaussian as the two procedures are equivalent. However, if the data are not gaussian, this equivalence is no longer valid. Also, if the normal equations are not linear, we make use of iterative methods (Newton-Raphson algorithm, Fisher, etc ...). In this work, we consider a particular case where the data are not normal and solving equations are not linear and that it leads to the equivalence between the maximum likelihood and the least squares methods, but the last is modified. In addition, we concluded by referring to the application of this modified method for solving the equations of Liang and Zeger. At the end of the work, we showed the existing relationship between the maximum likelihood for the GLM and the GEE resolution method which is the iteratively reweighted least squares ones (IRLS).

Discrete multiple orthogonal polynomials are useful extension of discrete orthogonal polynomials. The theory of discrete orthogonal polynomials on a linear lattice were extended to such polynomials by J. Arvesu, J. Coussement and W. Van Assche. In this study, we introduce a new family of discrete multiple orthogonal polynomials, namely \(\omega\)-multiple Meixner polynomials of the first kind, where \(\omega\) is a positive real number. Some structural properties of this family, such as raising operator, Rodrigue’s type formula and explicit representation are derived.The generating function for \(\omega\)-multiple Meixner polynomials of the first kind is obtained and by use of this generating function we reach to several consequences for these polynomials. One of them is a lowering operator which will be helpful for obtaining a difference equation. We obtain the difference equation which has the \(\omega\)-multiple Meixner polynomials of first kind as a solution. Also it is shown that for the special case \(\omega\) = 1, the obtained results coincide with the existing results for multiple Meixner polynomials of the first kind. In the last section as an illustrated example we consider the special case when \(\omega\) = 1/2 and for the 1/2- multiple Meixner polynomials of the first kind, we state the corresponding result for the main theorems. Overall, this study contributes to the understanding of these polynomial families and provides valuable insights into their properties and applications.

The assignment problem (AP) is a distinct form of a transportation problem where the key objective is to find an assignment schedule in a job where n jobs are assigned to n workers and each worker accepts exactly just one job so that the entire assignment cost must be minimum. In the existing method throughout the entire procedure they applied Hungarian method based on the computation of intervals. A new approach namely, center point method is proposed to solve an optimal interval assignment cost for fully interval assignment problems (FIAP). In the proposed method, the given FIAP is decomposed into a crisp assignment problem (AP) with the help of midpoint technique, solving it with the existing technique and by using its optimal solutions; an optimal interval assignment cost to the given FIAP is obtained. Comparison of intervals and partial ordering techniques were not used and there is no restriction on the elements of coefficient interval matrix in the proposed method. The proposed method is easier and also, simply because of the linear programming techniques. Using numerical examples the above method is illustrated.

The Power Series Method (PSM) is used as the numerical framework for estimating the Black-Scholes partial differential equation. The Black-Scholes model is a powerful tool for valuation of equity options. The Black-Scholes model is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option’s lifetime. PSM offers several advantages over traditional finite difference methods. The PSM is more stable than explicit methods and thus computationally more efficient. It is as accurate as hybrid approaches like Crank Nicolson and faster to compute. It is more accurate over a far wider spectrum of time steps. Finally, and importantly, it can be expressed analytically thus offering the capability of performing comparative statics in a far more stable and accurate environment. For more complex application this last advantage may have wide implications in producing hedge ratios for synthetic replication purposes. This study concludes that PSM is an excellent alternative to the numerical finance literature.

The concept of the fuzzy graph structure under the operation union is introduced and applied in the field of human trafficking. Types of human trafficking are treated as points and causes as relations.

Background and Aims: The modernization of legacy computational systems in computer science is an area that has been changing for a long time. The available systems, often decades old, face challenges such as a lack of compatibility with newer technologies, scalability issues, security vulnerabilities, and high maintenance costs. As technology evolves rapidly, organizations find themselves at a crossroads to modernize these legacy systems or continue to pour resources into increasingly obsolete technology. This research offers an in-depth exploration of the hurdles organizations face during legacy application modernization. The investigation delves into the primary motivations behind modernization, delineates the associated challenges, and proposes viable strategies and best practices to mitigate these issues.

Study Design: This is a Review Article that synthesizes and critically assesses a broad array of sources relevant to legacy application modernization. It amalgamates insights from various studies, offering a comprehensive overview and analysis of existing literature to derive meaningful conclusions and recommendations. Through this approach, the article provides a holistic understanding of the challenges and strategies associated with modernizing legacy systems.

Methodology: This global study was conducted over eight years, from January 2016 to August 2023. This research uses a literature review to collect data. In the literature review process, a comprehensive array of data collection methods is strategically employed to ensure the acquisition of a diverse and pertinent body of knowledge concerning the challenges associated with modernizing legacy applications and the effective strategies and best practices to address them. Systematic reviews and meta-analyses give structured synthesis, while manual searches collect real-world case studies. Thematic Analysis sorts findings, whereas Data Management arranges data, and the Critical Appraisal Skills programme evaluates the credibility of sources. This approach is an important starting point for modernizing legacy systems and developing effective policies and guidelines.

Results: The research identifies business necessities and technological advancements as the predominant catalysts for modernization. Legacy application modernization is not just a technological necessity; it's a strategic endeavor that paves the way for a future-ready, sustainable, and efficient organization. It further elucidates the obstacles encountered by organizations during this transition, such as the intricacies of data migration, the complexity inherent in legacy systems, and issues related to user acceptance and integration. The investigation also delves into potential strategies and best practices to navigate these challenges, emphasizing the significance of selecting the right modernization approach.

Conclusion: The existing research underscores that although the path to modernizing legacy applications has obstacles, they can be navigated successfully through astute planning, strategic decision-making, and adept execution. In doing so, organizations have the potential to metamorphose their dated systems into valuable tools that resonate with current business demands and the latest technological advancements.

In this paper, a finite capacity queueing model, encouraged arrivals, and reverse reneging customers with controllable arrival rates are considered. The term encouraged arrivals emerged from the situation that a system experiences after the release of offers and discounts by firms. Encouraged arrivals are a new addition to existing customer behaviour in queuing theory. The steady state solutions of system size are derived explicitly. The analytical results are numerically illustrated and relevant conclusions are presented.

The investigation of mathematical models with fractional derivatives has become a fruitful area of research in recent decades. For the first time, a new dissipationpreserving scheme is proposed and analyzed to solve a Caputo–Riesz time-spacefractional multidimensional nonlinear wave equation with generalized potential. Classically, Caputo-like fractional operators have found potential applications in studying systems with memory effects. In the present work, we consider a fractional extension of the nonlinear Klein–Gordon equation with damping, which involves Caputo temporal derivatives and Riesz spatial derivatives. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hypercube. We introduce an energy-type function and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employed a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carried out a number of numerical simulations to verify the validity of our theoretical results.

Aims/Objectives: To develop an intuitive guide for enhanced students’ understanding of the classical one-dimensional wave equation, bridging the gap between theoretical derivations and practical applications. The focus was on understanding wave propagation by modeling the elastic properties of a beam structure as a one-dimensional string.

Background: The study mainly focus on innovative teaching techniques and the integration of theoretical and practical approaches can significantly contribute to improving students’ understanding of mathematical theory and its applications in engineering, making it a valuable resource for educators and researchers in the field.

Study Design: The study employed foundational principles and theoretical derivations, and extended into the application of Fourier series techniques to elucidate concepts not typically covered in engineering mathematical textbooks.

Methodology: Analytical and numerical methods were utilised to reinforce critical concepts, making abstract ideas tangible for students. Numerical analysis aids in understanding the theory by demonstrating the evolution of wave patterns, aligning with the analytical solution.
Results: The comparison of analytical and numerical solutions revealed that different time step values (\(\Delta\)t) influence the numerical solution only by shifting the function, f(x), in amplitude, but its shape and agreement with the analytical solution was maintained.
Conclusion: This research showcased how innovative teaching techniques, combining analytical and numerical methods, can be used to enhance students’ understanding of mathematical theory and its applications in engineering.