Galoitica: Journal of Mathematical Structures and Applications

Initial value problems (IVPs) of ordinary differential equations (ODEs) are ubiquitous in science and engineering applications, and the classical fourth-order Runge–Kutta (RK4) method is by far the most popular solver due to its good accuracy-to-cost ratio. Among all four-stage fourth-order explicit RK methods there are two free node parameters left after satisfying the eight B-series order conditions, thus allowing further systematic enhancements. Here we employ a hybrid multi-seed Particle Swarm Optimization (PSO)-Nelder–Mead algorithm to search for optimal RK node parameters (c₂, c₃) with respect to a minimax normalized objective over eight commonly used nonlinear benchmark problems. The resulting PSO-RK4 method with (c₂ = 0.323665, c₃ = 0.653527) retains both the exact same order of convergence and absolute stability region as the classical RK method called the 3/8-rule, but exhibits reduced maximum global error on each of the eight benchmarks when N = 300; average improvement of 27.9% with gains up to 46.0% on the Bernoulli equation, 29.7% on logistic growth, and 21.3% on exponential. Robustness of these gains with respect to multi-step-size (N = 100, 200, 300, 500) is demonstrated.

This is the complete study of Lagrange equations, a basic formulation in mathematical analysis and classical mechanics. In this report, we present derivation and classification as well as analytical solution techniques for Lagrange type differential equations. These include the Lagrange equations of motion from the calculus of variations, Lagrange multipliers for constrained optimization, and Lagrange interpolating polynomials. All types all begin with the mathematical proof and step to solution algorithms. Several relevant examples are provided to demonstrate the application of these equations in pure & applied mathematics, along with their detailed solutions.

A MetaStructure is a higher-level framework that treats entire collections of structures as single objects, equipped with natural operations that preserve isomorphisms across different domains. The term “Struc- ture” here refers broadly to mathematical systems as well as real-world models. An Iterated MetaStructure generalizes this idea recursively, generating successive layers in which structures of structures form deeper hierarchical meta-levels. In this work, we extend and investigate the properties of Algebra, Topology, Lattices, Queues, Markov Chains, and Intervals through the lens of MetaStructures and Iterated MetaStructures.

This work is informed by a time analysis procedure to investigate evaluation of the study variables from 2005 to 2023, employing of time- series models to reveal the relationship between governance and corruption through application of EViews13. The corruption perceptions index is considered as dependent variable, government effectiveness and rule of law are assumed independent variables, whereas the corruption growth rate is control variable. To describe model variables, time series regression models are employed subsequently, the graphical analysis and Augmented Dickey-Fuller (ADF) test are applied to know time series stationarity. Also, the two tests are utilized to examine degree of integration, nature of stochastic approach for variables, in case dependent or independent, over study time. This is done to ensure that the time series of the variable is stationary, meaning that it has a constant mean across its values, a constant variance, and no trend. Moreover, the tests conducted at first and second differences to assess stationarity at a certain difference, ensuring achievement stationary series adequate for modeling. Furthermore, robustness and validity of the estimated model are evaluated by autocorrelation tests and diagnostic tests of residuals. The findings show that the graphical method applied to examine stationarity is not highly precise as it appears severe oscillations in all variables under study. This involves conducting a stationarity test for the time series in according to equation of constant and trend by using the ADF test under the first and second differences. At the second difference I(2) , all variables are stationary as results show and the model can estimate  a statistically significance model, with a coefficient (R2 =96.3%), referring that model can explain the changes in corruption perceptions and has a high explanatory potential. Also value of F- statistic is high, reaching (183.3882). The time-series results further demonstrate presence of a statistically significant relationship between corruption and governance. It is revealed that advancement of government effectiveness exhibited statistical significance at 99% confidence level, and it is considered one of crucial independent variables in decreasing growth of corruption. The results show that a one-unit increase in government effectiveness results an increase corruption reduction by approximately (53.43) points. Concerning acceleration of rule of law, it has major influence over reducing the corruption. The outcomes state a one-unit increase of advancing rule of law deceleration of corruption acceleration by approximately (26.42) points, at a 99% confidence level. Furthermore, value of the constant of estimated equation is ( -0.07) at the 95% confidence level, suggests that there is an escalation of corruption because of absence of governance indicators, which driven by endogenous factors of Iraqi society. This characterizes the preceding value of corruption before adopting of governance regulations.  In the model residuals, lack of autocorrelation is characterized by the Durbin-Watson, while value of S.E. of regression reveals a high level of estimation precision. Hence, it can be deduced that governance stringent procedures contribute to decreasing corruption rate over the study period.

Breast cancer is a common type of cancers and the main reason of increased death of women universally. Recently, ML methods have become important in varying fields, such as Logistic Regression, Elastic Net Logistic, Decision Tree, Random Forest, Boosting, Naive Bayes and K Nearest Neighbor. The aim of the current study is to know and predict the type of cancerous tumor whether it is benign or malignant. These above techniques are expected to be helpful. Breast tumor type diagnosis using numerous performance metrics i.e. accuracy, classification error, sensitivity and specificity, both certified and trained models were assessed. The models were developed to determine which model would provide the best performance and comparisons were done. A separate data set from the one used to create the models was utilized to confirm every model. According to the analysis, the findings showed that elastic net logistic model had the highest performance in accurate classification rate (accuracy), classification error and sensitivity. Making it the best classifier for predicting the kind of breast cancer among all other models, privacy and it was also distinguished by reduce the high dimensionality and multicollinearity problems.