This paper investigates the geometric and dynamical properties of Euclidean space endowed with a torsion tensor that framework is known as metric-affine geometry with a flat metric.We analyze the consequences of relaxing the torsion-free condition in Euclidean geometry. While classical Euclidean geometry assumes a symmetric, metric-compatible connection (Levi-Civita), we allow the connection to have an antisymmetric part determined by an independent torsion tensor. This simple yet rich extension reveals profound departures from Riemannian intuition: geodesics may deviate from straight lines, parallel transport becomes path-dependent even in flat space, and curvature can emerge purely from torsion. We derive explicit formulas for the connection, curvature, and geodesic equations in Cartesian coordinates. Through variational principles, we obtain field equations that couple torsion to the geometry, illustrating how torsion can be interpreted as an independent physical field (e.g., electromagnetic). The results highlight the operational meaning of torsion through closure failure of infinitesimal parallelograms and provide a clear pedagogical foundation for more advanced theories such as Einstein–Cartan, teleparallel, and metric-affine gravity.

KEYWORDS: Metric-affine geometry; Torsion tensor; Euclidean space; Geodesics; Curvature; Field equations; Variational principle; Einstein–Cartan theory; Teleparallel gravity.
MSC 2020: 53B05, 53B20, 83C05, 83D05 PACS 2010: 04.20.Cv, 04.50.Kd

The statistical techniques such as multivariate regression analysis and Autoregres-sive moving average (ARIMA) have been commonly applied to study plant phe-nology over decades. However, there has been inadequacies in understanding com-plex spatiotemporal big data that exists. In the recent past, Recurrent Neural Net-work, Convolutional Neural Network (CNN), Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) have been applied to the study of crop pro-ductivity, however, these models when used individually have certain shortfalls for instance, LSTM and GRU belong to the Recurrent Neural Network family but differ
in architecture and complexity. Therefore, there is need to leverage on the unique properties of each model, or duo models for instance the combined CNN-LSTM hy-brid model. In this note, we develop and analyze a multi-faceted ARIMA-CNN-GRU (ACG)-hybrid model for soybean productivity and future projections.

KEYWORDS: ARIMA, ACG-hybrid model, Soybean, Projection.

Some classes of harmonic-like quasi variational inequalities, which can be viewed as novel important generalization of quasi variational equalities, are introduced and investigated. Using various techniques such as projection methods, auxiliary principle, dynamical systems coupled with finite difference approach, we suggest and analyze a number of new and known numerical techniques for solving quasi harmonic-like variational inequalities. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. One can obtain a  umber of new classes of quasi harmonic-like variational inequalities by interchanging the role of operators. Various special cases are discussed as applications of the main results. Several open problems are suggested for future research.

KEYWORDS: Variational inequalities, Convex functions, fixed point, iterative methods, auxiliary principle, convergence, dynamical systems, sensitivity analysis.

This paper studies the complexity of some problems related to the satisfiability of Boolean formulas and consisting in asking about the uniqueness of a solu-tion. More precisely, we show that the following decision problems: Unique Satis-fiability (U-SAT), Unique k-Satisfiability (U-k-SAT) for k > 3 and Unique One-in-Three Satisfiability (U-1-3-SAT) have equivalent complexities by establishing expli-cit reductions relating each problem to each other; these reductions are polynomial (with respect to the binary size of the transformed instances) and keep the unique-ness of the solution when the solution is unique. As a consequence, these problems are all US-complete, and hence are co-NP-hard and belong to DP.

KEYWORDS: Complexity Theory, classes DP and US, US-completeness, Uniqueness of Solution,Boolean Satisfiability.

In this paper, we firstly introduce a new second-order generalized weak subdifferential for a set-valued mapping by means of the second-order generalized Studniarski epiderivative and scalarization. We secondly discuss an existence theorem and some properties of the second-order generalized weak subdifferential. Finally, we obtain the optimality conditions for locally strict efficient solutions of order two in a composite set-valued optimization problem. Several main results improve and generalize the corresponding results in the literature.

KEYWORDS: Composite set-valued optimization problem, Optimality conditions, Second-order generalized weak subdifferential, Locally strict efficient solutions of order two.

In this article, we consider the two-sample testing for sparse functional data, which is widely used in functional data analysis and longitudinal data analysis. Due to irregular observations and small sample sizes, the testing problem for sparse functional data is challenging and complicated. We propose several non-parametric testing methods using Cram´er-von Mises test statistics and Energy statistics based on conditional functional principal component analysis. The numerical simulation results show that the developed procedures have good performance in controlling two types of errors. Finally, empirical analysis is conducted on the two-sample tests for a carbon monoxide concentration dataset.

KEYWORDS: Cram´er-von Mises statistics, conditional principal components analysis, energy statistics, sparse functional data, two-sample tests.

This paper addresses a nonconvex optimization problem, the packing problem with inequality constraints simulation on DC (Difference of Convex) functions. We formulate a general class of packing problems as DC programming and use DC local search techniques. The DC local search method leverages the DC decomposition of the objective and constraint functions, allowing for iterative refinement through convex sub-problems. Each iteration involves solving a convex approximation of the original problem, guided by sub-differential information to ensure descent and feasibility. We demonstrate the effectiveness of this approach through various geometric packing scenarios, especially circle packing, showing significant improvements in packing density and computational efficiency. The DC local search method provides a flexible and scalable framework for tackling complex packing configurations in two dimensions. In test problem, we used Sangaku optimization problem, the packing 6 circle into rectangle of 1:1.934798 size. Computational results were obtained using Python in a Jupyter Notebook.

KEYWORDS: Packing problems, Difference of Convex functions, Local search method.

The objective of this work is to propose Topological Principal Component Analysis (TPCA) and Topological Discriminant Analysis (TDA) according to the type of data: quantitative, qualitative, or mixed variables.

Large datasets are increasingly common across many disciplines. The exponential growth of data requires the development of more advanced data analysis methods in order to process information efficiently. To better visualize data, many methods, such as PCA, allow the extraction of a low-dimensional structure from high-dimensional datasets. The proposed TPCA approach is a multidimensional descriptive method that studies a homogeneous set of continuous variables defined on the same set of individuals.

Topological Discriminant Analysis (TDA) is a supervised classification method that aims to uncover the intrinsic structures and discriminative information embedded in the data. Many predictive techniques exist and are widely applied to a variety of problems across numerous fields. Classification is the process that assigns each individual in a population to one of several predefined classes based on their characteristics, which are considered explanatory variables.

The two proposed topological approaches, TPCA and TDA, are based on the notion of neighborhood graphs: one in an exploratory and descriptive context, and the other in a decision-making and predictive context.

The variables may be more or less correlated or related depending on their type—quantitative, qualitative, or mixed. These topological methods analyze the structure of correlations or dependencies observed among the variables under consideration.

A real-data example illustrates the two topological methods. The results of TPCA are compared with those of classical PCA, while the results of TDA are compared with those of various machine learning predictive models.

KEYWORDS: proximity measure, neighborhood graph, adjacency matrix, principal components analysis, discriminant analysis, predictive modeling, machine learning models.

This study investigates the geometric properties of magnetic striction lines on ruled surfaces in uenced by magnetic fields, with particular emphasis on the Fermi-Walker derivative and energy of magnetic curves defined in the context of geodesic Frenet frames. That is, the Fermi-Walker derivatives and energy calculations of magnetic curves called magnetic striction lines on ruled surfaces, using the geodesic Frenet frame examined. Theorems are presented that describe when the Fermi-Walker derivatives become zero indicate geometric or physical equilibrium. Analytical for-mulas for the energy of vector fields are derived based on previously established geometric relations.

KEYWORDS: Fermi-Walker derivatives, Lorentz force, magnetic curves, striction lines.

This study presents a comprehensive evaluation of the size and power properties of several coefficient of variation (CV) tests under varying distributional assumptions and sample sizes. Using extensive Monte Carlo simulations, we assess the performance of both classical and modified CV tests across normal, gamma, chi-square, and log-normal distributions. The findings reveal that while many tests perform adequately under normality with large sample sizes, only a few maintain robustness under non-normal conditions and small samples. Notably, the median-modified versions of the Sharma and Krishna, Curto and Pinto, and ADJ tests consistently demonstrate strong power and effective size control across diverse settings. The parametric bootstrap test also performs well, particularly for skewed distributions such as the log-normal. In contrast, the classical t-test remains overly conservative, and the standard versions of the Miller, Vangel, and Panichkitkosolkul tests exhibit weak power. These results highlight the importance of using modified CV tests tailored to distributional characteristics and sample size, thereby improving the reliability of statistical inference in applied research.

Keywords: Coefficient of variation; Gamma distribution; Hypothesis testing; Log-normal distribution; Nominal size; Simulation study; Statistical power; Skewness.

Ranked set sampling (RSS) is a statistical technique in which multiple random samples are drawn, ranked with respect to characteristic of interest and then specific units from each sample are quantified to estimate the population mean. This method is a cost-effective sampling procedure designed to provide more efficient estimators of population parameters such as mean and can be more powerful than Simple Random Sampling (SRS) even in the presence of ranking errors. It was originally introduced by McIntyre (1952), this method is especially useful when measuring sampling units is difficult or expensive, but visual ranking them or expert judgment or any cheap auxiliary variable is easy. In this method, after selecting m2 units randomly from an infinite population and arranged them into m sets each containing m units, a unit is randomly selected from each set. Each so obtained unit is then ranked between 1 and m (both inclusive) with respect to variable of interest or on the basis of highly correlated auxiliary variable is then quantified. Sample is selected through a pre-assigned pattern as from the first set, select the smallest unit; from the second set, select the second smallest unit and this process is continued until the largest unit is selected from the ????????th set. For obtaining larger set size repeat the entire process ???????? times to obtain a sample size of ???????? = ???????????? . Performance of RSS further improves when appropriate unequal allocation is used instead of equal allocation based on Neyman’s approach with the same set size. The ranked set sample mean is an unbiased estimator of the population mean even with a smaller variance than the simple random sample mean of the same size. The cost-effectiveness of RSSachieved because only a smaller portion of units???????????? are measured, so costs are lower than taking a full rm2sample are used as in the SRS. The present paper focuses on the overviewand continued development on the RSS estimator and its broad applications particularly effective in agriculture, forestry, ecological studies and in environmental studies where visual ranking is possible. These procedures are illustrated using a real data set regarding the yield of potato. The technique is more useful to those who look for cost-effective technique for estimating agricultural products that are grown underground such as potato, ginger, turmeric, garlic, onion, beetroot, peanut, etc.

Keywords: Equal allocation, unequal allocation, cost-effective, relative precision, relative cost.

In educational experiments student’s performance is not only influenced by their study methods or teaching methods but also by several other factors like time period, environment and time of day. These additional sources of variation may influence the results and make it difficult to determine the true impact of study or teaching methods. This problem leads to the need of a design of experiments that can control two sources of variation along with determining the effect of the primary factor which is Latin Square Design. The objective of this research was to determine the effect of four (4) study methods (i.e., watching online lectures, solving practice questions, making notes/summaries, and re-reading notes/ textbooks), times of day, and the length of study session on academic performance of undergraduate students. A 4×4 Latin Square Design was utilized in this study, the data for which was collected through an online survey. Semester Grade Point Average (SGPA) was used as the response variable and analysis of variance (ANOVA) technique was used for analysis. The study was conducted at 5%level of significance.

This study highlights the practical applicability of Latin Square Design in educational research and explains how its structured approach helps in managing multiple influencing factors simultaneously. The findings contribute to understanding how study-related variables influence the academic performance.

KEYWORDS: Latin Square Design (LSD), Factor, Analysis of Variance (ANOVA), Study Methods, Semester Grade Point Average (SGPA).