Matrix Science Mathematic (MSMK)

This paper presents the formulation of higher order diagonally implicit 2-point super class of block extended backward differentiation formula (2DSBEBDF) for solving first order stiff initial value problems. The order of the 2DSBEBDF method is derived and found to be four. The Stability analysis of the method shows that the method is zero-stable and its absolute stability region shows that the method is A-stable within the stiff stability interval -1≤ρ<1. The numerical experiments demonstrate the effectiveness of the 2DSBEBDF method in solving stiff initial value and oscillatory problems over the existing stiff solver found in the literature.

Soft set theory provides a mathematically robust and algebraically versatile framework for modeling systems characterized by epistemic indeterminacy, vagueness, and parameter-dependent variability—features that pervade foundational inquiries in decision theory, engineering, economics, and the information sciences. At the heart of this formalism lies an extensive suite of algebraic operations and binary product constructs that collectively confer a rich internal structure upon the universe of soft sets, capable of faithfully representing intricate parametric interrelations. Within this conceptual setting, we introduce and rigorously investigate a novel soft product, referred to as the soft union–theta product, defined over soft sets whose parameter spaces are endowed with an intrinsic group-theoretic structure. The operation is meticulously axiomatized to ensure compatibility with generalized soft subsethood and equality relations, thereby preserving the formal algebraic integrity of the resulting system. A comprehensive algebraic analysis is undertaken to characterize the operation’s fundamental properties—including closure, associativity, commutativity, idempotency, and interactions with identity and absorbing elements—as well as its behavior in relation to the null and absolute soft sets. In parallel, the proposed product is analytically juxtaposed with existing soft binary operations within the stratified hierarchy of soft subset classifications, offering deeper insights into their relative expressive capacities and mutual structural coherence. Our results affirm that the soft union–theta product respects the algebraic constraints imposed by group-parameterized domains while generating a coherent and structurally consistent algebraic system over the space of soft sets. Two core algebraic contributions emerge from this study: (i) the integration of this product fortifies the internal operational harmony of soft set theory by embedding it into an axiomatic framework that preserves and extends fundamental algebraic behaviors; and (ii) the operation lays the groundwork for a generalized soft group theory, wherein soft sets over group-based parameter domains replicate the axiomatic signatures of classical group structures under suitably defined soft operations. By addressing the critical need for algebraic operations grounded in semantically meaningful and structurally sound axioms, this work significantly advances the algebraic unification and generalization of soft set theory. Beyond its theoretical depth, the proposed operation enables the construction of abstract algebra-driven soft computational models, with direct implications for multi-criteria decision-making, algebraic classification mechanisms, and uncertainty-sensitive data analysis over group-structured semantic spaces. Consequently, the algebraic apparatus formulated herein not only extends the theoretical boundaries of soft algebra but also solidifies its foundational relevance in both abstract mathematical logic and applied analytical domains.

We aimed to develop a conjugate gradient method by reformulating parameters so that iterative optimization techniques perform more effectively. Instead of using a set of basic conjugate gradient formulas, the methodology introduces a new parameter that produces better convergence. The method is applied in Fortran to test how many iterations and how many evaluations of the function are needed, as listed in table 1. The behavior of convergence and the results used for comparison are created with Matplotlib on Python and Ggplot2 on R programming for the chart. We like to compare our method to the proven LS (Liu-Storey) method when checking how effective our proposed method is. We found that the technique provides better results with lower iteration numbers and better convergence speeds for many test cases, proving it can challenge conventional methods in optimizing problems without constraints.

Soft set theory constitutes a mathematically robust and structurally versatile formalism for modeling real-world systems characterized by epistemic uncertainty, vagueness, and parameter-contingent variability—ubiquitous features across decision theory, engineering, economics, and information sciences. At the core of this framework lies a spectrum of algebraic operations and binary product constructions that endow the soft set universe with a rich internal structure, capable of encapsulating intricate interdependencies among parameters. In this context, we introduce and investigate a novel product of soft sets, termed the soft intersection–symmetric difference product, formulated specifically for soft sets whose parameter domains are structured as groups. This product is rigorously defined and analyzed within an axiomatic framework that ensures compatibility with generalized soft subsethood and equality relations. The structural analysis of the soft intersection–symmetric difference product includes the examination of essential algebraic properties—such as closure, associativity, commutativity, and idempotency. In addition, the interplay between this product and pre-existing soft products, is explored to regarding the subsets. Theoretical investigations reveal that the operation not only respects the algebraic architecture of the underlying group-parameterized domain but also induces a cohesive and well-behaved algebraic system on the collection of soft sets. This analytical framework yields two central algebraic insights: (i) the internal algebraic cohesion of soft set theory is significantly enhanced by embedding the newly defined product into a logically sound and operation-preserving environment; and (ii) the product itself possesses the formal potential to serve as a foundational construct for a generalized soft group theory, wherein soft sets over group-parameter spaces mimic the axiomatic behavior of classical group-theoretic constructs through suitably defined soft operations. Given that the maturation of soft algebraic systems is contingent upon the rigorous formulation of operations satisfying structurally meaningful axioms, the contributions of this study represent a notable advancement in the algebraic consolidation of soft set theory. Beyond theoretical enrichment, the proposed operation offers tangible utility in the construction of abstract algebra-based soft computational models, with applications spanning multi-criteria decision-making, algebraically-driven classification mechanisms, and uncertainty-aware data analysis governed by group-parametrized semantic domains. Thus, the framework established herein not only extends the theoretical boundaries of soft algebra but also fortifies its role as a foundational tool in both pure and applied mathematical discourse

This study investigates the impact of Cabri 3D (C3D) simulations, a virtual reality-based dynamic geometry software, on Senior High students’ academic performance in 3D geometry concepts. The research addresses the persistent challenges students face in understanding abstract spatial relationships and geometric principles through traditional teaching methods. Grounded in constructivist theory, the study posits that immersive, interactive learning environments can enhance conceptual understanding, spatial visualization skills, and problem-solving abilities. A quasi-experimental design was employed, involving 122 SHS2students from a public Senior High School in Ghana, divided into an experimental group (n=69) taught usingC3D simulations and a control group (n=53) instructed via conventional methods. Pre-test and post-test assessments measured students’ performance before and after the intervention. Descriptive and inferential statistics (independent sample t-tests) were used to analyse the data. The results revealed a statistically significant improvement in the experimental group’s post-test scores (M=53.12, SD=5.635) compared to the control group (M=39.30, SD=4.432) with a p-value of 0.002 (p<0.005). This finding supports the hypothesis that C3D simulations enhance students’ comprehension of 3D geometry by fostering engagement, motivation,and spatial visualization skills. The study aligns with prior research, highlighting the efficacy of technologyintegratedlearning in mathematics education. The study recommends the adoption of Cabri 3D and similardynamic geometry tools in SHS curricula to address learning gaps in abstract mathematical concepts. Schoolsare encouraged to equip ICT labs with relevant resources and train educators to leverage these technologieseffectively. Future research could explore long-term retention and scalability of such interventions acrossdiverse educational contexts.

This paper presents the formulation of higher order diagonally implicit 2-point super class of block extended backward differentiation formula (2DSBEBDF) for solving first order stiff initial value problems. The order of the 2DSBEBDF method is derived and found to be four. The Stability analysis of the method shows that the method is zero-stable and its absolute stability region shows that the method is A-stable within the stiff stability interval -1≤ρ<1. The numerical experiments demonstrate the effectiveness of the 2DSBEBDF method in solving stiff initial value and oscillatory problems over the existing stiff solver found in the literature.

A new classification of functions is presented in this work in relation to ordinary differential equations. A function for which the n’th derivative of it can be expressed in terms of linear combinations of lower derivatives and the function itself is defined as the linear differential function where n is the lowest derivative for such a relation to hold. The functions that do not obey the rule are defined as nonlinear differential functions. The properties of the differential functions are discussed through theorems and examples. One elementary application is the method of undetermined coefficients where the non-homogenous function should be a linear differential function. Variable coefficient equations are also treated within the context of linear differential functions. The approximation of nonlinear differential functions in terms of linear differential functions are discussed. An application to the perturbation solutions is given. The ideas and definitions presented in this work will add to the understanding of differential equations and their solutions as functions.

This article will begin with the claim that Hamilton spent a great deal of time trying to figure out the three-dimensional complex numbers. He was never able to accomplish that.

Complex numbers are in the form a+bi where ‘a’ is a real part and ‘bi’ an imaginary with 𝑖= √−1 The motive behind the claim is that both 𝑖2 and 𝑗2= −1; The failure may also have been caused by the lack of a proper definition for the field of complex numbers. To address this issue, the author of this article offers his own definition of the field of complex numbers with key vectots i,j,k taking values (-1,0,1) respectively Of course the field of complex numbers remains unchanged with 𝑋 ⃗⃗⃗ =𝑥+𝑦𝑖+𝑧𝑗 under transformation becoming 𝑋 ⃗⃗⃗ =𝑥𝑖+𝑦𝑗+𝑧𝑘 vector but it has it’s correspondence in the field of real numbers and it’s a vector (-1,0,1) The entire process of transition between fields, in author’s opinion, is possible thanks to the matrix of transformation It’s form has already been explored earlier on by the same author in his ‘Ternary Mathematics and 3D Placement of Logical Elements Justification’.

Professor Juan Weisz (Doctor of Philosophy, Northeastern University, Argentina) generously proposed the concept of field transition, which allows for conversion between imaginary and real numbers without altering the field’s structure or the relationships between its constituent parts. In essense it should work for all entries just the same way it works for the entries of real numbers The fact that serves as the proof is in 𝐴′𝐴=1 and it works well for both Real and Imaginary number fields which is what we are aiming to prove.

This study evaluates the performance of various fuzzy membership functions (MFs) in predicting volume and bedload rate using sediment data from a bathymetric survey at Ikpoba Dam. Twelve cases with different membership functions: Gaussian, triangular, trapezoidal, and bell-shape were tested across different epochs. The models were assessed based on Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²) values for both training and testing datasets. The Gaussian membership function (Gaussmf), with 7 membership functions and 200 training epochs, outperformed the others, achieving the lowest RMSE of 0.568 (training) and 0.579 (testing), MAE of 0.437 (training) and 0.445 (testing), and highest R² values of 0.914 (training) and 0.932 (testing) for volume prediction. For bedload rate, it also achieved the lowest RMSE of 0.509 (training) and 0.517 (testing), MAE of 0.391 (training) and 0.397 (testing), and highest R² values of 0.9354 (training) and 0.9496 (testing). In contrast, the Trapezoidal membership function (Trapmf) showed the worst performance with RMSE values of 0.874 (training) and 0.905 (testing), MAE values of 0.652 (training) and 0.677 (testing), and R² values of 0.812 (training) and 0.804 (testing). These results emphasize the significance of membership function selection and training epochs in optimizing fuzzy models for environmental and geospatial applications.

Aims/ Objectives: The aim of the article is to provide the proof for use of trits in coding of the signal for Ternary processors The author substantiates the aforementioned use by formula and properties of Ternary Algebra In order to do so an analysis of the incoming signal is done and the wave is presented in the form of 𝑓(𝑥)=𝑎 sin⁡(bx+C)+D By converting this formula we obtain numeric value of the energy conveyed by the standing wave The same numeric value is the mathematical representation of a signal in the Ternary circuits (Pierce, 1973)