International Journal of Mathematical Analysis and Research
International Journal of Mathematical Analysis and Research (IJMAR) is a scientific, double-blind, peer-reviewed international journal that typically focuses on advanced mathematical theories and techniques. In the area of mathematical analysis, the journal aims to offer a venue for the publishing of innovative research. The journal aims to foster intellectual exchange among mathematicians, researchers, and practitioners worldwide by showcasing the latest developments, innovative methods, and theoretical advancements in mathematical analysis. IJMAR aspires to contribute to the growth and dissemination of knowledge in mathematical analysis seeks to advance and spread novel concepts, approaches, and solutions, bridging the gap between pure and applied mathematics. It also seeks to highlight interdisciplinary connections, showcasing how advanced mathematical analysis can be employed to solve problems in science, engineering, computer sciences and technology.
Mathematical Statistics Approaches for Explainable Artificial Intelligence Models
Explainable Artificial Intelligence (XAI) has emerged as a critical research area for improving the transparency, interpretability, fairness, and reliability of machine learning systems. As artificial intelligence models become increasingly complex, especially deep learning architectures, understanding the statistical foundations behind model behavior becomes essential for trustworthy decision-making. This paper explores the role of mathematical statistics in developing and evaluating explainable AI models. The study examines statistical inference, probability theory, hypothesis testing, Bayesian methods, regression analysis, feature importance estimation, uncertainty quantification, and causal inference as fundamental tools for interpretability. Furthermore, the paper discusses how statistical techniques contribute to local and global explanations, model validation, fairness assessment, and robustness analysis. Several widely used XAI methods, including SHAP, LIME, partial dependence plots, and counterfactual explanations, are analyzed from a statistical perspective. The paper also highlights challenges such as interpretability–accuracy trade-offs, bias propagation, high-dimensional data complexity, and uncertainty in explanations. Finally, future directions involving probabilistic explainability, statistical learning theory, and human-centered AI are presented. This work aims to bridge the gap between mathematical statistics and explainable artificial intelligence by providing a comprehensive framework for statistically grounded and trustworthy AI systems.
Explainable Artificial Intelligence (XAI)
Copyright (c) 2026 https://creativecommons.org/licenses/by-nc-nd/4.0
Deep Learning-Based Numerical Solutions for Nonlinear Partial Differential Equations in Fluid Dynamics
Nonlinear partial differential equations (PDEs) play a vital role in modeling complex fluid dynamics phenomena such as turbulence, heat transfer, and flow instability. Traditional numerical approaches, including finite difference and finite element methods, often require extensive computational resources and may struggle with large-scale or real-time simulations. Recent advancements in deep learning have introduced efficient alternatives for solving nonlinear PDEs through data-driven and physics-informed approaches. This paper presents a comprehensive study of deep learning-based numerical solutions for nonlinear PDEs in fluid dynamics. The proposed framework integrates neural network architectures with physics informed constraints to accurately approximate fluid behavior while reducing computational complexity. Various deep learning models, including Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and Fourier Neural Operators, are analyzed and compared with conventional numerical techniques. Experimental results demonstrate improved prediction accuracy, faster convergence, and enhanced scalability in solving complex fluid flow problems. The study further discusses practical applications, current limitations, and future research opportunities in AI-driven scientific computing for fluid dynamics.
Nonlinear Partial Differential Equations
Physics-Informed Neural Networks (PINNs)
Computational Mathematics Approaches for Dynamic Inventory Optimization
Dynamic inventory optimization has become a critical research area in modern supply chain management due to fluctuating market demand, uncertain supply conditions, and increasing operational complexity. Computational mathematics provides powerful techniques for modeling, analyzing, and optimizing inventory systems in dynamic environments. This paper explores various computational mathematics approaches for dynamic inventory optimization, including linear programming, integer programming, dynamic programming, stochastic modeling, and metaheuristic algorithms. The study also examines the integration of machine learning and predictive analytics for improving demand forecasting and adaptive inventory control. A mathematical framework is proposed to minimize inventory-related costs while maintaining high service levels and operational efficiency. Experimental analysis and comparative evaluations demonstrate the effectiveness of computational optimization methods in reducing holding costs, avoiding stock shortages, and improving decision-making accuracy. The paper further discusses practical applications in retail, manufacturing, healthcare, and e-commerce sectors, along with current challenges and future research opportunities in intelligent inventory management systems.
Development of a Reduced Fuzzy Soft Set Mathematical Framework for Transforming Fuzzified Acoustic Voice Data into Parameterized Diagnostic Structures
This paper analyses the development of a reduced fuzzy soft mathematical framework for modelling and detecting vocal disorder risk using clinical data. The paper presents a mathematically grounded fuzzy soft expert system for the quantitative assessment of vocal disorder risk based on acoustic and demographic parameters. Let = { denote a finite universe of patients and denote the parameter set comprising fundamental frequency, vocal perturbation index, and age. Clinical acoustic data were collected and represented as a dataset Linguistic variables were obtained by the process of fuzzification using the appropriate designed membership functions assessment model. For each, a fuzzy membership function is constructed using continuous triangular membership function to represent linguistic classifications. The fuzzified linguistic variables were further transformed into soft set using discrete α-cut levels. Discrete α-cut levels are employed to generate parameterized soft sets, which are subsequently reduced using redundancy elimination criteria derived from soft set theory. The reduced parameter sets preserve essential diagnostic information while minimizing computational complexity. The resulting mathematical framework provides a systematic method for organizing acoustic information without enforcing binary classification. The mathematical formulation preserves uncertainty, enables modelling of multiple parameter interaction and prepares the dataset for advanced diagnostic inference. The mathematically technique is consistent, scalable, and suitable for biomedical applications involving imprecise data.
Exploring Fuzzy Associative Models: A Fuzzy Logic Approach to Handling Complex, Uncertain Data
Fuzzy Associative Models (FAMs) leverage fuzzy logic to effectively represent uncertain and imprecise knowledge in a compact, interpretable form. By accommodating the inherent ambiguity of real-world data, FAMs excel in applications where relationships between variables are complex, nonlinear, and uncertain. This makes them a powerful tool for handling real-world complexities, enabling more nuanced decision-making and problem-solving in fields like AI, data analysis, and beyond.
Genetic Algorithm Optimized Fuzzy N-Policy Queue Using α-Cut based Performance Analysis
In real-life queueing systems, many parameters such as arrival rate and service rate are often uncertain or imprecise. To handle such uncertainty, fuzzy set theory provides an effective framework for modelling queueing systems. In this paper, an N-policy M/M/1 queueing model is studied in a fuzzy environment where system parameters are represented as fuzzy numbers. The α-cut approach is employed to transform the fuzzy parameters into interval values, allowing the evaluation of system performance measures such as the expected queue length and waiting time. To obtain the optimal operating policy, a Genetic Algorithm (GA) is applied to determine the optimal threshold value of that minimizes the total system cost. The proposed method combines fuzzy modeling with evolutionary optimization to improve decision-making under uncertainty. Numerical illustrations and graphical analysis are provided to demonstrate the effectiveness of the proposed approach. The results show that the integration of α-cut based fuzzy analysis with Genetic Algorithm optimization provides a flexible and efficient method for determining optimal policies in uncertain queueing environments.
Data-Driven Software Quality Assurance: Leveraging Machine Learning for Risk Prediction and Test Optimization
Software quality assurance is undergoing a structural transition from manually orchestrated verification toward continuously adaptive, data-driven assurance. Contemporary software systems evolve under tight release cadences, heterogeneous architectures, volatile workloads, and increasing security exposure, making uniform testing strategies economically unsustainable and technically inefficient. This paper develops a literature-grounded research framework for data-driven software quality assurance that integrates machine learning based risk prediction with intelligent test optimization. The central argument is that modern quality engineering should no longer treat defect prediction, regression testing, observability, and governance as separate activities. Instead, these capabilities should operate as a closed feedback system in which code, process, runtime, and business-context signals are fused to prioritize the highest-risk components and allocate testing effort dynamically. Building on prior work in software defect prediction, test prioritization, observability, AI governance, enterprise automation, and domain-specific intelligent systems, the paper proposes a multi-layer architecture that links feature engineering, risk scoring, test selection, feedback learning, and operational controls. It further shows that patterns demonstrated in cloud-native pharmacy, healthcare, manufacturing, finance, and cyber-physical settings provide transferable insights for software assurance, particularly around traceability, secure microservices, anomaly monitoring, and deployment optimization. The paper contributes a unified conceptual model, a risk-to-test prioritization formulation, an implementation-oriented pipeline, and a research agenda for robust, explainable, and operationally grounded machine learning in quality engineering. The framework is intended for practitioners and researchers seeking to improve defect discovery efficiency, reduce waste in testing, and align software verification with real delivery risk.
A Mixed-Integer Linear Programming Model for Green Vehicle Routing Problems
The growing environmental impact of transportation and logistics has prompted increasing attention to sustainable routing practices. This study addresses the Green Vehicle Routing Problem (G-VRP) by developing a Mixed-Integer Linear Programming (MILP) model that simultaneously minimizes travel distance, fuel consumption, and carbon emissions while satisfying operational constraints such as vehicle capacity, route feasibility, and time windows. The proposed model is tested on benchmark instances and evaluated against traditional vehicle routing solutions, demonstrating substantial reductions in fuel use and environmental emissions without compromising service efficiency. Computational experiments highlight the trade-offs between operational efficiency and environmental sustainability, offering actionable insights for fleet managers seeking to implement eco-friendly routing strategies. The results also identify the scalability challenges and potential for integrating heuristic methods in larger transportation networks. This research contributes to both the theoretical understanding of green routing optimization and practical applications in sustainable logistics, providing a foundation for future studies on multi-objective, real-time, and electric vehicle routing systems.
Copyright (c) 2025 https://creativecommons.org/licenses/by-nc-nd/4.0
A detailed proof of the incorrectness of standard field theory (vector analysis) is proposed. The correct methodological basis for this proof is the unity of formal logic and rational dialectics. The unity of formal logic and rational dialectics is the only correct criterion of truth. The proof leads to the following irrefutable statement: standard field theory (vector analysis) is a gross error. Gross errors are as follows: (1) field theory is based on differential and integral calculus, which is an incorrect theory; (2) field theory is based on vector calculus, which is an incorrect theory; (3) field theory is formulated within the framework of a geometric coordinate system. But mathematical and physical quantities have no dimension “meter” and cannot be presented (be defined, exist) within the framework of a geometric coordinate system; (4) in the point of view of formal logic and dialectics, the concepts “field in abstracto” and “mathematical field” are identical and meaningless concepts. “Field in abstracto” and “mathematical field” have neither physical properties nor geometric properties; (5) the standard definition of a field is: “A field is a part of space, each point of which corresponds (conforms) to a certain value of some physical quantity”. In the point of view of formal logic, the term “correspondence” is meaningless. The subject-predicate conjunction in a definition must be either “is” or “is not”. Replacement of a dimensionless quantity in mathematical definitions and expressions by a dimensional quantity is an inadmissible operation; 6) a physical field has no points (i.e., values of a physical quantity) in a geometric coordinate system. A material point has coordinates, but a material point is not a point of a physical field. The error is in the assertion that the coordinates of a material point determine (define, identify) the value of a physical quantity; that the value of a physical quantity determines (defines, identifies) the coordinates of a material point; (7) the absurdity is that a dimensionless (i.e., mathematical) quantity is identical to a dimensional (i.e., physical) quantity.
Total Edge Irregularity Strength of Some Graphs
An edge irregular total k-labeling of a graph G = (V,E) is a labeling of vertices and edges of G in such a way that for any two different edges uv and u1v1 their weight and are distinct. The total edge irregularity strength, tes(G), is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we study the total edge irregularity strength of joint-Wheel graph WHn, < Wn : Wm > and path union of graph.
(1, 2)*-Weakly Delta Generalized Beta-Compactness and Connectedness in Bitopological Spaces
In this chapter, the concepts of (1,2)*-weakly delta generalized beta-compactness and (1,2)*-weakly delta-generalized beta-connectedness are introduced, and some characterizations of (1,2)*-weakly delta-generalized beta-compactness and (1,2)*-weakly delta-generalized beta-connectedness are investigated. Also, their properties are discussed. Further, it is proven that (1,2)*-weakly delta-generalized beta-connectedness is preserved under (1,2)*-weakly delta-generalized beta-irresolute mappings.
In this paper, we introduce a new labeling edge total mean labeling. An edge total mean labeling f: V ∪ E → {1, 2, ..., p + q} of a graph G = G(V, E) is a labeling of vertices and edges of a graph in such a way that for any two different edges uv and u' v' their mean and are distinct and the result in edge total means varies from 1,2,. . . ,q. A graph G is edge total mean graph if it admits edge total mean labeling. In this paper, we introduce a concept of edge total mean labeling of some graphs.
Multiobjective Optimization in Fractional Calculus: Theoretical and Computational Advances
Multiobjective optimization in fractional calculus has made important developments, mainly due to the addition of fractional operators to complex optimization methods. Some recent research has applied fractional differential-difference operators to generalize traditional optimization ideas, forming new groups of functions known as local fractional Univex functions. As a result, people can build strong, weak, and converse duality theorems for multiobjective fractional problems, which provide advancement in solving mathematically complex non-convex and non-smooth optimization problems. Robust optimization methods have been applied to multiobjective fractional programming to consider uncertain data and to introduce conditions called ε-optimality and robust ε-saddle points for weakly efficient solutions. With regard to computation, using Branch-and-Cut methods has made it possible to speed up optimization on linear fractional functions by removing inadequate solutions from efficient sets of integer quadratic problems. Such numerical methods have made it easier to solve fractional derivatives with great precision and now support applications in wave propagation, models of viscoelastic materials, and machine learning. New optimization techniques make it easier to solve multiple challenges by lowering computational effort without sacrificing the accuracy of results. All these changes together join theory and practice, giving new tools to areas where mismatched objectives and fractional mathematics are important.
Mathematical Modelling of Quantum Information Systems Using Operator Algebra Techniques
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Analytical Framework for Hybrid Systems Using Discrete-Time Markov Chain Approaches
Continuous or discrete dynamics with their interaction have been called hybrid systems and are ubiquitous in the design of many complex systems today, including cyber-physical systems, automated control systems and real-time embedded systems. The twofold nature of such state evolution presents these systems with peculiar problems of verification, control, and analysis. Also in this paper, I present an analytical model that uses the Discrete-Time Markov Chains (DTMCs) to model, analyze and predict the behaviour of the hybrid systems. DTMCs offer a probabilistic modeling framework which is capable of describing the stochastic dynamics of discrete hybrids efficiently and in a very elegant manner. The framework suggests a new approach to discretizing the ongoing dynamics and incorporating them in a DTMC-based analysis framework. The suggested approach is building transition probability matrices, a probabilistic state-space description and verification with the help of model checking techniques. The framework also uses a method of partitioning the continuous state space, resulting in abstract states which become an input to the DTMC model. The relevance of the framework is proved, inter alia, by several case studies of building temperature control systems, robotic navigation, and fault detection of automated systems. The results of the simulation depict the high level of accuracy and efficiency of the simulations compared to the traditional methods of hybrid system solvers. The key contributions of the paper can be listed as the follows: (1) A generalization of DTMC-based modeling strategy to hybrid systems, (2) A probabilistic abstraction approach to continuous systems, (3) Probabilistic verification approaches using PCTL (Probabilistic Computation Tree Logic), and (4) A case study evaluation. The paper is a contribution towards the reconciliation of deterministic modeling of hybrid systems with probabilistic verification methods and is of use when put to practice by engineers and researchers working in the field of reliability and performance analysis of systems.
A Multi-Scale Computational Study on the Fracture Mechanics of Fiber-Reinforced Geopolymer Composites for Seismic Applications
GPCs (geopolymer composites with fibers) are now being recognized as smart replacements for traditional cement products, especially in seismic situations that need more fracture resistance and energy absorption. To investigate the fracture behavior of fiber-reinforced GPCs, this study examines them through micro-scale simulations, meso-scale analysis and large-scale simulations. There is an assessment of steel, glass, polypropylene and hybrid fibers to understand their effect on how the matrix interacts, how Cracks Bridge and the failure patterns during cyclic loading. Experimental results are used to test if the simulation accurately predicts tensile strength, fracture energy and crack development. The impacts of fiber alignment, content and interface bonding on both stress and damage are simulated at a micro-scale. Meso-scale modeling points out that fiber pullout and bridging improve crack resistance, whereas macro-scale analysis measures how well structural elements deal with load and seismic pressure. The work includes using parametric analysis to explore the effects that different types of fibers have on the structure and proves the connection between its microstructure and how the structure functions. Data shows that fiber-reinforced GPCs perform much better under fracture than traditional concrete, making them suitable for earthquake-proof structures. Based on the findings, engineers can improve and design better sustainable composite materials used in civil engineering.
A Hybrid Analytic-Numeric Framework For Optimization In Constrained Multivariate Systems
This paper explains a method that uses both calculation and analysis to handle optimization problems in restricted multi-variable systems. By combining global stochastic methods with local algorithms, the framework takes advantage of the benefits of Bayesian optimization for global searching and interior-point algorithms for smooth convergence. Gaussian process regression is used to form surrogate models, which allows for exploring the design space probabilistically, while IPOPT is applied to refine candidate solutions that look promising. Making use of both global and local approaches helps resolve the obstacles of each (such as large costs for global and limited success for local approaches). Through case studies on both constrained Ackley functions and industrial energy systems, it is shown that the proposed framework performs exceptionally well, as it is 40% faster than traditional strategies. Penalty terms are used to process nonlinear constraints, and the method also uses nesting to reduce dimensions, allowing for fast processing of big data. These results support the view that Q-learning succeeds in managing how it learns new behaviors versus how it takes advantage of learned strategies in domains with both nonconvex goals and traces of boundaries.