Journal of Statistics and Mathematical Engineering

Element Invertibility and Structural Equivalence: A Task Analysis for Introductory Algebra

V. Rangaveni — Sat, 19 Sep 2026 00:00:00 +0000

This paper examines a recurring difficulty in introductory algebra: the word inverse is used both for an element within a structure and for a map between structures. Although these ideas are related, they answer different mathematical questions. A transformation may be irreversible, while the monoid to which it belongs can still be isomorphic to another monoid. Similarly, every element of a group may have an inverse even when a homomorphism from that group is not one-to-one. Using a two-state transformation system, this theoretical commentary develops a connected set of examples covering semigroups, monoids, groups, subgroups, homomorphisms, and isomorphisms. The examples include reversible and irreversible transformations, a semigroup without identity, an isomorphism produced by relabeling states, modular reduction, and a bijection that does not preserve an operation. These contrasts clarify the distinctions among element invertibility, operation preservation, and bijectivity. The paper proposes a task-analysis framework that identifies the mathematical evidence required for each claim, such as an inverse, a homomorphism condition, or a counterexample. The aim is to support more precise teaching and assessment of algebraic structures in second-year engineering mathematics and introductory abstract algebra.

A Unified Disparate Impact Ratio (DIR) Framework for Data Science: Assessing Fairness, Model Fit, and Predictive Risk

Padma Lochan Pradhan, Sandeep Dwarakanath Pande — Sat, 03 Oct 2026 00:00:00 +0000

The study focuses on how the increasing adoption of data science, machine learning, and artificial intelligence in decision-making has created a need for evaluation methods that address not only predictive performance but also fairness and prediction-related risk. This study proposes a Unified Disparate Impact Ratio (DIR) Framework for Data Science that integrates three complementary dimensions: fairness, model fit, and predictive risk. The framework uses DIR to examine differences in favorable outcome rates between a focal group and a reference group, while incorporating conventional model-fit measures, including accuracy, precision, recall, F1-score, AUC, and calibration. Predictive risk is assessed through false-positive rates, false-negative rates, error disparities, and uncertainty-related indicators. The proposed framework represents model evaluation as a multidimensional vector (UDIR = [DIR, MF, PR]), enabling simultaneous examination of outcome disparities, predictive effectiveness, and subgroup-specific prediction risks. A performance life-cycle model is further proposed to support data preparation, model development, fairness assessment, validation, deployment, and continuous monitoring. The framework emphasizes that DIR should be interpreted alongside model performance and predictive-risk measures rather than treated as a standalone determination of fairness. The proposed approach provides a structured foundation for fairness-aware, performance-aware, and risk-aware data science and can be extended across domains including finance, healthcare, education, employment, insurance, cybersecurity, and public-sector analytics. Future empirical research can validate the framework using real-world datasets, multiple algorithms, dynamic monitoring, explainable AI, and causal fairness analysis.

Testing Multivariate Exponentiality Against MHNBUE Alternatives - A Laplace-Transform Statistic with Closed-Form Null Variance for Arbitrary Dimension

A. Touseef Ahmed, B. Towfeeq Ahmed, Zahiruddeen — Mon, 05 Oct 2026 00:00:00 +0000

This article develops a -dimensional Laplace-transform test of the hypothesis that a non-negative random vector has independent exponential margins, against the multivariate harmonic new better than used in expectation (MHNBUE) alternative. Setting , the practitioner standardizes to before applying the statistic. For the standardised observations, this paper constructs a test statistic from the -fold Laplace transform of the multivariate equilibrium function, derives a closed-form expression for its null variance valid for every dimension , prove a -dimensional Laplace-uniqueness characterisation of the null hypothesis, and establish its asymptotic normality. To the best of the authors’ knowledge, no previously published test for the MHNBUE alternative has been established for existing work covers only Monte Carlo upper and lower critical values, Pitman asymptotic efficacy, and empirical power are evaluated for against linear failure rate, Makeham, and Weibull alternatives. Numerical evidence shows that positive Clayton dependence pushes the statistic negative at a rate increasing with dimension; motivated by this, a pre-specified lower-tailed test is applied to the cross-validated American Football League scoring-times data, yielding rejection at the 5% significance level, interpreted as departure from bivariate exponentiality driven primarily by positive dependence rather than by marginal ageing.

A Fuzzy Logic Framework for AI Decision-Making Under Uncertainty

ravi jayapal — Fri, 09 Oct 2026 00:00:00 +0000

Artificial intelligence systems are increasingly required to operate in complex, dynamic environments characterized by incomplete, imprecise, ambiguous, and conflicting information. Classical binary logic and purely probabilistic models frequently prove inadequate when confronted with linguistic uncertainty, graded membership of concepts, and the need for human-like approximate reasoning. This paper presents a comprehensive fuzzy-logical framework specifically designed for AI-based decision-making under uncertainty. The framework systematically integrates fuzzy set theory, multi-style fuzzy inference systems (Mamdani, Sugeno, and Tsukamoto), and hybrid neuro-fuzzy architectures to support robust, transparent, and adaptive decision processes. This study formalizes the representation of multiple forms of uncertainty, details the complete inference pipeline from fuzzification through defuzzification, and introduces adaptive learning mechanisms for membership functions and rule weights. A formal decision model is derived that ranks alternatives using defuzzified scores, possibility measures, and fuzzy ranking indices. An experimental study on a simulated multi-criteria medical risk-assessment task and an autonomous navigation scenario demonstrates that the proposed framework achieves higher interpretability and competitive predictive performance compared with pure neural baselines, while remaining data-efficient through the incorporation of expert linguistic knowledge. The approach enhances transparency, resilience, and human–AI collaboration in domains such as autonomous systems, clinical decision support, financial risk assessment, and intelligent control.

Simulation Functions and Fixed-Point Theorems in Orthogonal Bi-b-Metric Spaces with Applications to Volterra Integral Equations

Gajendra Singh Dangi, Uma Shankar Singh — Tue, 13 Oct 2026 00:00:00 +0000

Fixed point methods provide a useful framework for studying existence and uniqueness problems in nonlinear analysis. This work presents the development of common fixed-point results for two self-mappings defined on orthogonal bi-b-metric spaces. The proposed framework combines an orthogonality relation, a pair of b-metric structures, a simulation function, and an implicit contractive relation. Under suitable completeness, continuity, and preservation assumptions, the study establishes the existence and uniqueness of a common fixed point. The proof is constructed through an orthogonal iterative sequence and a Cauchy-type argument adapted to the bi-b-metric setting. A numerical example is included to illustrate the assumptions of the main theorem and to demonstrate how the simulation-function condition can be verified. As an application, the fixed point result is used to study a system of nonlinear Volterra integral equations. The associated operators are formulated on an appropriate space of continuous functions, and the hypotheses of the fixed point theorem are used to obtain existence and uniqueness of a solution. The proposed formulation brings together orthogonality, dual b-metric structure, and simulation functions in a single framework and provides a basis for further work on generalized metric spaces and nonlinear integral or fractional equations.