The Babylonian Journal of Mathematics (BJM) E-ISSN: 3006-113X, serves as a testament to ancient mathematical heritage. Embracing the legacy of Babylonian mathematical achievements, this journal invites global contributions, aiming to extend and honor the rich mathematical traditions of the Babylonian civilization through modern research and discoveries.
Global convergence guarantees for adaptive gradient algorithms with Barzilai–Borwein and alternative step-length strategies
Motivated by recent progress in adaptive schemes for convex optimization, this work develops a proximal-gradient framework that enforces global convergence without resorting to linesearch procedures. The proposed approach accommodates widely used step-length rules, including Barzilai–Borwein updates and one-dimensional Anderson-type acceleration. Importantly, the analysis applies to problems where the smooth component admits only local Hölder continuity of its gradient. The resulting theory unifies and strengthens several existing results, while numerical experiments confirm the practical benefits of coupling aggressive step-length selection with adaptive safeguarding mechanisms.
Copyright (c) 2026 Babylonian Journal of Mathematics
Gradient-Regularized Deep Ritz Networks for Solving Elliptic PDEs: Application to the Poisson Equation
In this work we propose a gradient-regularized Deep Ritz Network (GR-DRN) for solving elliptic partial differential equations, which we shall concretely apply to the Poisson equation on the unit square domain. We then propose a method that generalizes the classical Deep Ritz formulation by adding a gradient-penalty term to achieve smoother solution and lower non-physical oscillations during training. The model is evaluated against three benchmark datasets that are analytically generated, representing a smooth solution, a case with high-frequency oscillatory solution, and a variable-coefficient setting that reflects heterogeneous diffusion. Our numerical results show that GR-DRN consistently obtains lower L² and H¹ errors than the standard Deep Ritz method, especially on the difficult problems with high-frequency or variable-coefficient. It also produces more accurate gradient fields and stable convergence shift. Overall this shows that gradient regularization is an easy but effective improvement to neural variational solvers for elliptic PDEs.
Deep Ritz Method; Gradient Regularization,
Physics-Informed Neural Networks for Solving PDEs Using Gradient-Regularized Loss Functions: Application to the Nonlinear Burgers’ Equation
Through the use of loss function restrictions within neural networks, Physics-Informed Neural Networks (PINNs) have emerged as a potent method for solving partial differential equations (PDEs) by incorporating physical rules into the learning stages. However, when the residuals and provided data are out of balance or when the collocation network noise overstates appropriateness, the stability and convergence rate for PINNs are frequently low. In order to increase the precision and stability of PINN training, this study offers a qualified examination of enhanced loss formulations using gradient regularization. In order to control physical solutions and overfitting, the suggested gradient-regularized loss imposes a smoothness penalty on network parameters that is comparable to the classical regularization of Tikhonov mothed. Improved convergence robustness and improvements lower the mean squared residual error to the standard PINN baseline, according to the case studies.
Approximate Solution of Two-Dimensional Linear Systems of Fractional Order Partial Integro-Differential Equations Using Variational Iteration Method
The main objective of this paper is to introduce and solve two-dimension linear system of fractional order partial integro-differential equations using the most well-known approximation methods, which is the variational iteration method. After that, the convergence of the derived sequence of iterative approximations to the exact solution is proved, which is must be assumed to be exist according to existence and uniqueness theorem of partial integro-differential equations. Two illustrative examples are considered utilizing the computer software (PTC Mathcad) and then comparison between the exact and the approximate solutions for a test examples are given to show the efficiency and reliability of the proposed method.
Copyright (c) 2025 Babylonian Journal of Mathematics
Graph-Theoretic Characterizations of Quasi-Idempotents in Full Order-Preserving Transformation Semigroup
This paper presents a digraph-theoretic extension of the characterization of quasi-idempotent in the semigroup On of full order-preserving transformations on a finite chain. Building on earlier results . that describe quasi-idempotent as those transformations α ∈ On satisfying α≠α^2=α^4, we provide a novel interpretation using the functional digraphs of such maps. We show that a transformation is quasi-idempotent if and only if each vertex in its associated digraph is either fixed or maps directly into a fixed point, and every non-trivial strongly connected component forms a 2-cycle. Furthermore, we prove that no directed path of the form v1 → v2 → v3 exists where all vertices are non-stationary. These findings offer a new perspective on the structure of On, bridging algebraic properties with graphical structure, and set the stage for visual and computational analysis of quasi-idempotent generation in transformation semigroups.
Some Expansions to The Weibull Distribution Families with Two Parameters: A Review
This study is concerned with reviewing the different expansions of the two parameters Weibull distribution families, which are considered popular probability distributions used in many fields, such as reliability analysis, systems engineering, and fault studies. The research aims to provide a comprehensive review of the scientific literature related to these expansions, focusing on the theoretical frameworks, practical applications, and limitations facing each type of these expansions. The main points of this study are summarized in a mathematical review of methods by which the expansions were developed to improve their suitability to various data and discussing the improvements that were added to address the problems that arise when applying the traditional version of the distribution. Integrating these expansions with the exponential distribution as a baseline to discuss the performance of each expansion, a Monte Carlo simulation was presented for each expansion and choose the best expansion using some criteria this is to calculate the efficient of estimating parameters for each expansion. As for the practical application on real data, these expansions were applied to real data to decide the issues of performance accuracy in practical application, and compare the performance between different expansions using measures such as statistical accuracy and ease of interpretation.
Spectral Realization of the Nontrivial Zeros of the Riemann Zeta Function via a Hermitian Operator Framework
We present a spectral construction of a Hermitian operator whose spectrum coincides exactly with the imaginary parts of the nontrivial zeros of the Riemann zeta function The operator, denoted H∞, is defined on a discrete geometric space modeled by a 20-vertex dodecahedral graph, incorporating a discrete Laplacian, an entropy-based coherence potential, and a prime-indexed infinite-order algebraic term derived from Infinity Algebra. We show that H∞ is self-adjoint, spectrally complete, and compatible with the analytic continuation and functional symmetry of ζ(s). A spectral determinant constructed from its eigenvalues matches the Hadamard product representation of ζ 1 + it , and no extraneous roots appear off the critical line. Numerical approximations from a truncated version of the operator validate this correspondence. The construction yields a functional-analytic framework that supports a spectral- theoretic resolution of the Riemann Hypothesis
Existence and Uniqueness Theorem of Multi-Dimensional Integro-Differential Equations With Fractional Differointegrations
The aim of this paper is to establish the existence and uniqueness theorems of multi-dimensional conformable fractional partial integro-differential equations theorem for generalized contraction mappings with respect to w-distances in complete metric spaces.
Enhanced Parameter Estimation for the Modified Gompertz-Makeham Model in Nonhomogeneous Poisson Processes Using Modified Likelihood and Swarm Intelligence Approaches
This research introduces a new method to estimate Nonhomogeneous Poisson Processes (NHPP) parameters through integration of Modified Gompertz-Makeham Process (MGMP) with statistical methods and optimization techniques. The Modified Maximum Likelihood Estimator (MMLE) constitutes our proposed estimation solution because traditional Maximum Likelihood Estimation (MLE) proves inadequate when analyzing complex systems and precise parameter assessment particularly under conditions of low sample counts or nonlinear conditions. The method adds structural constraints together with supplementary information to the likelihood function to achieve enhanced inference accuracy. The highly non-linear likelihood equations require solution through the implementation of Particle Swarm Optimization (PSO) as a biologically based metaheuristic algorithm which shows excellence when dealing with challenging parameter spaces. The combination of MMLE with PSO results in a hybrid framework which demonstrates its performance against standard techniques through simulated data series and factory failure records taken from the Badush Cement Factory. The MMLE-PSO method proves its superior accuracy levels proven by tests utilizing RMSE, MSE, PRR and PP metrics. The study presents compelling evidence for the combination of statistical models with intelligent optimization methods which improves reliability studies and survival analytics and failure prediction. The research findings present statistical validity alongside computational feasibility which provides an effective parameter estimation methodology for changing stochastic processes.
Correcting Convexity Proofs for Two Signomial Functions in Geometric Programming
Geometric programming (GP) is a powerful framework for optimizing posynomial and signomial functions, which are widely used in engineering design, economics, and other fields. A critical aspect of solving geometric programs efficiently lies in understanding the convexity properties of the functions involved, as convexity ensures that local minima are also global minima. This work focuses on correcting and verifying the convexity proofs for two specific signomial functions in geometric programming: f 1 (x)=x 1 2 x 2 −1 +x 2 3 −x 1 x 2 and f₂(x) = x₁³x₂⁻² + x₁⁻¹x₂² - x₁x₂⁻¹. Through a detailed analysis, we demonstrate that f1(x) is **not convex** over the domain (x1,x2)> 0, as its Hessian matrix fails to be positive semi-definite for all positive values of x1and x2. For f2(x), the convexity **cannot be conclusively determined** based on the analysis, as the logarithmic transformation does not universally preserve convexity for signomial functions with negative coefficients. Specific cases suggest local convexity, but a general proof for all (x1,x2)> 0 remains elusive. These results highlight the challenges in analyzing signomial functions and emphasize the importance of rigorous convexity verification in geometric programming. The findings have significant implications for optimization feasibility and modeling considerations, particularly in applications where signomial functions with negative coefficients arise. This work provides a foundation for further research into advanced techniques for analyzing and optimizing such functions, ensuring more robust and reliable solutions in geometric programming.
An Analytical Approach to Bimonotone Linear Inequalities and Sublattice Structures
This paper presents a comprehensive analytical approach to the study of bimonotone linear inequalities and their relationship with sublattice structures in Rn. Bimonotone inequalities, which are linear constraints characterized by coordinate-wise monotonicity, naturally arise in optimization problems, economics, and combinatorial geometry. We explore the geometric properties of convex sublattices and demonstrate how they can be efficiently represented as the solution sets of systems of bimonotone inequalities. By analyzing the algebraic and geometric structures of these sublattices, we provide new insights into their behavior under various operations, such as intersection and projection. Additionally, the paper discusses the application of these concepts in optimization, game theory, and machine learning, where monotonicity constraints are commonly employed. The results contribute to a deeper understanding of how bimonotone inequalities define convex sets with lattice structures, offering a valuable tool for solving high-dimensional optimization problems with monotonicity constraints
An Enhanced Strategy for Computing Minimum Path and Cut Sets to Improve Complex Network Reliability
Improving the reliability of complex systems is an urgent requirement due to their wide applications in various sciences. This study presents the development of two methods to compute minimum paths and minimum cut sets analytically using the connectivity matrix. The proposed method is characterized by its simplified procedures free of complex computational procedures that may be subject to errors, in addition to the speed of completion. The accuracy of calculating these paths ensures that the reliability value of the system components can be extracted and thus facilitates the optimization process with a reliable result. Moreover, the proposed method was applied to two types of complex systems and the results were very accurate. Finally, this method opens up prospects for improving the reliability of highly complex systems more smoothly and with promising results.
Chemical Reaction and Activation Energy Impacts on Unsteady Radiative Flow of Sisko Fluid
In this work, we study the two-dimensional boundary layer flow of Sisko fluid over stretching surface. Further we have observed, the effect of chemical reaction and activation energy on unsteady radiative flow of Sisko fluid. The suitable transformations are used to convert non-linear partial differential equations (PDEs) into ordinary differential equations (ODEs). Numerical solutions are obtained for the velocity, temperature and concentration profiles by utilizing the bvp4c technique. The effect of various parameters on the heat transfer and concentration profiles are depicted in the graphs and tables are briefly discussed.
Fuzzy Soc–Small Two-Absorbing Modules and Related Concepts
This study presents the fuzzy socle small two-Absorbing module and compares it to other fuzzy modules, including fuzzy small prime, fuzzy small two-Absorbing, and fuzzy two-Absorbing. Numerous aspects found during the investigation and discussion supported the new ideas, which fall within the fuzzy hollow, fuzzy chain, and fuzzy multiplication module classes. Discuss the connection between this concept and the fuzzy socle small two-Absorbing ideal. To create a new fuzzy socle small two-Absorbing, these findings are required.
Copyright (c) 2024 Babylonian Journal of Mathematics
New Generalized Extended Hardy-type Dynamic Inequalities on Time Scales
This research will demonstrate several novel dynamic inequalities of the Hardy type on time scales. These inequalities will be generalized and expanded. Several new integral and difference inequalities are produced as a consequence of our findings in both continuous and discrete scenarios. The dynamic Holder ¨ inequality, the integration by parts formula on time scales, and Keller’s chain rule on time scales will be utilized to demonstrate the primary findings. To apply the primary findings, we shall employ discrete calculus, quantum calculus, and continuous calculus and treat them as special cases.
This work found the circular segmentation (CS) for , and 19 to (2, ), after we find the (RRCT) for each group and diagonal the matrix of this (RRCT) if we suppose that the terms of this basic diagonal are then the (CS) is . Let vector void on F, indicate whole linear isomo. of V upon same, a representation of G for representation void is a homo. of to . A representation model of is a homo. of to , where is the degree of the representation model. Whole invertible ´ model form a group on a field F indicate ( ,F). A homo. of GL(n,F) to F-{0} is the determinant of these model, ( ,F) indicate the kernel of it. Thus ( ,F) is a subgp. of ( ,F) include whole models of determinant 1 on F.
The Weyl module resolution studied by Buchsbaum where the Weyl module is the image of the Weyl map for the skew-partition and is a free module defined on a commutative ring with identity; where runs over all partitions . There are a number of classical formulas that express the formal character of the representation in terms of standard symmetric polynomials. Such formulas are also valid for the more general representation modules associated to skew partition ; where , where the set of all irreducible polynomial representations of general linear group of degree is described by the module The reduction from the terms of the characteristic-free of Weyl module resolution to the terms of the Lascoux resolution found in this work for the partition (9,8,3) by using the boundary maps and prove that the sequence of the reduction terms is exact.
Let be a free module defined on the commutative ring with identity. Buchsbaum studied the Weyl module resolution where the Weyl module is the image of the Weyl map for the skew-partition ; where runs over all partitions . There are a number of classical formulas that express the formal character of the representation in terms of standard symmetric polynomials. Such formulas are also valid for the more general representation modules associated to skew partition ; where , where the set of all irreducible polynomial representations of general linear group of degree is described by the module For the partition (9,9,3) by applying the boundary maps we reduction the terms of the characteristic-free of Weyl module resolution to the terms of the Lascoux resolution and prove that the sequence of the reduction terms is exact.
The set of all irreducible polynomial representations of general linear group of degree is described by the module ; where runs over all partitions . There are a number of classical formulas that express the formal character of the representation in terms of standard symmetric polynomials. Such formulas are also valid for the more general representation modules associated to skew partition ; where . Let be a commutative ring with identity and a free -module. The Weyl module resolution studied by Buchsbaum where the Weyl module is the image of the Weyl map for the skew-partition . Reduction the terms of the resolution of the characteristic-free of Weyl module to the terms of the resolution of Lascoux by employing the boundary maps for the partition (9,6,3) and prove that the sequence of the reduction terms is exact.
The collection of whole - account grade maps respecting a restricted group G of commutative group cf (G,Z) beneath spot wise addendum. Into these one group own - account generalized characters a subgroup indicate R(G). Whole invertible ´ model form a group on a field F indicate ( ,F). A homo. of GL(n,F) to F-{0} is the determinant of these model, ( ,F) indicate the kernel of it. Thus ( ,F) is a subgp. of ( ,F) include whole models of determinant 1 on F. Let vector void on F, indicate whole linear isomo. of V upon same, a representation of G for representation void is a homo. of to . A representation model of is a homo. of to , where is the degree of the representation model. From the rational representations character table (CTRR) we compute the cyclic decomposition (CD) for , and 7 to (2, ).
Investigation of Bounded Modules and Some related Concepts
The definition of a class of module as well as submodule namely (bounded module and bounded submodule) has introduced in various ways. In this study, we concentrate on some comparisons and analyzations of bounded module’s properties and its relation specifically with prime module and other types of modules and submodules such as: scalar, multiplication, and cyclic modules. In addition, we scan the behavior of this concept under some conditions in order to reach other types of modules and given some counterexamples for several unsatisfying relationships. Finally, we add some new results about some types of modules that are related to bounded modules and open a new path that encourage other researchers about this topic.
Fuzzy Semi and Fuzzy Strongly Semi Two-Absorbing Second Sub-modules
In this work, fuzzy semi Two-Absorbing second sub-modules and fuzzy strongly semi Two-Absorbing second sub-modules are presented. The connections between fuzzy semi prime (fuzzy strongly semi prime) second sub-modules, fuzzy quasi prime (fuzzy strongly quasi prime) second sub-modules, and fuzzy Two-Absorbing (fuzzy strongly Two-Absorbing) second sub-modules are also covered. Here are some fundamental characteristics and attributes of these ideas.
Fuzzy Two-Absorbing (fuzzy strongly Two-Absorbing) second sub-modules
Fuzzy semi prime (fuzzy strongly semi prime) second sub-modules
Fuzzy quasi prime (fuzzy strongly quasi prime) second sub-modules
Fuzzy Classical Two-Absorbing Second (Secondary) Sub-modules and Strongly Classical Two-Absorbing Second Sub-modules
Fuzzy classical Two-Absorbing second sub-modules, Fuzzy classical Two-Absorbing secondary sub-modules and strongly classical Two-Absorbing second sub-modules are presented in this search under the names of comultiplication fuzzy modules, cocyclic fuzzy modules, and multiplication fuzzy modules, along with some basic traits and qualities of these concepts. The relationship between fuzzy classical Two-Absorbing second sub-modules, fuzzy classical Two-Absorbing secondary sub-modules, fuzzy strongly classical Two-Absorbing second sub-modules, and other fuzzy second sub-module concepts is also covered.
fuzzy classical Two-Absorbing secondary sub-modules
fuzzy classical Two-Absorbing sub-module
In this paper, the notion of nearly Endo quasi prime sub-modules is introduced together with several fundamental features and theorems. It provides several properties and descriptions of these ideas, and additional diagrams and examples are used to show how the study's findings translate into nearly Endo quasi prime and nearly Endo prime sub-modules
Modified Technique to Solve Degeneracy in Linear Programming Problems
This paper addresses degeneracy that has been analyzed in the simplex method in Linear programming (LP) issues. The newly changed technique is proposed for choosing particular pivot rows from leaving variables. This method offers greater beneficial and faster consequences in the assessment of the prevailing classical approach. The proposed set of rules is a higher preference to avoid the confusion of taking an arbitrary ratio to pick out leaving variables and for this reason, the proposed method is robust to solve degeneracy in linear programming (LP) problems.
Bio-Convective Thermally Radiated Casson Fluid Model PDEs Past over a Darcy-Forchheimer Porous Stretched Sheet via OHAM Strategy
The current study examines the topic of continuous flow of nanofluids over two-directional boundary level employing Casson heat transmission across a linearly stretched sheet. Firstly, the partial differential equations are transformed into non-linear ordinary differential equations with the help of similarity parameters. These non-linear ordinary differential equations are solved with the given boundary conditions by applying the BVPh2.0 method on Mathematica software. The effects of magnetic impact, radiation parameter, porosity number, Brownian motion parameter, thermophoresis parameter, Casson fluid parameter, Schmidt number, Prandtl number, peclet number, bioconvection on Velocity, temperature and concentration profiles is observed. It is noted that the concentration and temperature profiles increase by expanding values of thermophoresis parameter also the temperature increase by increment in Brownian motion while a reverse result obtained on concentration profile. Also, the influence of Casson fluid, thermophoresis and Brownian motion on skin friction, Sherwood number and Nusselt number is noted and check the behavior of these numbers by increasing or decreasing values of thermophoresis parameter, Casson fluid parameter and Brownian motion. And, calculated that by increment in thermophoresis and Brownian motion, the Nusselt number decreases. The graphs of temperature profile, velocity profile and concentration profile are drawn and also other results are tabulated.
On commutativity of alternative rings with [xy^n x±yx^n y,x]=0
Let R be a n-torsion free with identity 1 ,In this article we investigate and prove the commutativity of alternative ring of the property(p_1 ), (p_2) and(p_3 ) under suitable constraints. (p_1 ) [xy^n x±yx^n y,x]=0 (p_2 ) [xy^n x±yx^n y,y]=0 (p_3) [x(〖xy)〗^2+(〖xy)〗^2 x,x] = 0. ∀ x,y ∈R.
Restricted visible submodules and fully restricted visible module
In this article, the concept of Restricted visible (for short; Res-visible) submodule and fully restricted visible (for short fully Res-visible) module have been introduced which are considered the generalizations of the concepts of visible submodules and fully visible module respectively where every visible submodule (fully visible module) is Res-visible submodule (fully Res-visible) module, but the converse is not be true. Examples have been presented illustrating those relationships.
Optimizing Network Reliability in Dynamic or Rapidly-Changing Topologies
With communication, power and transportation networks facing ever-greater dynamics, optimizing reliability poses significant modeling and computational challenges. Most existing reliability optimization research assumes static topologies with predefined failure probabilities and connectivity demands. This work puts forth a time-varying mathematical program to maximize expected network reliability under shifting topological uncertainties. Sets, parameters, decision variables and constraints are defined as functions of time to capture variability in links, capacities, risks and demands. The formulation adapts to detected changes through re-optimization triggered by model error thresholds. Approximation techniques based on constraint sampling and decomposition address solve efficiency for large fluid networks. Evaluations on simulated dynamic test cases demonstrate superior reliability versus periodic and static optimization approaches while fulfilling budget limits. Accuracy metrics assess model fidelity over increasing volatility levels. Implementation case studies exhibit optimized resilience in software-defined communication architectures, smart grid reconfiguration, and adaptive transportation maintenance scenarios. The mathematical programming foundation provides a pathway to achieve connectivity resilience for critical infrastructure networks facing intensifying dynamics. The integration of optimization, prediction and adaptive response provides a paradigm for decision making under modern uncertain conditions.
Copyright (c) 2023 Babylonian Journal of Mathematics
Biomathematics and its Applications in Modeling COVID-19 Infections
The emerging field of biomathematics utilizes mathematical techniques to gain quantitative insights into biological systems. Mathematical modeling and computational simulations are becoming increasingly vital tools for elucidating complex infectious disease dynamics. The COVID-19 pandemic has highlighted the applicability of biomathematical methods for informing public health policy decisions. Here we provide an overview of major modeling approaches used to analyze SARS-CoV-2 transmission, forecast COVID-19 trajectories, evaluate intervention strategies, and understand viral evolution. Key model types discussed include compartmental models, agent-based models, and immunological models. We highlight critical model parameters and how their estimation from noisy data presents core challenges. The continued integration of biological mechanisms and epidemiological evidence through iteratively refined models promises more realistic representations of the intricate system interactions underlying the COVID-19 pandemic. While assumptions and uncertainties pervade, mathematical abstractions have already provided invaluable guidance for real-time decision-making in this global health emergency. Collaborative development of versatile biomathematical frameworks for emerging infectious diseases can equip policymakers with prescient, science-based insights for navigating future outbreaks.