Punjab University Journal of Mathematics
Punjab University Journal of Mathematics (PUJM) is a distinguished publication of the Institute of Mathematics, University of the Punjab, Lahore. It is indexed in the Master Journal List – Emerging Sources Citation Index (ESCI). In the Journal Citation Reports by Thomson Reuters 2023, PUJM achieved an impressive impact factor of 1.0, reflecting its rising prominence and scholarly contribution in the field of mathematical sciences. The journal is published on a monthly basis and is dedicated to disseminating high-quality research in mathematics. The official abbreviated title, as assigned by the ISSN authority, is Punjab Univ. J. Math. The first issue of the journal was published in 1967 by Prof. Dr. Syed Manzur Hussain (Late). Article Processing Charges: It is pertinent to mention that from January 2025 the following Article Processing Charges (APC) will be implemented on every article accepted for publication in PUJM. For International Authors: $100 For Local Authors: Rs.15000/- per paper For Local and Foreign Joint Authorship: $100 Prof. Dr. Ghazala Akram, PhD & PostDoc (Editor in Chief) Institute of Mathematics, University of the Punjab, Lahore, Pakistan. Dr. Saima Arshed, PhD (Managing Editor) Institute of Mathematics, University of the Punjab, Lahore, Pakistan.
On the Elliptic Sombor and Euler Sombor indices of corona product of certain graphs
Graph theory has wide-ranging applications in chemistry, nano science and networking problems. The corona product operation effec tively represents compounds like dendrimers, where a central molecule is encircled by multiple repeating units. By representing atoms as vertices and bonds as edges, graph theory provides a clear and structured way to analyze and understand the architecture of intricate chemical compounds through various graph operations. The corona product of two graphs G and H gives a new graph obtained by taking one copy of G and |V (G)| copies of H by joining ith vertex of G to every ith copy of H. Elliptic Sombor and Euler Sombor indices are recently defined topo logical indices using Sombor index. Elliptic sombor index is defined as ESO(G) = uv∈E(G)(du +dv) d2 u +d2 v and Euler Sombor index is defined as EU(G) = uv∈E(G) d2 u + d2 v + dudv, where du and dv are degrees of vertices u and v in graph G. In this article, we compute the Elliptic Sombor and Euler Sombor indices of few resultant graphs using the operations join and corona product on standard graphs like path, cycle and complete graphs.
Copyright (c) 2024 Kirana B, Shanmukha M C, Usha A https://creativecommons.org/licenses/by/4.0
Computation of RDM-Polynomial, Operators, and Topological Descriptors for HexDerived Networks
In this paper, we formulate a reverse degree RDM-Polynomial for graphs, along with differential operators and integral operators derived from it. Reverse degree topological indices are also calculated using the computed RDM-Polynomials. Numerical invariants known as topological descriptors (TD’s) are essential for describing the molecular topology of a particular molecular graph. This method has been utilized to develop topological descriptors based on reverse degree. The present study analyses and computes the relevant indices for triangular and rectangular type-3 hex-derived networks. The results of this study provide researchers and academics with new insights that pave the way for further research and related discoveries.
Copyright (c) 2024 Abdul Rauf Khan, Saad Amin Bhatti, Yilun Shang https://creativecommons.org/licenses/by/4.0
Picard-Tikhonov-Mann (P TM) iteration process with applications
In mathematics, a fixed point of a function is the process of finding a solution to an equation that can be written as Υx = x for a suitable function Υ. Fixed point theory has many applications. For instance, in compilers (computer programs), fixed point computations are used for program analysis. An example of this is the data-flow analysis that is often required to optimize code. The vector of PageRank values of web pages is the fixed point of a derived linear transformation of the world wide web connectivity structure. Our main focus in this paper is to consider a novelalgorithm for finding the fixed point in some abstract spaces. Some applications are presented regarding split feasibility/constrained minimization problems/signal enhancement. Furthermore, we perform the numerical illustrations to investigate the basic techniques using Matlab R2016a. The proposed novel algorithm demonstrates strong convergence to fixed points of the considered mapping, with practical applications in solving split feasibility and constrained minimization problems. Numerical results confirm the effectiveness of the method in signal enhancement tasks. This approach offers promising potential for further research in abstract metric spaces and iterative solution techniques.
Copyright (c) 2024 Esra Yolac¸an https://creativecommons.org/licenses/by/4.0
On the Numerical Evaluation for Studying the Time Fractional Partial Delay Differential Equations
An important challenge in mathematical modeling is to find a model that captures the fundamental physics of a system and is simple enough to allow for mathematical analysis. In the physical sciences,physiology, ecology, and other practical research domains, fractional delay differential equations (FDDEs) are frequently used. Most fractional delay differential equations can only be solved numerically because they lack analytic solutions. In contrast to the Adomian decomposition method, a newmethodfor solving delay differential equations called the new Fractional Novel Analytical Scheme (FNAS) is presented in this paper. A fractional novel analytical scheme is built on the fractional Taylor series. The calculation is done using the Caputo derivative. Three well-known physical models such as advection-dispersion equation of fractional-order, nonlinear gas-dynamics equation of fractional-order and convection-diffusion equation of fractional-order with proportional time-delay are solved by using the proposed technique to demonstrate the performance and efficiency the FNAS. By graphing absolute error values and contrasting results to numerous existing solutions, the correctness of the proposed technique is presented. In addition to being straightforward, the suggested strategy is accurate and logical in the difficulties it solves
Fractional Partial Delay Differential Equations
Fractional Novel Analytical Scheme (FNAS)
Copyright (c) 2024 Uroosa Arshad , Mariam Sultana https://creativecommons.org/licenses/by/4.0
Study of ternary nanoparticles in Oldroyd-B fluid using ethylene glycol
This paper presents the expansion of an analytical model for the performance of Oldroyd-B nanofluid, where ethylene glycol is assumed as the base fluid and copper (Cu) tri-nanoparticles are introduced for nanofluid preparation. A systematic strategy is established by converting the related nonlinear governing equations into dimensionless form. To find analytical solutions for the research problem, the Laplace transformation is used as a key tool. This research includes a review of the effects of important variables on the temperature and velocity patterns inside thefluid. In particular, for greater values of the volume fraction of nanoparticles, the temperature is enhanced while a decline in velocity is observed. For comparison is done by choosing different base fluids like kerosene and ethylene glycol and velocities are tested at various temperatures and velocities. The profile can be enhanced for kerosene. Comparative analysis is done by comparing the obtained results to those produced by the mathematical software Mathcad.
Copyright (c) 2024 Muhammad Haris Butt, Muhammad Imran Asjad, Muhammad Armaghan Sadiq, Abid Hussanan https://creativecommons.org/licenses/by/4.0
An Introduction to Dynamic Soft Sets: A Framework for Modeling Temporal Uncertainty
In engineering, healthcare, and decision-making systems, effectively handling uncertainty remains a fundamental research challenge. Existing frameworks, including fuzzy sets, rough sets, and soft sets, typically operate on static parameter spaces with fixed approximation mappings, which limits their ability to model real-world environments where relevant parameters and their interpretations evolve over time. To address this limitation, this paper introduces a dynamic extension of soft set theory and makes the following principal contributions. (1) A formal definition of Dynamic Soft Sets is proposed, in which both the active parameter subsets At ⊆ E and the approximation mappings Ft : At → P(U) are explicitly indexed by time or context t ∈ T, enabling the representation of parameters that activate, deactivate, or change relevance over time as a natural extension of Molodtsov’s soft sets. (2) This framework is further generalized to Dynamic Hypersoft Sets, allowing time-varying multi-attribute Cartesian product parameter spaces At = At1 ×···×Atn, thereby extending classical hypersoft sets to dynamic environments. (3) A complete set-theoretic operator framework for Dynamic Soft Sets is developed, including union, intersection, complement, restricted and extended operators, as well as logical AND/OR operations, all defined in a time-dependent setting with fundamental algebraic properties established. (4) A systematic comparative analysis is presented to distinguish Dynamic Soft Sets from classical soft sets and hypersoft sets, highlighting differences in parameter dynamics and structural flexibility through struc tured tables. (5) The practical applicability of the proposed framework is demonstrated through a healthcare monitoring case study, showing how time-varying sensor availability and evolving clinical conditions can be naturally modeled using Dynamic Soft Sets. Unlike classical soft sets with static parameters and hypersoft sets with fixed Cartesian product domains, Dynamic Soft Sets introduce explicit time- or context-indexing of both parameter subsets and approximation mappings, achieving dual structural and temporal dynamism. The proposed framework provides a unified andmathematically rigorous foundation for time- and context-adaptive uncer tainty modeling and decision-making in dynamic real-world applications.
Copyright (c) 2026 Muhammad Saeed https://creativecommons.org/licenses/by/4.0
Enhanced multi-criteria decision-making through fuzzy soft set parameter reduction and score optimization
Parameter reduction is a crucial task in multi-criteria decision making (MCDM), particularly when dealing with high-dimensional and uncertain data. Fuzzy soft set (FSS) theory, which integrates the param eterization capability of soft sets with the uncertainty-handling strength of fuzzy sets, provides an effective framework for such problems. In this paper, we focus on parameter reduction in FSS and its impact on score-based decision-making. Existing approaches, including S-normal and I-S-normal parameter reduction methods, often lead to identical or indistinguishable decision scores and involve considerable computational complexity. To overcome these limitations, a new and efficient decision making algorithm is proposed within the fuzzy soft set framework, yielding unique and more discriminative scores after parameter reduction. The proposed algorithm is applied to illustrative examples and real-life decision making problems, demonstrating improved accuracy and reduced com putational effort. A comparative analysis with the TOPSIS method further confirms the effectiveness and reliability of the proposed approachfor MCDMunder uncertainty.
Copyright (c) 2026 Tabasam Rashid, Muhammad Amman, Asif Ali https://creativecommons.org/licenses/by/4.0
Intuitionistic Fuzzy Credibility Implicative Ideals in BCK-algebra for Efficiency Evaluation, Supply Chain and Risk Analysis
This study proposes a robust multi-criteria decision-making (MCDM)frameworkforefficiencyevaluation, supplychain, andrisk anal ysis based on intuitionistic fuzzy credibility sets (IFCSs) and intuitionis tic fuzzy credibility implicative ideals (IFCIIs) in BCK-algebras. An in tegrated LOPCOW–SWARA–ERUNS framework is developed to handle decision-making under uncertainty. The proposed approach is applied to evaluate leading companies, including Amazon, Walmart, and Alibaba, by considering key factors such as cybersecurity, geopolitical instability, en vironmental impacts, economic conditions, technological aspects, social issues, and logistics-related risks. Conventional MCDM methods often struggle to address uncertainty and subjective bias, resulting in unreliable outcomes for complex decision problems. To overcome these challenges, SWARAisemployed to determine subjective weights, while LOPCOW is used to compute objective weights. The ERUNS method is then applied to rank the alternatives, ensuring a comprehensive and reliable evaluation. Sensitivity and comparative analysis are conducted to assess the robust ness of the proposed framework. The results confirm that the proposed method significantly enhances MCDM accuracy by effectively managing uncertainty.
Copyright (c) 2026 Laraib Hayat, Asim Naseem, Muhammad Riaz https://creativecommons.org/licenses/by/4.0
A note on the Fekete-Szego ¨ Conjecture for Usual Subclasses of Univalent Functions
One of the most fundamental problems in the theory of univa lent functions is the Bieberbach problem, also known as the Bieberbach conjecture, which claims that the modulus of the nth Taylor coefficient of a function f in class S is bounded by n (that is, f ∈ S, n ≥ 2 (n ∈ N = {1,2,3,...}), |cn| ≤ n). In general, studies on the Bieberbach conjec ture, the proof of which occupied mathematicians for a long time, have evolved—as a result of the intellectual efforts in the field—from certain bounds on the modulus of individual Taylor coefficients to the investi gation of bounds on the modulus of functionals formed by combinations of these coefficients. In the present article, the Fekete–Szeg¨ o conjecture, which involves one such important functional, is considered, and certain bounds are determined for a parameter η with 0 ≤ η ≤ 1
Copyright (c) 2026 Alaattin Akyar https://creativecommons.org/licenses/by/4.0
Numerical Investigation with Stability Assessment of Semi-Analytical Scheme for Time-Fractional Order Heat Type Emden-Fowler Equations
In the present paper, time-fractional order linear and nonlinear heat type Emden-Fowler equations are reformulated from existing classical equations by applying Caputo-Fabrizio time-fractional derivative. Then, a semi-analytical scheme, that is an amalgamation of Laplace trans formation and Picard’s iterative technique, is exploited to simulate singular initial value problems for corresponding time-fractional order heat type Emden-Fowler equations. Further, the stability of developed scheme is also assessed by exploiting R-stable mapping and Banach contraction principle. Numerical results, error estimation, and comparison of obtained results with exact solutions are presented through graphs and tables to exhibit the efficiency of time-fractional order derivative and implemented semi-analytical scheme.
Caputo-Fabrizio time-fractional deriva tive
Copyright (c) 2026 Saif Ullah, Noor Fatima https://creativecommons.org/licenses/by/4.0
A Multi-Criteria Decision-Making Technique Based on Generalized Bipolar Fuzzy Prioritized Operators and Their Application in Supply Chain Management
The evaluation and assessment of AI-driven models for supply chain management require complex decision-making frameworks that can cope with uncertainty, indecision, and conflicting expert opinions. The expressiveness of traditional fuzzy decision-making models is limited and is not able to represent both positive and negative sides of the criteria. In addition, existing generalized aggregation operators (AOs) rely on arbitrary weights of criteria that introduce subjectivity and reduce the accuracy of the findings. This paper addresses these shortcomings by suggesting a new multi-criteria decision-making (MCDM) method based on generalized bipolar fuzzy prioritized operators (G-BFPR). We propose four new operators, namely, generalized bipolar fuzzy prioritized average (G-BFPRA), generalized bipolar fuzzy prioritized weighted average (G-BFPRWA), generalized bipolar fuzzy prioritized geometric (GBFPRG), and generalized bipolar fuzzy prioritized weighted geometric (G-BFPRWG) operators. This framework, in a systematic manner, computes weights using priority relations, eliminating subjectivity and reflecting on positive and negative preferences in uncertain settings. Our approach provides a context-specific, balanced assessment mechanism of AI-based supply chain models, which is demonstrated by a case study that validates the superiority of our approach to the current theories. Theproposed bipolar fuzzy MCDM method offers a holistic solution to the ranking of optimization techniques in supply chain management, which enhances operational performance, minimizes costs, and enhances sus tainability. Furthermore, the needs and benefits of the proposed work are disclosed in this article, as it includes a comparative analysis.
Copyright (c) 2026 Ubaid ur Rehman, Abaid ur Rehman Virk, Tahir Mahmood, Hafiz Muhammad Waqas https://creativecommons.org/licenses/by/4.0
AHolomorphic Bundle Criterion for Compactness of Pseudoconcave Solvmanifolds
Weprovethatapseudoconcavecomplexhomogeneousspaceof a connected solvable linear algebraic group is necessarily compact. This resolves a central conjecture in the theory, showing that pseudoconcavity characterizes compactness for this large class of solvmanifolds. The proof combines new pluriharmonic obstructions for C∗-bundles with the structure theory of solvable groups, demonstrating that any noncompact solvmanifold admits a nonconstant pluriharmonic function, contradicting pseudoconcavity. Our result unifies and extends all known partial classifications for this class of solvable linear algebraic groups, establishing pseudoconcavity as a definitive geometric property that forces compactness in the solvable setting.
Copyright (c) 2026 Raheel Farooki https://creativecommons.org/licenses/by/4.0
Mathematical Modeling for Transmission Dynamics of Hepatitis B Virus
Hepatitis B virus (HBV) remains a major global public health concern, motivating the use of mathematical models to better understand its transmission dynamics and control strategies. In this study, a compart mental mathematical model based on a system of linear differential equa tion is formulated to describe the spread of HBV in a population. The basic reproduction number R0 is derived to characterized the transmis sion potential of the disease. Analytical results show that the disease-freeequilibrium (DFE) is locally asymptotically stable when R0 < 1 and become unstable when R0 > 1, while the endemic equilibrium (EE) existsand is stable for R0 > 1. To investigate the dynamic behavior of the model numerically, both the standard finite difference (SFD) scheme and a non-standard finite difference (NSFD) scheme are implemented. The SFDscheme exhibits conditional convergence and may produce nonphysical results for certain step sizes, whereas the proposed NSFD scheme preserves essential qualitative properties of the continuous model, including positivity and stability of solutions. A stability analysis of the NSFD scheme is also presented. Numerical simulations and comparative analysis of both schemes validate the theoretical findings and demonstrate the superior performance of the NSFD method in accurately capturing the transmission dynamics of HBV.
Copyright (c) 2026 Shah Zeb, Muhammad Bilal, Muhammad Rafiq, Siti Ainor Mohd Yatim, Affan Ahmad, Muhammad Irfan, Muhammad Sarwar Ehsan https://creativecommons.org/licenses/by/4.0
Decision Analysis Using Energy of Neutrosophic Hesitant Fuzzy Soft Sets
In this paper, we introduce a novel hybrid structure called the neutrosophic hesitant fuzzy soft set (NHFSS), which integrates the flex ibility of hesitant fuzzy information with the expressive capacity of neutrosophic and soft set theories. Motivated by the need to handle multiple layers of uncertainty in decision-making problems, we propose a newframework for modeling such environments. Building upon this structure, we define two complementary measures: the pessimistic energy and the optimistic energy of an NHFSS, which capture lower-bound and upper bound contributions of the hesitant and neutrosophic elements, respec tively. These measures are then synthesized to derive the total energy of the NHFSS. We develop a decision-making algorithm based on this energy characterization and apply it to a real-world example to demonstrate its effectiveness. A comparative analysis is conducted to highlight the advantages of the proposed approach over existing models. Our findings suggest that the energy-based treatment of NHFSSs provides a robust and discriminative framework for multi-criteria decision-making under complex uncertainty.
Copyright (c) 2026 Milica Dabi´ c, Nenad Stojanovi https://creativecommons.org/licenses/by/4.0
Innovative Insights and Fractal-View of Phi-Four Equations with Time Fractal Derivative
In this work, a new analytical technique recognized as the homotopy perturbation method is utilized to identify the general solutions of Phi-4 nonlinear partial differential equations of fractional order with respect to time. The first order general solution is determined by HPM and then compared with the exact result. It can be seen that the result obtained using HPM shows a large convergence rate for Phi-4 partial differential equations with time fractal derivative. The exact graphic solutions have been shown, and their respective errors have also been represented usingtables.
Fractal Phi-four equation with respect to time
Riemann-Liouville fractional integral operator
Copyright (c) 2026 Rashida Qayyum Khan, H. M. Younas, Shauket Ali Tahir https://creativecommons.org/licenses/by/4.0
Insights into Engineering Shapes through Curve and Surface Modeling withthe 4-Point Quaternary Subdivision Scheme
Curve and surface modeling is an essential area of computer graphics that involves the creation of complex geometries and shapes used in various industries and engineering disciplines, such as automotive, architecture, and entertainment. Subdivision schemes are among the best known and most widely applied methods for generating smooth shapes. Convexity and monotonicity are important geometric properties in shape design for engineering applications, as they help ensure that shapes are structurally sound, manufacturable, and exhibit desirable performance characteristics. While the smoothness of a shape is measured in terms of the order of continuity, preserving geometric properties such as convexity and monotonicity remains a challenging task. Keeping this in view, a 4-point quaternary subdivision scheme is proposed, and its geometric properties are analyzed for a certain range of parameters. Several examples, includ ing real-life and engineering-based graphical objects, are presented on curves and surfaces to demonstrate the smoothness and shape-preserving capability of the proposed scheme.
Copyright (c) 2025 Muhammad Tahseen Iqbal, Rakib Mustafa, Ghulam Mustafa, Humaira Mustanira Tariq, Samsul Ariffin Abdul Karim https://creativecommons.org/licenses/by/4.0
Entropy-Driven Decision-Making for Cybersecurity Risk Assessment using PULq-ROFS and the CODAS Technique
Multi-criteria group decision-making (MCGDM) is a significant procedure because it facilitates and enhances decision-making (DM) by incorporating diverse variables and professional viewpoints, producing more adequate results. A set of flexibility to manage confusion and un-predictable information is improved by adding probability which makes choices more predictable. The present work suggests a technique that utilizes fuzziness to handle MCGDM issues that frequently arise in cybersecurity risk assessment. This strategy overcomes the fundamental difficulties in privacy and security information by using the probabilistic uncertain linguistic q-rung orthopair fuzzy set (PULq-ROFS). Comparedwith other fuzzy collections, such as statistical tentative, linguistically in-tuitionistic, and linguistically Euclidean imprecise collections that effec-tively include erratic and non erraticproblems, the PULq-ROFS provides multiple characteristics. To advance this framework, we propose two new operators the PULq-ROF einstein weighted average (PULq-ROFEWA) and PULq-ROF Einstein order weighted average (PULq-ROFEOWA) that efficiently integrate the statistical lexical choice data. These operations introduce a creative concept within the PULq-ROF context. Furthermore, by applying the entropy technique, we determine the relative weights of parameters based on their informational contribution to the study. In addition, we employ the combinative distance-based assessment (CODAS) method to evaluate choices according to their distance from the least op-timal solution, thereby ensuring a more accurate and reliable decision-making procedure. The suggested PULq-ROF-CODAS technique effi-ciency is illustrated by its implementation in cybersecurity risk assess-ments, where managing ambiguities and communication judgment are critical. The results support the theoretical frameworks capacity to rank cybersecurity threats based on significance while maintaining agreement between specialist perspectives and collective evaluations, ultimately con-tributing to stronger and more well-planned cybersecurity mechanisms.
Copyright (c) 2025 Uzma Ahmad, Saira Hameed, Rehman Khan, Ayesha Khan https://creativecommons.org/licenses/by/4.0
Characterization of Amenability in Non-Expansive Dynamical Systems by the Schauder Fixed Point Property
In the following manuscript, we are addressing the question:“Which amenability property of dynamical system is characterized by the Schauder fixed point property (Sfpp)?”. We have first introduced the property FC and then we provided suitable answers to this problem for the respective class ELA (extremely left amenable) of semitopological semigroups by developing a relation between properties FC , FE and Sfpp.
Copyright (c) 2025 Amna Kalsoom, Sana Siddique, Maliha Rashid https://creativecommons.org/licenses/by/4.0
Generalized CRF Geometry on Matsuki Orbits in Complex Grassmannians
From the ambient complex-type generalized complex (GC) structure on Z = Grk(Cn), we prove an Inheritance Theorem establishing that each Matsuki orbit M = (G0 · z) ∩ (K · z) inherits a canonical generalized CRF structure (in the sense of Vaisman) ΦM via eigenbun dle restriction; the construction is intrinsic and invariant under closed B fields. We establish a restricted Darboux normal form splitting (M,ΦM) locally into a complex block and a symplectic transverse factor, yielding the invariants type and transverse symplectic rank. Thus we obtain a natural generalized CRFK structure on each Matsuki orbit in the Grassmanian, together with explicit formulas for its type and transverse symplectic rank. Our results fit into the project of describing the complex geometry of lower-dimensional G0-orbits in complex flag manifolds. For G0 = SU(p,q) these are computed directly from the signature data (a,b,r) of h|W, with detailed calculations for SU(2,2) Gr2(C4); in purespinor terms a generator is ϕM = ηtCR,0∧eiωT withωT thetransverseLeviform. These results connect classical CR geometry of real orbits with generalized complex/Dirac geometry and provide the first explicit homogeneous examples of generalized CRFK manifolds from real group actions, opening avenues for deformation, cohomology, and quantization.
Copyright (c) 2025 Irfan Ullah https://creativecommons.org/licenses/by/4.0
An Online Detection Method for Identifying True Change Points
Time series data can be analysed using the singular spectrum analysis (SSA) technique by decomposing the data into its most basic elements, such as trends, recurrent patterns, and noise. Change point detection, a sequential application of SSA, is a procedure that uncovers the locations in time series data when sudden changes in its attributes occur. In an offline setting, the objective is to detect change points by analyzing the full dataset simultaneously. On the other hand, in an online (sequential) setting, the goal is to detect changes as fast as feasible using streaming data points. In this study, a novel online change point detection algorithm is proposed, which uniquely combines a sequential application of SSA with an integrated alarm mechanism. This combination allows for real time detection while explicitly differentiating between genuine changes (true alarms) and fake changes (false alarms). Several Experiments are performed in MATLAB and it is found that the window width parameter N is essential for detection procedure. The selection of the appropriate parameter is critical to guaranty accurate online change point detection. Since extremely small values of N cause false detections by mistaking outliers for change points, whereas exceptionally large values of N fre quently miss true change points. Thus, the best value of N for striking a balance between sensitivity and accuracy is either medium-sized or large. The study concludes that the newly developed SSA-based algorithm han dles both synthetic and real-world data, offering a reliable tool for real time data analysis with a low number of false alarms and precise detection of notable changes.
Copyright (c) 2025 Saba Shakeel, Shazia Javed, Uzma Bashir https://creativecommons.org/licenses/by/4.0
Quotient Geometry of the A6-Action on P2(C): Local Linear Models, InvariantAlgebras, and Orbifold Invariants
We provide an explicit analytic description of the quotient sur face Y = P2(C)/A6 under the Valentiner action. Using Cartan’s lin earization and invariant algebras, we classify all local quotient models. Cyclic subgroups C2,C3,C4,C5 yield classical cyclic singularities, while D8 and D10 give non-cyclic singularities. Isotropy from V4, E9, and S3 corresponds to smooth points. Gluing these local models produces a nor mal projective surface with a completely determined singularity structure.
Copyright (c) 2025 Muhammad Bin Nasir https://creativecommons.org/licenses/by/4.0
Modeling Micromagnetorotational Dynamics in Magneto-Micropolar Fluids withThermal Relaxation Effects
Conventional models of magneto-micropolar fluids often over look the complexities of anisotropic particle rotation and depend on the physically restrictive Fourier’s law, which assumes instantaneous heat trans port. To overcome these limitations, the present study comprehensively investigates the boundary-layer flow of a magneto-micropolar fluid by integrating two key physical mechanisms: the anisotropic Micromagne torotation (MMR) constitutive relation and the Cattaneo–Christov (CC) heat-flux model accounting for thermal relaxation. The resulting nonlinear governing partial differential equations are solved numerically using the shooting method. The findings reveal that the MMR parameter exerts a dual influence—broadening the momentum boundary layer while thinning the thermal boundary layer. Moreover, the inclusion of the CC model introduces a physically consistent correction by predicting a measurable reduction in the local Nusselt number relative to the classical Fourier for mulation. The outcomes of this research provide fundamental insights essential for the accurate modeling, design, and optimization of micro cooling systems and magnetic nanoparticle transport technologies
Copyright (c) 2025 Atif Ali Zafar, Saba Ashfaq, Mahnoor Asif, Hadia Parveen https://creativecommons.org/licenses/by/4.0
Suppose that the sequences {Pi}i≥0, {Ri}i≥0, and {Ji}i≥0 correspond to the Padovan numbers, Perrin numbers, and Jacobsthal num bers, respectively. In this paper, we determine all Padovan and Perrin numbers that can be written as the sum of two Jacobsthal numbers by ap plying Baker’s method for linear forms in logarithms, combined with the reduction technique of Dujella and Peth˝o.
Sequences(Padovan,Perrin,and Jacobsthal)
Copyright (c) 2025 Mustafa Ismail, Ahmed Gaber, Mohammed Abdul Azim Seoud, Mohammed Anwar https://creativecommons.org/licenses/by/4.0
Degree-Based Index Optimization in Trees with Pendant Constraints
Graphs are used in mathematics to mathematically depict net works, which are essentially collections of interconnected things. The topology and structure of networks and molecular graphs can be better understood with the help of topological indices which are practical math ematical tools. In this article, we examine lower bounds topological in dices, including the Gourva, hyper Gourava, Forgotten and hyper Forgot ten indices, in a unified manner within the group of trees with η vertices and η1 pendent vertices. Our goal is to derive sharp inequality and de scribe the associated extremal graphs. Lower bounds for a number of vertex-degree-based topological indices are provided by the primary find ings. We make use of some graphs transformations to compute the T I for fixed pendant vertices and order n. These limitations are novel even for the Gourva, hyper Gourava, Forgotten and hyper Forgotten indices. Trees with a fixed number of pendant vertices maximize scalability, energy efficiency, and multicast communication in networks like peer-to-peer sys tems and Wireless Sensor Networks (WSNs). Fixed pendant vertex trees aid in the modeling of molecular structures with certain bonding proper ties in chemical graph theory, improving stability and medication effec tiveness.
Copyright (c) 2025 Salma Kanwal, Khadija Sarwar https://creativecommons.org/licenses/by/4.0
Hilfer-Hadamard Volterra Fractional Integro-Differential Equations with Initial Conditions
Thefocalpointof thisresearchistheinvestigationofHilfer Hadamard. Volterra fractional integro-differential equations subject to nonlocalinitialconditions.Thestudyaddressestheanalyticalchallenges inestablishingtheexistenceanduniquenessofsolutionsunderappropriateassumptions.Tothisend,Krasnoselskii’sfixedpointtheoremandtheBa nachcontractionprincipleareemployedtoguaranteethewell-posedness oftheproposedsystem. Furthermore, theUlam-HyersandUlam-Hyers Rassiasstabilityconceptsareutilizedtoanalyzethestabilitypropertiesof theobtainedsolutions,offeringdeeperinsightsintotherobustnessofthe model. Finally,severalillustrativeexamplesareprovidedtosubstantiatethetheoreticalresultsandtodemonstratetheaccuracy,applicability,and efficiencyoftheproposedapproach
Volterra integro-differential equations VIDEs
Copyright (c) 2025 Abduljawad K. Anwar, Shayma Adil Murad https://creativecommons.org/licenses/by/4.0
Anoptimisation over the M¨ obius group for an optimal solution in image registration
Image registration, the process of aligning two images to ensure their appearances closely match, is a field profoundly influenced by the pioneering contributions of D’Arcy Wentworth Thompson. In his influen tial work On Growth and Form, Thompson introduced the concept of im age transformation across species, emphasising the idea that deformations between closely related species should exhibit simplicity. Specifically, Thompson explored isogonal transformations—now known as conformal transformations—suggesting that such deformations could be described by finite dimensional groups. While the full set of diffeomorphisms or the conformal group are too complex to be considered simple (as they are in finite), the eight-dimensional M¨obius group, a subgroup of the conformal group, provides a more manageable framework. The M¨obius group com prises four fundamental transformations: scaling, rotation, translation, and inversion, offering a balance between simplicity and mathematical expres siveness. In this study, we propose a novel approach to image registration using M¨obius diffeomorphisms. Additionally, an innovative method for se lecting the initial guess, a critical component in the optimisation process, particularly within the gradient descent framework, has been introduced. Through a series of numerical examples, we demonstrate the effectiveness of this approach in achieving precise and efficient M¨obius-based image registration. Our results highlight the potential of M¨obius transformations as a robust tool for image alignment, bridging historical theoretical in sights with contemporary computational methodologies. This work not only advances the field of image registration but also underscores the en during relevance of Thompson’s foundational ideas in modern scientific research.
Copyright (c) 2025 M. Y. Tufail, S. Gul https://creativecommons.org/licenses/by/4.0
Design and Analysis of a Combined 4-Point Ternary Refinement Scheme with C4-Smoothness
In this paper, we introduce a new combined four-point ternary refinement scheme capable of generating both interpolating and approx-imating curves. The scheme is constructed by translating a quintic B-Spline refinement scheme to new positions using displacement vectors that incorporate four shape control parameters. We demonstrate that this combined scheme produces limit curves with up to C4 continuity and al-so provides significant flexibility in crafting smooth curves with reduced support size. Furthermore, we establish that various well-known inter-polatory and approximating refinement schemes are special cases of our proposed approach. Some key properties, including polynomial gener ation, reproduction, and support width, are examined. Additionally, we investigate the fractal generation characteristics of the refinement scheme, revealing that specific parameter choices enable effective construction of fractal curves. Through a series of numerical experiments, we validate that the limit curves produced by our scheme accurately reflect the associ-ated control polygons, yielding more realistic results compared to existing methods.
Copyright (c) 2025 Pakeeza Ashraf, Fatima Rasheed https://creativecommons.org/licenses/by/4.0
AConcise Bond-Distance Summation Descriptor for Effective Melting PointPrediction of Boronic Acids
Predicting the melting points of boronic acids is crucial for guiding synthetic strategies and understanding their physicochemical be haviors. In this study, we introduce a novel bond-distance summation descriptor, a concise 20-component vector that numerically encodes the molecular structure by summing atomic numbers over the shortest paths from the boron atom. We benchmarked this descriptor against four es tablished feature extraction methods Coulomb Matrix, Mordred, Morgan Fingerprints, and Molecular ACCess System (MACCS) and evaluated the predictive accuracy of five machine learning models: Decision Tree, Ran dom Forest, XGBoost, LightGBM, and Support Vector Machine. Despite having far fewer features than the high-dimensional Mordred and Mor gan representations, our 20-length descriptor achieves competitive results, particularly when paired with XGBoost, which consistently exhibits supe rior performance in terms of Mean Absolute Error (MAE) and R2 score. These findings underscore the potential of a concise, interpretable descrip tor for effective melting point prediction, paving the way for the future integration of this scheme into broader cheminformatics applications.
Copyright (c) 2025 Muhammad Zia Afzal, Shahid Saeed Siddiqi https://creativecommons.org/licenses/by/4.0
On Commutants of Toeplitz Operators with Generalized BiharmonicPoly-Quasihomogeneous Symbols
This paper introduces generalized biharmonic poly-quasihomogeneous functions and studies Toeplitz operators with such symbols on the Bergman space. We characterize the commutants of these operators and investigate their spectral properties, generalizing previous work on biharmonic and poly quasihomogeneous symbols. We provide comprehensive proofs, examples, and extend our results to related classes of symbols.
Copyright (c) 2025 Raja’a Al-Naimi https://creativecommons.org/licenses/by/4.0
Designing by constructing Cubic Trigonometric Nu B-Spline with Arabic Font
In stark contrast to the Latin script used in English, Arabic script is a cursive writing system that is written from right to left. The structure of the Arabic script and the relationships between the charac ters must be thoroughly understood while designing Arabic fonts. In this research, a novel freeform spline method is built for the designing and modeling of Arabic characters and letters, with practical interpolative and approximation features. Authors have constructed a new trigonometric nuB-spline technique which is different in construction from the simple B-spline and contains shape parameters that help in designing. As in this paper, Arabic fonts are formatted by using suggested spline techniques. The proposed technique possesses geometric continuity of order 2 and satisfies B-spline properties such as property of local support, property of positivity, and property of unity. Moreover, different Arabic typefaces are used to demonstrate geometric features such as affine invariance, variation decreasing, and convex hull. Likewise, shape properties are successfully employed at points, intervals, and each side by using a single parameter family. The goal of smoothness and flexibility has also been achieved by the shape features. Numerous data points are collected through human contact, which takes a little bit longer than scanning illustrations yet leads to excellent results.
Copyright (c) 2025 Jaweria Aslam, Hijaz Ahmad, Shamaila Samreen, Humaal Shafiq, Imran Siddique https://creativecommons.org/licenses/by/4.0