https://jmmrc.uk.ac.ir/ Journal of Mahani Mathematical Research en daily 1 Mon, 01 Jun 2026 00:00:00 +0330 Mon, 01 Jun 2026 00:00:00 +0330 https://jmmrc.uk.ac.ir/article_4871.html Knowledge acquisition is the most important challenge in building an expert system in any field, and one of the sources of knowledge will be the data collected in that field. Traditionally, the data collection process is assumed to have a symmetric cost. For example, this assumption will not be acceptable in the medical due to various expenses. Designing a cost-sensitive classification and a cost-sensitive feature selection method are two approaches to considering cost factors. Cost-effective feature selection improves financial return by significantly saving feature data cost as well as limiting credit losses and this can be used in different areas, for example, computer imaging and medical diagnosis which also have a large number of features that may be irrelevant or redundant. Analysis of the research reviewed in this study shows that cost-sensitive feature selection focuses on selecting a feature subset with minimum total cost while achieving a classification accuracy that is as high as possible. The review of selected studies showed a downward trend in using heuristic methods in this field, Wrapper methods are in the first rank regarding usage in evaluation criteria, and 76\% of selected studies are in the single-objective category. Most of the studies were classified in the single-label category based on the number of determined labels. https://jmmrc.uk.ac.ir/article_4947.html Binary Particle Swarm Optimization (BPSO), proposed by Kennedy and Eberhart, extends PSO to binary search spaces. However, BPSO suffers from computational complexity due to velocity-based position updates and limited scalability in high-dimensional problems. In this paper, we introduce Expected BPSO (EBPSO), a simplified and faster variant that removes the velocity component and directly uses a probabilistic position update mechanism inspired by expected particle behavior. We theoretically analyze EBPSO’s convergence and evaluate its performance across two domains: (1) ten scalable binary benchmark functions (F1–F10) and (2) feature selection for classification using four real-world datasets (Breast Cancer, Iris, Wine, and Digits). EBPSO consistently outperforms BPSO, Binary GA, and other recent binary metaheuristics (e.g., BDO, BSCA, BGWO, BRKO) in both solution quality and runtime. For example, EBPSO achieved up to 15× speedup over BPSO and maintained a competitive advantage across all tested dimensions. In the feature selection task, EBPSO was used within a wrapper model using an SVM classifier. It reached accuracies of 99.07\% on Digits, 99.44\% on Wine, and 98.42\% on Breast Cancer datasets while selecting fewer features than other methods. Statistical significance was confirmed using paired t-tests and Wilcoxon signed-rank tests, both yielding p-values < 0.01 across all evaluations. Overall, EBPSO demonstrates superior performance, scalability, and statistical robustness, making it a promising tool for large-scale binary optimization and efficient feature selection. https://jmmrc.uk.ac.ir/article_4904.html In recent years, due to the increasing growth of technology and new technologies, data is obtained in more complex structures as the main component in analysis. One of these complex structures is tensors. Therefore, in order to answer this need (analysis of data with tensor structure), it is necessary to expand statistical concepts and methods in the field of data with tensor structure. On the other hand, in reality, we may also encounter skew data. Therefore, in this article, we have introduced the skew normal tensor distribution and obtained some of its important statistical properties. Subsequently, we employed the EM algorithm to obtain maximum likelihood estimates of the parameters and assessed their accuracy through simulation studies. Finally, we have shown the effectiveness of the obtained estimators with real data. https://jmmrc.uk.ac.ir/article_5335.html The existence of a natural hierarchy in all the multilevel problems are required for the proper exploration as specialized analytical ‎tools.‎ ‎Multilevel analytical tools provide the proper estimation of such type of problems‎. ‎The variance components models‎, ‎t‎he hierarchy linear models and random coefficient models are known as multilevel regression‎. ‎In this paper‎, ‎we consider random intercept and slope model and estimate parameters of model using the EM algorithm‎. ‎Also‎, ‎we propose a boosting algorithm based on the EM algorithm‎. ‎The performance of the proposed algorithm is studied and is compared to usual variable selection methods using simulation studies‎. ‎A real data analysis carried out to illustrate the procedures obtained theoretically. https://jmmrc.uk.ac.ir/article_5056.html Feature selection plays a crucial role in facial image classification by reducing dimensionality and improving robustness to variations in expression, pose, and lighting. However, researchers face challenges when selecting features from high-dimensional, unlabeled data due to the nonlinear manifold structure of facial images. To address this, this paper proposes UFSOLPP, a novel unsupervised feature selection method that consists of three main stages. First, the method employs Orthogonal Locality Preserving Projections (OLPP) for feature extraction, aiming to preserve local data structures and enforce orthogonality without dimensionality reduction. Unlike conventional OLPP, which uses heat kernel to measure similarity, this paper replaces it with cosine distance to better capture angular relationships that are for facial image discrimination. Second, it measures the similarity between the original and orthogonal features using the Pearson correlation distance. Third, it models both feature sets as vertices in a weighted bipartite graph. The edge weights are computed using the Pearson correlation similarity, and the method uses the Hungarian algorithm to compute maximum matching. The method selects the original features involved in the maximum matching as the final subset. This strategy removes noisy, correlated, and redundant features effectively, while preserving interpretability and discriminative power. Experiments demonstrate that UFSOLPP outperforms eight state-of-the-art methods. It achieves 96.00% accuracy and 0.9800 NMI on Jaffe, 68.66% accuracy and 0.7532 NMI on ORL, and 82.33% accuracy and 0.8557 NMI on the high-dimensional Pixraw10P dataset. These results highlight the practical value of UFSOLPP and its ability to handle high-dimensional data efficiently in unsupervised facial image analysis. https://jmmrc.uk.ac.ir/article_5419.html In unsupervised feature selection, the absence of class labels makes it necessary to rely on statistical measures like variance to assess feature importance. However, the presence of noise, outliers, and feature redundancy can significantly reduce the accuracy of this process, negatively affecting the performance of machine learning models. To address these issues, this paper proposes the UFSLRAGE algorithm, which begins by applying the LRAGE feature extraction method to minimize reconstruction error, adaptively updating the similarity between samples to remove noise and outliers while generating orthogonal and uncorrelated features. This paper then constructs a weighted bipartite graph that represents both original and extracted features, with cosine similarity determining the edge weights. The LAPJV algorithm is used to identify a maximum matching in this graph, where the vertices corresponding to the original features are selected as the most informative ones. This paper evaluates UFSLRAGE on five widely-used image datasets: Jaffe, Yale, ORL, COIL-20, and pixraw10P, using metrics such as accuracy, normalized mutual information (NMI), precision, recall, and F-measure. Experimental results demonstrate that UFSLRAGE consistently outperforms other state-of-the-art unsupervised feature selection methods, achieving an average NMI of 0.9358 and an average accuracy of 0.9333 on the pixraw10P face image dataset with 10,000 features. https://jmmrc.uk.ac.ir/article_5074.html In this paper, we define complex $(p,q)-$extension $\alpha-$Chebyshev differential equations on $|x|\leq 1$. Our consideration is focused on determining properties of generalized Chebyshev polynomials of the first, second, third and Fourth kind, sparking interest in constructing a theory similar to the classical one. We solve the complex $(p,q)-$extension $\alpha-$Chebyshev differential equations on $|x| \leq 1$. https://jmmrc.uk.ac.ir/article_5509.html ‎This paper explores a continuous-time economic model and proposes a technique for discretizing it into a discrete-time system using Euler's method‎. ‎By manipulating the parameter $\omega$‎, ‎which represents the savings proportion‎, ‎we determine the critical coefficient of bifurcations in the normal form using the central manifold‎. ‎The coefficient for flip and Neimark-Sacker bifurcations at the trivial fixed point is obtained through affine transformations‎, ‎while closed forms are employed for other types of bifurcations‎. ‎To analyze the conditions and coefficients of the normal forms for each bifurcation‎, ‎we employ both analytical and numerical methods‎. ‎Additionally‎, ‎we visualize the bifurcation curve in the parameter space and generate a bifurcation diagram using the MatcontM package.‎ https://jmmrc.uk.ac.ir/article_5217.html For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected $2$-distance graphs. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs.  For graphs whose diameter is greater than $2$, we define a maximal fine set, and by contracting $G$ with respect to these subsets, we obtain a new graph $\widehat{G}$  such that $D_2(G)$ is connected if and only if $D_2(\widehat{G})$ is connected. In particular, $D_2(G)$ is disconnected if and only if $\widehat{G}$ is bipartite. https://jmmrc.uk.ac.ir/article_5236.html This paper introduces the idea of second Ricci solitons. A second Ricci soliton is nothing but a steady hyperbolic Ricci soliton. We study the geometry of closed and compact second Ricci soliton manifolds.  Immersed submanifolds as second solitons also will be investigated. Finally, we investigate this structure on warped product manifolds. https://jmmrc.uk.ac.ir/article_5560.html In this paper, we introduce the notion of $H_v$-near fields as a new generalization of classical near fields within the framework of hyperstructure theory. The concept of an $H_v$-near field is introduced and defined here for the first time. The main contribution of this paper is the development of the foundational theory of $H_v$-near fields, including the investigation of their basic algebraic properties, structural characteristics, and construction methods. Several examples and construction techniques are presented to illustrate the existence and diversity of these hyperalgebraic structures. Furthermore, we define and study the fundamental relation $\theta^{*}$, which plays a central role in describing equivalence classes and the internal structure of $H_v$-near fields. Our approach is based on methods and techniques from hyperstructure theory. https://jmmrc.uk.ac.ir/article_5243.html This paper addresses the existence of solutions for a class of fractional differential boundary value problems. The main novelty of this paper lies in providing a new analytical framework to prove solution existence for this class of equations. Our approach is grounded in a combination of a Krasnoselskii-type fixed point theorem with the concept of the measure of weak noncompactness (MWNC) in the Banach space $L^{1}(I)$. This technique allows us to prove existence under more general growth conditions than in previous works that relied  on complete continuity. By establishing a set of sufficient conditions, we demonstrate that the boundary value problem under consideration admits at least one solution. Several illustrative examples are presented to highlight the applicability and effectiveness of the obtained results. https://jmmrc.uk.ac.ir/article_5561.html     This paper introduces a novel double integral transform and highlights its theoretical and practical significance in solving complex mathematical problems. The fundamental definitions, key properties, and its relationships with existing integral transforms are systematically developed. The novelty of the proposed transform lies in its generalized structure, which extends the classical Mellin framework and provides a unified approach for handling a broader class of functions. Several applications are presented, including the derivation of Mellin transforms for elementary functions and the solution of partial and fractional differential equations. In addition, a comparative analysis with existing generalized Mellin transforms is carried out to demonstrate the advantages and efficiency of the proposed method. Finally, a comprehensive table of transforms for functions frequently arising in mathematics, physics, and engineering is provided to facilitate practical implementation. https://jmmrc.uk.ac.ir/article_5244.html By the notion of $F_a$-factorable operators, we establish a new version of the Riesz Representation Theorem adapted to this class of operators. Also, for $F_a$-factorable operators $T$ and $T^{\prime}$, we derive some equivalent conditions such that the equation $T^{\prime}=TX$ has a unique solution in the class of $F_a$-factorable operators. Moreover, by using the concept of $F_a$-frame in $L^2(0,\infty)$, which is similar yet distinct from the traditional frame concept, we establish optimal $F_a$-frame bounds, and provide a complete characterization of all corresponding $F_a$-dual frames. https://jmmrc.uk.ac.ir/article_5562.html In this paper, we study properly biharmonic hypersurfaces in the Minkowski 4-space $\mathbb{E}_1^4$ that are invariant under the action of some Lie groups of isometries. Using the symmetries imposed by Lie groups, we provide a complete local classification of such hypersurfaces according to the causal character of the orbits and the ambient pseudouclidean metric. Explicit parametrizations are obtained for all non-degenerate cases, and geometric properties such as the mean curvature, shape operator, and biconservative condition are analyzed in detail. Several examples are presented to illustrate the theory. Our results generalize previous classifications of biconservative hypersurfaces in Euclidean and Lorentzian settings and highlight the interplay between group symmetry and higher-order geometric constraints. https://jmmrc.uk.ac.ir/article_5305.html Self-RAG enhances Retrieval-Augmented Generation (RAG) by enabling Large Language Models (LLMs)  to dynamically retrieve external knowledge and self-evaluate outputs. However, the original Self-RAG heavily relies on a manually tuned weighted-sum mechanism for combining critique scores, rendering the system brittle and poorly adaptable to diverse query distributions. To address these limitations, Pareto Front Enhanced Self-RAG (PFE-SELF-RAG) is proposed as a tuning-free Multi-Objective Optimization(MOO) framework. It first applies Maximal Marginal Relevance (MMR) to enrich context diversity, then incorporates two evaluation strategies: Pareto Front-based selection and Geometric Mean (GM) Aggregation. The primary significance of this approach lies in eliminating fragile manual weight tuning. By mathematically modeling the trade-off between factual accuracy and relevance, PFE-SELF-RAG enables adaptive candidate selection, allowing the number and quality of outputs to vary dynamically. This represents the first formal application of Pareto optimization to candidate ranking in self-reflective RAG systems, establishing a principled alternative to heuristic aggregation. Evaluations on PopQA, ARC Challenge, PubHealth, and TriviaQA demonstrate substantial impact. The Full Pareto Set strategy consistently outperforms the Self-RAG baseline, achieving %58.6 on PopQA (%+3.7), %68.0 on ARC Challenge (%+1.6), %73.0 on PubHealth (%+0.6), and %71.3 on TriviaQA (%+4.3). These improvements underscore the practical impact of replacing brittle heuristics with principled optimization, establishing PFE-SELF-RAG as a robust and scalable standard for self-reflective RAG systems. https://jmmrc.uk.ac.ir/article_5600.html In the present paper a certain subclass of analytic functions related to parabolic domains is defined. We characterize duality of the class $ \small        M(\alpha) = \left\{ f \in \mathcal{A} : \operatorname{Re} \{ \alpha\frac{f(z)}{z} + (1 - \alpha) f'(z) \} >\left|\alpha\frac{f(z)}{z} + (1 - \alpha) f'(z) -1\right|,\ z \in \mathbb{D} \right\},$ which gives some interesting results such as neighborhood, necessary and sufficient condition for normalized analytic function $f$ in open unit disk $\mathbb{D}$ to be in $M(\alpha)$ in terms of convolution. https://jmmrc.uk.ac.ir/article_5320.html In the present paper, we introduce and investigate a new class of meromorphic functions analytic in the open unit disk and applying a $q-$derivative and $q-$differential integral operator associated with quantum calculus. Furthermore, by using the familiar Riesz-Dunford integral of a linear operator on Hilbert space H, a new class of univalent functions with a fixed point is introduced. Coefficient estimate, distortion bound and extreme points are obtained. https://jmmrc.uk.ac.ir/article_5601.html This paper introduces the notion of refined Pythagorean fuzzy superhypergraphs ($\textnormal{RPFS}$) as a mathematical model for clustering complex data under uncertainty. We define a fundamental relation $\beta$ on an RPFS and prove that its quotient is a Pythagorean fuzzy graph ($\textnormal{PFG}$). We also prove that every $\textnormal{PFG}$ can be embedded in an $\textnormal{RPFS}$, and any $\textnormal{RPFS}$ is a quotientable $\textnormal{PFG}$. A combinatorial formula is proven for the number of distinct $\textnormal{RPFS}$s on any given set and for complete-based $\textnormal{PFG}$s. This study presents the link-degree and superedge-degree points and investigates the relation of link-degree in fundamental $\textnormal{PFG}$ and superedge-degree in any related $\textnormal{RPFS}$. Furthermore, it is analyzed that the direct product of two $\textnormal{RPFS}$s is again an $\textnormal{RPFS}$, and its quotient equals the direct product of their quotients. In the categorical property, it is established that the set of all $\textnormal{RPFS}$s forms a commutative monoid under the direct product operation. Finally, we considered the relation between fuzzy data clustering and RPFS, and clustered, modeled, and combined some real-world problems via RPFS and the concept of its direct product. https://jmmrc.uk.ac.ir/article_5321.html The degree variance and the geometric mean of the degrees of the vertices for ‎graph $G‎$ are defined ‎as ‎‎‎$\text{Var}(G)=‎\frac{1}{n}\sum_{i=1}^{n}\Big(d(v_i)-‎\frac{2m}{n}‎ \Big)^2‎‎$‎‎ and ‎‎$‎\text{GM}(G)=\Big(\prod_{i=1}^{n}‎d(v_i)\Big)‎^‎\frac{1}{n}‎,‎$‎ respectively, ‎w‎here $n$, $m$ and $d(v)$ represent the number of vertices, edges and degree of vertex $v$. ‎Also,‎ the geometric degree variance of graph ‎$‎‎G$ ‎defined as ‎‎‎$‎‎\text{GVAR}(G)=‎\frac{1}{n}‎\sum_{i=1}^{n}(d(v_i)-\text{GM}(G))^2‎‎$‎‎. ‎‎‎ We determine the two first moment of degree variance in (uniform) random tree‏‎s.‎ We also show a convergence in probability associated with this quantity. ‎Finally, we present bounds for the ‎expected ‎value‎ of geometric mean of the degrees ‎and‎ geometric degree variance. https://jmmrc.uk.ac.ir/article_5326.html This study develops a systematic comparative framework for estimating Lorenz curves and Gini coefficients, addressing key methodological gaps in measuring income inequality. We employ the Generalized Mean Squared Error (GMSE) to compare several parametric models (such as polynomial, beta, and established functional forms) with isotonic regression as a non-parametric alternative. Extensive Monte Carlo simulations using log-normal and Pareto distributions show that isotonic regression consistently achieves higher accuracy than parametric approaches. An application to Iranian household income data (n = 18,809) further confirms these results. Based on data characteristics and research objectives, the findings offer practical guidance for selecting appropriate estimation methods. https://jmmrc.uk.ac.ir/article_5339.html In this paper, we present four nonparametric methods to fit some fuzzy regression models, when both the explanatory and response variables are fuzzy quantities. In this approach, we first introduced a distance between triangular fuzzy numbers. Then, two fuzzy nonparametric regression models are presented based on the extended version of K-nearest neighbors (KNN) method on fuzzy data (with the same/modified weights). In addition, a new method is investigated to fit two fuzzy nonparametric regression models based on the R-neighborhood radius (RNR) method on fuzzy data (with the same/modified weights). Among these methods, the two methods of KNN and RNR with the modified weights have the better performances than the methods with the same weights. To evaluate the proposed fuzzy nonparametric regression models, two measures of goodness of fit are presented. The application of the proposed methods are studied in modelling some data sets. https://jmmrc.uk.ac.ir/article_5349.html Community detection is a fundamental problem in the analysis of complex networks. It has applications across social, biological, and financial domains. Traditional algorithms, such as Louvain, LPA, and modularity optimization, often require manual parameter tuning. They also suffer from inaccurate cluster center selection. To address these challenges, we propose AlCMean’s, a novel algorithm. AlCMean’s combines Laplacian energy–based automatic center identification with DeepWalk embeddings for robust node representation. Unlike existing Laplacian-based and clustering methods, AlCMean’s eliminates the need to predefine the number of communities, enhances cluster center selection using structural importance, and leverages representation learning for more accurate and stable assignments. Experimental results on benchmark datasets demonstrate 10–20% higher NMI and ARI scores compared to Louvain, Newman–Girvan, LPA, Fast-Greedy, and a recent GNN-based competitor (MAGI, KDD’24). Additional evaluations with modularity and F1-scores confirm the superiority of AlCMean’s. Ablation studies highlight the critical contributions of each component. Despite its reliance on DeepWalk parameters and increased runtime relative to lightweight heuristics, AlCMean’s consistently outperforms state-of-the-art methods. This makes it a promising tool for real-world network analysis. The source code and datasets are publicly available at https://github.com/shahinmomenzadeh/ALCMeans.git https://jmmrc.uk.ac.ir/article_5368.html In this paper, we obtain some results concerning  vanishing, finiteness and artinianness of formal local cohomology modules. Also, we determine annihilators,  cosupport and the set of coassociated primes of these modules in some special cases. https://jmmrc.uk.ac.ir/article_5393.html In this article, we use the square root as a tool to study hoop algebras. To do so, we define square root and make the first attempt to explore the significance properties of this concept in this setting. Then, due to the key role of square roots in obtaining new hoop algebras, we apply them to the filters of hoop algebras, and show that the formation of square roots on quotient structures of hoop algebras by their filters is well-behaved. In addition, a new class of hoop algebras having square roots, so-called good hoop algebras, is introduced, and some relationships with other classes of ordered algebras such as Boolean algebras and Gödel algebras are explored. Several examples are provided as well. Ultimately, it is shown that the class of all (good) bounded V-hoop algebras with square roots is a variety. https://jmmrc.uk.ac.ir/article_5406.html In this paper, we study finite groups with a prime-power number of Sylow \(p\)-subgroups. Motivated by the work of Yang et al.~(2022), who characterized non-solvable groups with a prime-power number of Sylow $2$-subgroups, we investigate the corresponding problem for odd primes. We prove that if a finite group \(G\) has a non-abelian composition factor whose order is divisible by an odd prime \(p\), and the number of Sylow \(p\)-subgroups is a prime power, then \(p\) must be a Mersenne prime and \(n_p(G)=2^k\) for some integer \(k\ge 2\). https://jmmrc.uk.ac.ir/article_5423.html Yongting capability index is a suitable criterion for measuring and evaluating the eficiency of industrial processes to produce items conforming the fuzzy quality. This paper develops and applies several statistical estimation approaches for evaluating Yongting’s capability index based on the fuzzy quality and provides a comprehensive comparative study of their performance. This enhances the methodological toolkit available for researchers and practitioners engaged in evaluating and improving industrial production processes. The proposed and discussed approaches in this paper are: (1) Kernel density estimation, (2) Monte Carlo estimation, (3) method of moments estimation and (4) maximum likelihood estimation. The proposed estimation approaches are compared in a simulation case study to show the performance discussed approaches.  https://jmmrc.uk.ac.ir/article_5426.html ‎Let $R$ be a commutative ring with identity, and $M$ be an $R$-module. We denote by $\zeta(M)$ the semimodule consisting of all varieties of submodules of $M$ over the semiring $\zeta(R)$ of varieties of ideals of $R$. In this article, we introduce two functors from the category of $R$-modules to the category of $\zeta(R)$-semimodules and investigate conditions under which these functors preserve the short exact sequences of modules. They are the hom-functors $\operatorname{Hom}_{\zeta(R)}(\zeta(P),\zeta(-))$ and $\operatorname{Hom}_{\zeta(R)}(\zeta(-),\zeta(E))$ associated with $R$-modules $P$ and $E$, respectively. It is shown that $\zeta$ is exact and both $\operatorname{Hom}_{\zeta(R)}(\zeta(P),\zeta(-))$ and $\operatorname{Hom}_{\zeta(R)}(\zeta(-),\zeta(E))$ are left exact on any short exact sequence $0\rightarrow M' \xrightarrow{f} M\xrightarrow{g} M'' \rightarrow 0$ with $M''$ a radical module. In particular, we provide conditions under which the hom-functors preserve short exact sequences of modules.  https://jmmrc.uk.ac.ir/article_5433.html ‎We develop nonparametric inference methods for Tsallis-based cumulative residual entropy functionals when dealing with length-biased lifetime observations. The paper introduces kernel-type estimators for both the static measure and its time-dependent version, with explicit corrections for the sampling mechanism that systematically oversamples longer-lived units. We derive large-sample approximations for bias and variance under standard smoothness assumptions and appropriate bandwidth choices, establish weak and $L^2$-consistency, and prove central limit theorems. Numerical     experiments using exponential and Weibull distributions examine finite-sample behavior through bias, variance, and mean squared error calculations, while normality diagnostics validate the asymptotic approximations. We also apply the methodology to automotive component durability data, where results confirm the stable performance and practical     value in realistic length-biased scenarios where standard sampling assumptions break down. https://jmmrc.uk.ac.ir/article_5439.html In this paper, because semihoops are the most basic residuated structure that contain all logical algebras based on Galois connections, therefore we introduce a new type fuzzy graph and its complement based on semihoops, denoted by $SH$-graph $G$ and $G^\prime$, respectively. Then, the $SH$-graph automata related to the $SH$-graph $G$ and a minimum zero forcing set are introduced, denoted by $A(Z(G))$. After that, the concepts of isomorphism between two $SH$-graphs $G_1$ and $G_2$ and isomorphism between two $SH$-graph automata $A(Z(G_1))$ and $A(Z(G_2))$ are introduced. Then, we prove that if two $SH$-graphs $G_1$ and $G_2$ are isomorphic, then two $SH$-graph automata $A(Z(G_1))$ and $A(Z(G_2))$ are isomorphic; otherwise it is not true. In addition, the concept of equivalence of $SH$-graph automata is proposed. Moreover, we prove that $SH$-graph automata obtained from the same $SH$-graphs are equivalent under different zero forcing sets in some special $SH$-graphs. And then, we know that $SH$-graph automata $A(Z(G_1))$ and $A(Z(G_2))$ are equivalent can not be characterized by $SH$-graphs $G_1$ and $G_2$ being isomorphic. Finally, we introduce a several of simple practical applications of $SH$-graph and $SH$-graph automata. https://jmmrc.uk.ac.ir/article_5446.html In this article, a numerical method to solve multi-order fractional diff erential equations with Caputo derivatives is suggested. Shifted Chebyshev cardinal functions are employed as basic functions. The corresponding fractional derivative operator matrix for these cardinal functions is computed. By approximating the unknown expression of the problem in terms of the shifted Chebyshev cardinal functions, applying their fractional derivative operator matrix, and utilizing the collocation method, solving the equation under question is converted into solving a system of algebraic equations. Bysolving this system, the approximate solution of the problem is obtained. Finally, the accuracy and effi ciency of the proposed method are examined by solving several numerical examples. The results show that the method presented in this article is an effi cient and highly accurate method to solve such multi-order fractional diff erential equations. https://jmmrc.uk.ac.ir/article_5457.html Transportation problems are widely used decision-making models in logistics, production, and supply-chain management. In real-world applications, the input parameters such as costs, supplies, and demands are often uncertain or imprecise, making classical crisp formulations inadequate. To address this challenge, this study proposes a fuzzy multi-choice goal programming (FMCGP) model enhanced with fuzzy function-approximation techniques. Unlike previous works, where fuzzy transportation problems are treated using direct defuzzification or ranking approaches, our method integrates fuzzy least-squares linear regression and a fuzzy binary polynomial approximation to represent and approximate multi-choice fuzzy goals flexibly. This dual approach allows the decision-maker to simultaneously handle multiple fuzzy objectives and constraints within a unified framework. A key feature of the proposed methodology is that all comparisons between fuzzy and crisp values are evaluated using the necessity measure with a degree of 0.8, ensuring mathematically consistent and practically interpretable inequality relations. To demonstrate the model's applicability, we present a case study of a transportation planning problem under uncertainty. The numerical experiments illustrate how the proposed approach outperforms existing fuzzy transportation methods in terms of solution feasibility, interpretability, and computational efficiency. The results confirm that the FMCGP model with fuzzy function approximation provides a powerful and flexible tool for decision-making under uncertainty, offering improved accuracy and robustness compared with classical fuzzy transportation approaches. In addition, the framework is general enough to be extended to other types of fuzzy optimization problems beyond transportation. https://jmmrc.uk.ac.ir/article_5461.html In this paper, we explore the diverse applications and distinctive properties of Tsallis entropy by introducing generalized definitions of strong and weak secrecy. Tsallis entropy suggests that generalized weak secrecy and strong secrecy are commonly employed in information-theoretic security challenges. Additionally, we examine the interplay between Tsallis entropy and the criteria for strong and weak secrecy. The primary motivation behind this study is to elucidate the concept of “generalized weak secrecy,” a widely utilized notion. Also, this research delves into the precise relationship between conditional entropy and the minimum adversarial error probability, illustrating how generalized weak security can be translated into practical guarantees. For static and memoryless sources, it is demonstrated that the vanishing of the leakage rate requires the adversarial error probability to reach its upper bound. Moreover, generalized strong security, characterized by the vanishing of the variational distance, results in the complete operational failure of the adversary. These findings underscore the critical role of Tsallis entropy in assessing the security of systems. https://jmmrc.uk.ac.ir/article_5468.html In the present paper, the right dc-Noetherianity for $S$-posets is defined and studied. Some fundamental properties of right dc-Noetherian $S$-posets are presented, and the relation of right dc-Noetherian $S$-posets with sub $S$-posets, factor $S$-posets, products, and coproducts are studied. Finally, the relations between right po-Noetherian $S$-posets, right dc-Noetherian $S$-posets and right Noetherian ordered pomonoids are investigated. https://jmmrc.uk.ac.ir/article_5469.html We study Schatten class Toeplitz operators on weighted Bergman spaces induced by almost standard radial weights on the unit disk. We obtain a complete characterization of such operators generated by positive Borel measures. The characterization is given in terms of the Berezin transform, integrability of localized averages with respect to the M\"obius invariant measure, and discrete summability over pseudohyperbolic lattices. For Toeplitz operators generated by complex Borel measures, we establish sufficient conditions for Schatten class membership in terms of discrete lattice averages of the total variation, together with corresponding norm estimates. As an application, we derive Schatten class bounds for differences of such operators. https://jmmrc.uk.ac.ir/article_5470.html We present and explore the fundamental structural theory of e-groups, a generalization of groups introduced by Borumand et al. (2018). We introduce the notions of full e-subgroups and normal full e-subgroups, and we construct the quotient of an e-group under these conditions. Moreover, we define and investigate generated e-subgroups, establish their basic properties, and characterize cyclic e-subgroups. A detailed analysis of the kernel of an e-homomorphism reveals that the subset kernel is not suitable for isomorphism theorems; to resolve this, we adopt the universal algebraic perspective and employ the congruence kernel. Using this approach, we establish the First Isomorphism Theorem for e-groups and provide concrete examples illustrating the result. Furthermore, we discuss the formulation of the Second and Third Isomorphism Theorems within the congruence framework, and we examine the relationship between congruences and normal full e-subgroups.