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.
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.
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.
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.
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 \(k\ge 2\).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 $|x| \leq 1$.
