ABOUT (AIM AND SCOPE) Journal of Mathematical Extension (JME) is a bimonthly, peer-reviewed journal dedicated to the publication of original research and survey articles in pure and applied mathematics. The journal covers a broad range of areas, including classical analysis, operator theory, approximation theory, ordinary and partial differential equations, functional equations, optimization and mathematical programming, nonlinear analysis and its applications, algebra, geometry, topology, and related areas. JME operates a double-anonymized (double-blind) peer-review process. Authors are normally informed within 10–20 days whether their submission will proceed to external peer review. The complete peer-review process takes approximately 3–4 months on average, although the actual duration may vary depending on the manuscript and reviewer availability. The journal's current acceptance rate is approximately 18–22%. This journal does not charge publication/processing fees from Author(s). We also provide many author benefits, such as free PDFs and a liberal copyright policy. Please see our Guide for Authors for information on article submission. To see our full Editorial team click here. *) The standard abbreviation form for this title is "J. Math. Ext.". If you need to cite this journal, please use this abbreviated form. If you require any further information or help, click here. INDEXING AND ABSTRACTING JME is abstracted and indexed in the following databases: Clarivate (Master Journal List) Google Scholar Islamic World Science Citation Center(ISC) Zentralblatt MATH AMS Digital Mathematics Registry ٍEbsco Directory of Open Access Journals(DOAJ) Scientific Information Database (SID) According to the authentication letter numbered 3/18/311314 dated 25 February 2019 issued by the Iranian Ministry of Science, Research and Technology, Journal of Mathematical Extension is ranked as a "Scientific" Journal. This Journal cooperates with the Iranian DEA Society in the scope of reviewing and publishing DEA papers which significantly promotes the quality of papers in this field. The link to the letter of joint cooperation can be found here. Check us on LinkedIN, publons and JournalGuide.
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On Weakly S-Atrinian and Weakly S-Noetherian Dimension
Let S be a multiplicative subset of a ring R and let M be an R-module. In this article, we introduce and study the concepts of weakly S-Atrinian and weakly S-Noetherian dimension of an R-module M. These dimensions are ordinal numbers, and in essence, they measure the deviation of the R-module M from being weakly S-Artinian and weakly S-Noetherian. We observe some basic facts for modules with theses dimensions, similar to the basic properties of modules with the Krull dimension and Noetherian dimension.
Copyright (c) 2026 JOURNAL OF MATHEMATICAL EXTENSION
In this paper‎, ‎we introduce $E$-$K$-frames for a separable Hilbert space $\mathcal{H}$‎, ‎where $E$ is an infinite matrix of real or‎ ‎complex numbers‎, ‎and $K$ is a bounded operator on $\mathcal{H}$‎. ‎Actually‎, ‎$E$-$K$-frames are a kind of generalization of $K$ frames by infinite matrices‎. ‎Assuming an (invertible) infinite matrix $E$ as a mapping on the Hilbert space $\bigoplus _{n=1}^{\infty}\mathcal{H}_n$‎, ‎we‎ ‎aim to study the properties of $E$-$K$-frames‎, ‎define the notion of $E$-atomic system for an operator $K$ on $\mathcal{H}$ and address its relationship by $E$-$K$-frames‎.
Statistical Inference of Process Capability Indices With Fuzzy Limits for the Exponential Distributions Family
Statistical quality control relies on process capability indices (PCIs) to quantify the ability of a process. When specification limits (SLs) are modeled as fuzzy sets, conventional PCIs must be fuzzified. In this paper, we present confidence intervals for PCIs with fuzzy specification limits, specifically for the family of exponential distributions. This is achieved by presenting theoretical results and a numerical example for illustration. ‎Also‎, ‎we propose a hypothesis testing related to PCIs $C_{\tilde{p}}$ and $C_{\tilde{p}}^{'}$ with a numerical example‎.
This paper introduces and investigates two new classes of operators: \emph{topologically super weakly mixing operators} and \emph{locally topologically super weakly mixing operators}. We explore key topological properties of locally super topologically weakly mixing operators and provide an explicit example to demonstrate that the class of topologically weakly mixing operators is a proper subclass of the class of topologically super weakly mixing operators. Furthermore, we prove a theorem that offers a definitive negative answer to a long-standing open problem in the theory of \(J\)-class operators.
A Modeling Approach Integrating Data Envelopment Analysis and the K-means Algorithm for Performance Evaluation and Clustering of Decision Making Units
In increasingly competitive and resource-constrained environments, performance evaluation plays a vital role in improving organizational efficiency and supporting data-driven decision-making. Traditional Data Envelopment Analysis (DEA), while widely adopted, often struggles to account for structural heterogeneity among Decision-Making Units (DMUs), leading to biased comparisons and limited managerial insight. This study addresses these limitations by proposing a hybrid analytical framework that integrates K-Means clustering, DEA modeling, and performance gap analysis to enhance the accuracy, fairness, and practicality of efficiency assessments. The proposed methodology begins with unsupervised clustering using the K-Means algorithm to segment DMUs into operationally homogeneous groups based on input and output characteristics. DEA is then applied independently within each cluster using the input-oriented CCR model under constant returns to scale (CRS). To improve discriminatory power, super-efficiency and cross-efficiency evaluations are performed to rank efficient units and assess mutual performance consensus. Finally, a performance gap analysis identifies quantitative input reductions and output enhancement targets for inefficient units, offering concrete guidelines for improvement at both the unit and system levels. The empirical results demonstrate that clustering prior to DEA significantly improves the validity of benchmarking by isolating peer comparisons within structurally similar groups. Super-efficiency scores reveal meaningful variation among efficient DMUs, while cross-efficiency results provide a more socially consistent view of performance. Moreover, the gap analysis highlights the potential for a 14% reduction in workforce, 11% optimization of operational costs, and up to an 18% increase in total outputs through more effective resource utilization. These findings underscore the managerial relevance of the proposed model and its potential application across various sectors where performance and resource optimization are critical.
The Restricted Rung Wiener Index of Layered Graphs
In transportation and communication networks, inter-layer routing through fixed gateways inflates path costs compared to unrestricted shortest paths. To quantify this overhead, we introduce the Restricted Rung Wiener (RRW) index, a distance-based measure for layered graphs with mandatory crossover constraints. We derive closed-form expressions for ladder, grid, and circular prism graphs. Linear topologies exhibit quadratic overhead in displacement from central gateways. Grids incur an unavoidable $\Theta(N^3)$ penalty independent of layer count, even under optimal placement. Circular prisms, due to rotational symmetry, yield uniform overhead across all gateway choices. These results provide quantitative guidelines for gateway optimization in constrained multi-layer networks.
Multipliers as Gelfand-Representable Operators
Abstract. Using the Gelfand transform, we characterize multipliers T on commutative faithful Banach algebras via cTx = φT bx and show that for invertible T, the deformed algebra AT = (A, x · Ty) preserves both the maximal ideal space Δ(A) and multiplier algebra M(A). Specializing to L1(G), we prove twisted convolutions f ∗ (v ∗ g) with bv ̸= 0 are the only such deformations.
On the Cofinite Properties of Generalized Local Cohomology Modules
Let $A$ be a commutative Noetherian ring with non-zero identity and let $I$ be an ideal of $A$‎. ‎Consider $F$ as a finitely generated $A$-module and $C$ as an arbitrary $A$-module‎. ‎In this context‎, ‎for a non-negative integer $n$‎, ‎we establish that if the local cohomology modules $\operatorname{H}^{i}_{I}(C)$ are $(\operatorname{FD}_{< n}‎, ‎I)$-cofinite over $A$ for all integers $i$ and if the collection of such $(\operatorname{FD}_{< n}‎, ‎I)$-cofinite $A$-modules constitutes an Abelian category‎, ‎then the generalized local cohomology modules $\operatorname{H}^{i}_{I}(F‎, ‎C)$ are likewise $(\operatorname{FD}_{< n}‎, ‎I)$-cofinite over $A$ for all integers $i$‎.
Strategic Alliances and Partnerships Processes in a Two-Stage Network DEA Framework under Variable Returns to Scale
In this paper, in today’s competitive environment, strategic partnerships and alliances among decision-making units (DMUs) have emerged as effective mechanisms for achieving resource savings, cost reduction, and performance improvement. This paper proposes a novel framework for modeling strategic partnership and alliance processes within a two-stage network Data Envelopment Analysis (DEA) structure under Variable Returns to Scale (VRS). In the proposed approach, two partnering DMUs are allowed to mutually utilize each other’s excess resources, enabling them to improve their performance and move toward their target efficiency levels. A distinctive feature of the model is its ability to determine the new input levels allocated from the partner units within the alliance process, while preserving the internal two-stage network structure of each DMU. The model is formulated in a multiplier form, which provides valuable insights into the contribution of each stage of the network to the overall efficiency improvement resulting from the strategic alliance. To demonstrate the applicability and effectiveness of the proposed framework, it is applied to a set of cement manufacturing plants located in the southern region of the country. The empirical results indicate that strategic alliances can significantly reduce resource consumption, enhance stage-wise and overall efficiency, and improve operational performance of the participating DMUs. The proposed model offers a practical decision-support tool for managers and policymakers seeking to design, evaluate, and implement effective strategic partnerships in network production systems.
In this article, several types of subalgebras of an L-algebra ”L” are introduced. One of these subalgebras is fuzzy impotent subalgebra. After that, by using the concept of a t-norm ”T”, the definition of a fuzzy impotent subalgebra with respect to T is presented. The notion of soft impotent subalgebra, which is an extension of the notion of soft sets, is introduced and several examples and propositions are investigated during the article. Random set, falling shadow of a random set are defined and then falling fuzzy subalgebra and falling impotent subalgebra are introduced and some relations between them are investigated.
Unified Dynamic Initialization Framework for Theoretical and Practical Convergence in Non-Convex Optimization
Gradient Descent (GD) is a widely used first-order iterative optimization algorithm, known for its efficiency in minimizing convex objective functions. However, in real-world applications—particularly in artificial intelligence—optimization often involves non-convex loss functions, where GD can struggle with local minima and convergence issues. In this article, we examine the limitations of traditional GD in non-convex settings and introduce a novel strategy: dynamically updating the algorithm’s starting point at each iteration. This adaptive approach enhances the algorithm’s ability to navigate complex optimization landscapes, thereby significantly broadening the practical applicability of GD to a wider range of challenging problems.
On the Stability of Ternary AQ-Hom-Ders: Bridging Generalized AQ-Functional Equations
This paper investigates the relationship between ternary $AQ$-hom-ders and the generalized $AQ$-functional equation (generalized $AQ$-$\mathbb{FE}$) within the framework of ternary Banach algebras (\textsf{TB}\textrm{A}). By considering the specific cases of $t=1$ and $t=2$, we establish a systematic connection between additive/quadratic mappings and their corresponding ternary $AQ$-hom-ders. The non-trivial nature of these mappings is demonstrated through concrete examples on matrix algebras, highlighting their relevance in non-commutative settings. Furthermore, we derive the solutions for the generalized $AQ$-$\mathbb{FE}$ and establish their Hyers–Ulam ({\rm HU}) stability. Finally, the {\rm HU} stability of these ternary $AQ$-hom-ders is rigorously investigated in \textsf{TB}\textrm{A} using the fixed point (\textsf{FP}) approach.
P-value and Postrerior Probability for the Interval Hypothesis Testing
This article examines the behavior of frequentist $p$-values and Bayesian posterior probabilities for testing interval hypotheses in the Pareto distribution. Here, we consider the hypothesis $H_{0\epsilon}:|\theta-\theta_0|\leq \epsilon$ against $H_{1\epsilon}:|\theta-\theta_0|>\epsilon$, where $\theta$ is the shape parameter of the Pareto distribution. Our analysis reveals a fundamental divergence between classical and Bayesian approaches: while the $p$-value for the interval hypothesis converges to the point-null $p$-value as $\epsilon \to 0$, the Bayesian posterior probability exhibits no such convergence. This discrepancy persists across various prior distributions, including non-informative, exponential, and conjugate priors, and remains significant even for small values of $\epsilon$. We demonstrate that common approximation methods for reconciling these measures fail for the Pareto distribution, emphasizing the distribution-dependent nature of this problem. The results highlight important limitations in using point-null approximations for interval hypothesis testing, particularly within the Bayesian framework. Our findings contribute to the ongoing discussion about reconciling frequentist and Bayesian evidence and provide guidance for practitioners working with power-law distributions in various application domains.
Describing Parametric Families of Algebraic Points of Degree at most ‘ on the Superelliptic Curve Family Cm Defined by y^2 = x^5 + m^2
We give an explicit description of the set of algebraic points of arbitrary degree on the family of hyperelliptic curves Cm, studied in [21] by Jeong and al. and defined by the affine equation y2 = x5 + m2, where m is a square-free integer. This family is a special case of the superelliptic family of curves Cq;p;a defined by yq = xp + a, with q and p prime numbers and a an integer. This more general family was also studied in [22] by J¸edrzejak, who, using height theory, explicitly determined the set C(1)m (Q) of rational points (see Proposition 1.1). The purpose of this note is to extend those results to algebraic points of arbitrary degree. We first construct an explicit Q-basis of the vector spaces L(l infty) for l in N, and we provide an explicit description of the Mordell--Weil group of rational points on the Jacobian Jm(Q). We then apply a fundamental form of the Abel{Jacobi theorem [15, 17] to characterize principal divisors associated with rational functions under consideration. Finally, by introducing a parameter γ, we explicitly construct a union of families of algebraic points, denoted Cm(l)(Q).
Robust Asymmetric Censored Regression Model and Outlier Contaminated Data
This study introduces a robust regression modeling framework for censored observations based on the two-piece scale mixture of normal (TP-SMN) class. Using this distributional family enables exact maximum likelihood inference while simultaneously accommodating skewness, the presence of outliers, and censoring mechanisms. The effectiveness of the proposed methodology is assessed through comprehensive simulation experiments and an empirical analysis using real data, which indicate superior estimation precision and enhanced computational performance. These results highlight the advantages of the proposed modeling censored data with symmetric and asymmetric, as well as light- and heavy-tailed, behavior, offering both methodological flexibility and practical applicability.
Cost Congestion in Data Envelopment Analysis
Abstract. Identifying ineciency factors is crucial for evaluating the performance of Decision Making Units (DMUs) in Data Envelopment Analysis (DEA). Among these ineciencies, congestion is particularly important and should be prioritized for elimination once identied. In DEA literature, congestion typically refers to a situation where an in- crease in one or more inputs leads to a decrease in one or more outputs, or conversely, a decrease in inputs leads to an increase in outputs. This study extends the traditional notion of congestion by introducing the concept of cost congestion. Cost congestion occurs when an increase in expenditures on one or more components of the cost vector results in a decline in output, or when a reduction in cost components leads to an increase in output. Furthermore, this study analyzes cost ineciencies by considering factors such as congestion, technical ineciencies arising from inputs and input prices, and also provides a decomposition of lost opportunities attributable to various ineciency sources.
Boundedness Theorems for Composition-Differentiation Operators Acting on Classical Function Spaces on Bidisk
We establish boundedness criteria for products of composition and differentiation operators acting from weighted Bergman space $ A^2_{\alpha_{1}, \alpha_{2} }(\mathbb{D}^2)$ on the bidisk into the weighted Bergman space $A^2_\beta(\mathbb{D})$ on the unit disk. We then derive sufficient conditions on the symbol functions to guarantee the boundedness of composition-differentiation operators acting from the Hardy space $H^2(\mathbb{D}^2)$ on the bidisk into the Hardy space $H^2(\mathbb{D})$ on the unit disk.
In this paper, we introduce and investigate some properties of ϕ-δ-S-primary submodules, which is a generalization of the ϕ-δ-primary submodules and prime submodules in general. We extend several main results about ϕ-δ-primary submodule and prime submodule into the general framework. Moreover, we show that there is a one-to-one correspondence between the set of all ϕ-δ-S-primary submodules of an Rmodule M containing K and the set of all ϕ-δ-S-primary submodules of M/K. In the end, we present some results concerning ϕ▷◁-δ▷◁-S▷◁-primary submodules in the amalgamation Modules.
On the Energy and Nullity of Uniform Semigraphs
The energy of the semigraph is defined as the sum of the absolute eigenvalues of its adjacency matrix. The study of semigraph energy provides a valuable framework for analyzing the stability, connectivity, and structural properties of hyper-networks. The nullity of a graph encodes both algebraic and structural information. We extend this concept to semigraphs, an ordered-tuple generalisation of graphs, and derive closed-form formulas for the nullity and energy of (i) Path-Uniform Semigraphs $P_{k+1}^{\,n}$, (ii) Cycle-Uniform Semigraphs $C_{k}^{\,n}$, (iii) Star-type Semigraphs $S_{2,n}^{3}$ and their $3$-Uniform counterparts $S_{n}^{3}$ and (iv) Rooted $3$-Uniform Semigraph trees $T_{n}^{3}$, Sharp lower and upper bounds are established in terms of pure middle and isolated vertices, showing that every non-trivial semigraph has nullity at most $n-2$. Beyond enriching semigraph theory with spectral techniques, our results supply new tools for analysing controllability, energy, and redundancy in higher-order network models.
Applications of Pythagorean Fuzzy Superhypergraph Entropy in Uncertainty Optimization
This paper introduces the Pythagorean fuzzy supervertices, superedges, links, and Pythagorean fuzzy superhypergraphs (PFSh). We further present the class of Pythagorean fuzzy quasi-superhypergraphs (PFQS), classifying them into smooth, soft, rough, and coverable subcategories. Crucially, we prove these structures are a direct generalization of traditional Pythagorean fuzzy graphs. We analyze the properties of these models, investigating the impact of Pythagorean $(\alpha, \beta)$-level subsets and establishing key relationships between Pythagorean fuzzy sets and quasi-superhypergraphs. The concept of a strong superhypergraph is introduced, with conditions for its formation from cut-level sets provided. To quantify uncertainty, we pioneer definitions for Pythagorean fuzzy entropy of supervertices, superedges, and links, proving the resulting superhypergraph entropy is a valid measure. Algorithmically, we adapt the Fuzzy Kruskal's Algorithm to minimize superedge entropy via a spanning superhypertree, denoted $E_{min}(\mathcal{E})$. The framework's utility is demonstrated through applications in business superhypernetworks and social complex hypernetworks, bridging fuzzy set theory and network science.
O‎‎n‎ Wardowski ‎T‎yp‎e ‎M‎ulti-valued ‎C‎ontractions ‎‎‎ on ‎C‎omple‎x ‎V‎alued ‎M‎etric ‎S‎paces and ‎R‎elated ‎F‎‎ixed ‎P‎oint ‎R‎esults
Wardowski ‎typ‎e ‎admissible multi-valued ‎contractions on ‎comple‎x ‎valued‎ metric ‎spaces ‎‎are ‎introduced and some fixed point results in this direction are ‎proved.‎ ‎The new ‎results‎ is illustrated with an example‎.
A Bi-objective Model for Collaborative Uncapacitated arc Routing Problem Including Coalition Satisfaction
This paper deals with a carrier collaboration problem in which carriers form a coalition to share and exchange their transportation requests located on the arcs of the network. This problem has recently gained a significant attention to enhance the operational efficiency. However, the existing problem formulations are single objective optimization problems solely focusing on maximizing the profit and overlooking satisfaction criterion which is crucial to the coalition’s long-term survival. This paper proposes a bi-objective integer linear programming model aiming at maximizing profit as well as coalition satisfaction. In order to ensure satisfaction among all members of the coalition, an efficient metric is proposed using proper mathematical norms (L 1 -Norm and L ∞ -Norm) , encouraging a well-distribution of satisfaction among members. Moreover, this paper shows how the carriers’ overtime can be introduced in the problem formulation which could potentially improve the total profitability of the system and coalition satisfaction. The proposed models are tested on a publicly available data set of 1416 instances and the performance of the models are analyzed.
Determining an Assurance Non-Archimedean Epsilon in FDH Models with Variable Returns to Scale
The use of epsilon form in free disposal hull (FDH) models is essential for evaluating the performance of decision-making units (DMUs). However, an inappropriate choice of the non-Archimedean epsilon value may cause the multiplier side model to be infeasible and therefore the envelopment side model to become unbounded. In this paper, a method is presented to determine an assurance value of non- Archimedean epsilon in FDH models with input orientation and assuming variable returns to scale (FDH-VRS models). This value of epsilon is calculated using only basic arithmetic on the input data, and guarantees that the multiplier side become feasible and the envelopment side is bounded. Following that, by providing numerical examples, the role of epsilon in improving the accuracy of the evaluation of DMUs as well as the computational efficiency of the model is shown. The results of running the proposed model on real data from 25 supermarkets in Finland show that this epsilon-based FDH model with VRS technology produces the same results as the base model without epsilon with the same accuracy, with fewer calculations (equal to the number of DMUs). This research provides a practical and mathematical solution to overcome the challenge of epsilon selection in non-convex FDH environments.
Discrete-Time Series Based on Asymmetric Distributions
The goal of this study is to introduce a flexible intervention autoregressive process that is modified based on random autoregressive coefficients and asymmetric innovations. The transfer function of the proposed process is designed to follow a dynamic step change structure. In intervention analysis, outlier or influential observations significantly impact statistical inference. Therefore, we discuss Bayesian marginal effect analysis to evaluate the influence of errors in response variables, priors, and contemporaneous disturbances concerning the Bayesian factor evaluator. By considering Markov Chain Monte Carlo samples, the proposed marginal effects and diagnostic measures can be easily obtained. Real data on new weekly COVID-19 cases from Greece, covering the period from March 1, 2020, to December 17, 2023, confirms the effectiveness of the presented methods.
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Utilization of Fermatean Fuzzy subsets to $Q$-Ideals
This paper introduces the notions of $ALQ$-algebra and $CL$$Q$-algebra and shows that $ALQ$-algebras and $CL$$Q$-algebras are isomorphic‎. ‎It is presented in the‎ ‎the notions of Fermatean fuzzy $Q$-algebras and ($3$-regular) Fermatean fuzzy $Q$-ideals as a generalization of fuzzy $Q$-algebras and fuzzy $Q$-ideals‎. ‎It investigates the connection between Fermatean fuzzy $Q$-algebras and Fermatean fuzzy $Q$-ideals and is shown that Fermatean fuzzy $Q$-ideals are a subclass of Fermatean fuzzy $Q$-subalgebras and $3$-regular Fermatean fuzzy $Q$-subalgebras are Fermatean fuzzy $Q$-ideals‎. ‎In this study‎, ‎the concepts of nil radical Fermatean fuzzy $Q$-subsets are introduced‎, ‎and it is stabilized that nil radical Fermatean fuzzy $Q$-subsets are Fermatean fuzzy $Q$-subalgebra‎. ‎Also‎, ‎the Fermatean fuzzy $Q$-algebras(ideals) are extended by‎, ‎intersection‎, ‎union‎, ‎combination‎, ‎and isomorphisms and $\mathbb{J}$-coverd conditions in any given $Q$-algebra‎.
Evaluation of Decision-Making Units with Nonhomogeneous Outputs in Different Time: An Application of Network Data Envelopment Analysis
Data Envelopment Analysis is an efficient method for evaluating the performance of decision making units (DMUs). This method is used to evaluate the efficiency of units using homogeneous inputs and outputs under a given technology. However, in practice, achieving complete homogeneity is often difficult, and organizations frequently face nonhomogeneous inputs and outputs. This study examines the efficiency evaluation of (DMUs) with homogeneous inputs and nonhomogeneous outputs over multiple time periods. Furthermore, in this study, the concept of nonhomogeneity, is examined from a different perspective, in the internal structure of the network. Using the proposed approach, the performance of (DMUs) can be more accurately evaluated, in situations where, their outputs change over different periods. This evaluation includes cases, where, in subsequent time periods, some (DMUs) may not produce some of the previous period’s outputs or may produce new outputs, and some DMUs may not exist, in subsequent periods, or new DMUs may emerge. The main advantage of the proposed method in evaluating nonhomogeneous (DMUs) is the nonhomogeneity of time periods. Finally, the proposed models for the evaluation of efficiency are applied in the banking industry, which, can improve the management and Decision Making processes in this industry.
Structure and Applications of $(p, q)$-Derivations in Algebra
In this paper, we investigate $(p, q)$-derivations in both general algebras and Banach algebras. Our main result extends the Singer-Wermer theorem to $(p, q)$-derivations, proving that their ranges are contained in the radical of the algebra. As a direct consequence, we prove that the range of any left derivation lies within the radical of the algebra even without assuming continuity. This improves a result due to Bra\v{s}er and Vukman. For Banach algebras, we show that primitive ideals remain invariant under bounded $(p, q)$-derivations. Finally, we study $(p, q)$-derivations of group algebras and give an answer to the question posed.
Investigating some Properties of G-frames in Krein Spaces
This paper presents a new definition for g-frames in Kreinspaces, based on the properties of Krein Space. It is shown that a g-frame in a Krein space is also a g-frame in the associated Hilbertspace. However, the converse is not true unless the span of the J-adjoint of positive operators is uniformly J-positive, and the span ofthe J-adjoint of negative operators is uniformly J-negative. The J-g-frame operator is defined for g-frames in Krein spaces, and its properties,such as invertibility and boundedness, are discussed. Additionally, a J-g-dual sequence is introduced for g-frames in Krein spaces and shownto be a g-frame in the associated Hilbert space, but not necessarily inthe Krein space. Finally, the relationship between a g-frame in a Kreinspace and the sequence induced by it in the associated Hilbert space isexamined.
Exploring the Presence of Positive Solutions in a Fractional Equation Featuring Dual Integral and Derivative Boundary Conditions under the p-Laplacian Operator
In this article‎, ‎we investigate the existence of a positive solution for the fractional differential equation with dual derivative and integral boundary conditions and the uniqueness of this solution‎. ‎In this regard‎, ‎we use fixed point theorems in collective operators of Banach spaces‎. ‎Using the results of the main theorem‎, ‎in addition to finding a positive solution‎, ‎we can use an iterative scheme to approximate this differential equation and show this issue with an example‎.
We introduce pointfree counterparts of countably disconnectivity, specifically countably basically disconnected frames (briefly, $c$-basically disconnected frames) and countably $F$-frames (briefly, $F_c$-frames), along with their frame-theoretic characterizations. We present characterizations of zero-dimensional extremally disconnected frames, zero-dimensional $c$-basically disconnected frames, and zero-dimensional $F_c$-frames $L$ using the ring-theoretic properties of the ring $\mathcal{R}_cL$ of continuous real-valued functions with countable images on $L$. We show that a zero-dimensional frame $L$ is extremally disconnected if and only if any annihilator ideal of $\mathcal{R}_cL$ is generated by an idempotent, $c$-basically disconnected if and only if the annihilator of each element of $\mathcal{R}_cL$ is generated by an idempotent, and an $F_c$-frame if and only if the annihilator of each element of $\mathcal{R}_cL$ is pure.
Study on a New Family of Close-to-Convex Functions in a Vertical Strip Involving Subordination and Convolution
In this present paper,we consider a subclass of close-to-convex functions associated with a vertical strip domain. We obtain several results concerned with integral coefficient inequalities,convolutions, and representations for functions belonging to this class $ \mathbf{MC_c}$. Furthermore, we consider the Fekete-Szego, inclusion relations and radius problems involving certain classes of strongly close-to-convex functions, parabolic close-to-convex functions, and other types of close-to-convex functions.
Approximation Solution of the Differential Equation with Fractional Derivatives Using Fuzzy Transformations
In this research, we consider the real fractional partial differential equation. Many numerical methods for fractional partial differential equations have been presented. In this article, we seek to calculate the continuous approximate solution of the equation, and for this purpose, we use the inverse phase transformation method, which is a new technique for calculating the approximation function using real data. We use the inverse fuzzy transformation method because, firstly, it is convergent, and secondly, to approximate a function in certain intervals, unlike other methods that only use a limited number of points, in this method, all points are used with a degree of accuracy. Space fractional equation has been discretized using explicit finite difference method using the space caputo derivative with order 1<a<2. The efficiency of the proposed method is shown by two test problems for various values of t. Tables and graphs are used to describe the results, which ensures that results are in an excellent agreement with analytical solution.