# Caspian Journal of Mathematical Sciences Volume:11 Issue: 2, Summer Autumn 2022

• تاریخ انتشار: 1401/12/15
• تعداد عناوین: 16
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• Hamid Bigdeli *, Hassan Hassanpour Pages 368-380
Defender-attacker game is a model for conflicting between a defender and an attacker. Defender tries to prevent attacking an opponent by assigning limited security resources. In real world the utility values of the defender-attacker game are assigned by experts which usually are uncertain. According to that the assigned values by several experts may be slightly different and conflicting, we consider a set of all their viewpoints. This approach is similar to hesitant fuzzy environment. Also, each of the experts may have the different weights; AHP method is used to determine the weights of each of the experts. A weighted sum method is applied to obtain a game with aggregated payoffs. An expected value of the fuzzy numbers is introduced to convert the problem into defender-attacker game with interval payoffs. According to this, we proposed a method to solve security game in fuzzy environment. It is shown that the optimal solution of the expected value model is the optimal solution of the original model. Finally, a practical example is illustrated to solve by the proposed method.
Keywords: Defender-attacker game, Fuzzy sets, Bilevel programming
• Omer Unsal *, Zeynep Sakartepe Pages 381-396
In this study, we focus on extended (3+1)-dimensional Jimbo-Miwa equations in constructing complexiton solutions. On this way, we use modified double sub-equation method which presents different solutions from ones obtained through double sub-equation method. Modified double sub-equation method employs two wave transformations to reach expected solutions. In literature, this method is given in a different way and considered as generalization of double sub-equation method.
Keywords: Complexiton solution, modified double sub-equation method, extended (3+1)-dimensional Jimbo-Miwa equations
• Verda Gurdal * Pages 397-409
In this paper, we define and study the concept of $\mathcal{I}_{3}$-limit points and $\mathcal{I}_{3}$-cluster points of triple sequences in the topology induced by random $2$-normed spaces. We discuss the relationship between $\mathcal{I}_{3}$-cluster points and limit points and prove some important results.
Keywords: random $2$-normed space, $t$-norm, ideal convergence, triple sequence, $F$-topology
• Milutin Obradovic, Nikola Tuneski * Pages 410-416
Let $f$ be function that is analytic in the unit disk $D=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, $\left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<\lambda \quad\quad (z\in D),$then $f$ belongs to the class $U(\lambda)$, $0<\lambda\le1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from $U(\lambda)$ in the case when $0.400436\ldots \le\lambda\le1$.
Keywords: Univalent functions, Class U($, lambda$), fifth coefficient, sharp estimate
• Alireza Pourmoslemi *, Marjan Adib, Tahere Nazari Pages 417-429
In this paper, after introducing inner products on groups, first, we define a Hilbert group using the inner products and in the last section, we present some fixed points for closed and midconvex subgroups of such Hilbert groups.
Keywords: Fixed points, Hilbert groups, Normed groups, Midconvex subgroups
• Rasul Rasuli * Pages 430-438
We introduce the concept of Q-soft R- submodules over a commutative ring. Some Properties of Q-soft R- submodules areinvestigated. In particular, we consider properties of intersection and direct sum for Q-soft R- submodules.
Keywords: Theory of modules, submodules, maps of modules, fuzzy set theory, direct sum
• Reha Yapali *, Harun Polat Pages 439-447
In this paper, weighted mean methods of summability are given in intuitionistic fuzzy normed spaces IFNS. Also, some Tauberian conditions are defined for the weighted mean methods of summability in IFNS.
Keywords: intuitionistic fuzzy normed space, weighted mean summability, slow oscillation, Tauberian theorem
• Hadi Ghasemi, Tayebe LaL Shateri * Pages 448-460

In this paper, we introduce the notion of continuous ∗-frames and ∗-C-controlled frames in Hilbert C∗-modules. We present some results of continuous frames in the view of ∗-C-controlled frames in Hilbert C∗-modules. Also we define ∗-(C, C')-frames and investigate multiplier operators for these frames.

Keywords: Hilbert C∗-module, ∗-controlled frame, continuous ∗-controlled frame, multiplier operator
• Mohammad Bhat *, Aamir Dar Pages 461-479
A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we are interested in the dual wavelets whose construction depends on nonuniform multiresolution analysis associated with linear canonical transform. Here we prove that if the translates of the scaling functions of two multiresolution analyses in linear canonical transform settings are biorthogonal, so are the wavelet families which are associated with them. Under mild assumptions on the scaling functions and the wavelets, we also show that the wavelets generate Riesz bases
Keywords: Nonuniform, Biorthogonal, Scaling function, Linear Canonical Transform
• Masoumeh Firouzjah, Bashir Naderi *, Yousef Edrisi Tabriz Pages 480-494
In this study, the problem of consensus of multi-agent chaotic systems of fractional order is considered. Using the fractional order derivative in Caputo's sense and the classical stability theorem of linear fractional order systems as well as algebraic graph theory, sufficient conditions are provided to ensure consensus for fractional multi-agent systems. The distributed adaptive protocols of each agent are designed using local information and a detailed analysis of the leader-following consensus is presented. Some numerical simulation examples are provided to show the effectiveness of the proposed results.
Keywords: Consensus, Fractional-order, Chaos, Multi-agent systems, Algebraic graph theory
• Tahere Nozari *, Fateme Movahedi Pages 495-507
The study of hyperstructures derived from particular mathematical objects is very important and interesting. Graph theory has been established as a fundamental and important tool for solving practical problems in other branches of mathematics. This paper can be considered as one of the connections between hyperstructures and graph theory. In this way, by using the dominating set notion of a graph, we define a hyperoperation on verticals of it and study its properties and then we construct a hypergroup based on this hyperoperation. This hypergroup is presented for some classes of graphs.
Keywords: Semihypergroup, hypergroup, graph, domonating set
• Elham Basiri * Pages 508-517
‎In this paper‎, ‎we consider the progressively Type-II censoring and ‎the sample size is assumed as a random variable from a Poisson distribution. The optimal sample size is determined by considering ‎a‎ cost constraint‎. ‎Towards this end, ‎‎‎we first introduce a cost function and then the optimal parameter of Poisson distribution is obtained so that the cost function does not exceed a pre-fixed value‎. ‎In the following‎, ‎through a simulation study‎, ‎the results are evaluated‎. ‎Finally‎, ‎the conclusion of the article is presented‎.
Keywords: Random Sample Size‎, ‎Cost Function‎, ‎Optimization
• Paulraj Gnanachandra *, Saeid Jafari, N. Rajesh Pages 518-525
In this paper, we introduce and study a class of topologized groups called beta -I-topological groups.
Keywords: Topological group, beta-I-open sets, beta-I- topological group
• Vahid Sadri *, Gholamreza Rahimlou Pages 526-539
Regarding the applications of the fusion frames and generalization of them in data proceeding, their iterative is of particular importance when one of their members is deleted. In this note, a method for reconstruction of continuous generalized  usion frames and the error operator with its upper bound are presented. Also, the approximation operator for these frames will be introduced.
Keywords: Perseval frame, c-frame, cg-fusion frame, Bochner integrable function
• Dawood Hassanzadeh-Lelekaami *, Hajar Roshan-Shekalgourabi Pages 540-549
As a generalization of the Artinian module, we introduce the class of pseudo Artinian modules. We explore some algebraic properties of this class, and we study some topological properties of the prime spectrum of pseudo Artinian modules.
Keywords: Artinian modules, pseudo Artinian modules, prime submodule, Noetherian space, Spectral space