نشریه پژوهشهای ریاضی
سال نهم شماره 1 (پیاپی 24، بهار 1402)

In this paper, a new method for identifying outlier time series based on GARCH model by exponential distance approach is presented in three steps: first fuzzy and hard clustering methods are implemented on time series, then the outlier time series are detected and removed from the dataset. After removeing outlying time series, clustering algorithms are applied for dataset again. The 30 stocks of the top active, lucrative and profitable stocks in the Iranian stock market are used to evaluate the presented methods. By computing the Silhouette and Xe-Beni indexes, the accuracy of the clustering methods are compared and finally, it is shown that by removong the outlier time series, the GARCH model based on the exponential distance approach has the best performans.

In this paper, a time-fractional diffusion equation of distributed order including the Caputo-Prabhakar fractional derivative is studied. We use a numerical method based on the linear B-spline interpolation and finite difference method to study the solutions of these types of fractional equations. Finally, some numerical examples are presented for the performance and accuracy of the proposed numerical method.

Algebraic structures on topological spaces can’t distinguish all the non-homeomorphic spaces. Recently, by equipping some of these algebraic structures with topology, one can separate the non-homeomorphic spaces with the same algebraic structures. Particularly, the fundamental group is equipped by the various types of topologies which some of them make it a topological group and some of them don't. Virk and Zastrow has studied the generalizations of the existing topologies on the fundamentalgroup to the universal path spacew ith a proper comparison. If we remove the condition α(0) = 𝑥0 from the definition of the universal path space, we obtain an object that have been discussed in mathematical literatures under the name “ fundamental groupoid”. Indeed, fundamental groupoid denoted by 𝜋𝑋, is the category of homotopy classes of paths in 𝑋 as the morphisms and has the set 𝑋 as the objects set. For any 𝑥, 𝑦 ∈ 𝑋, the set 𝜋𝑋(𝑥, 𝑦) is the set of homotopy classes of paths in 𝑋 from 𝑥 to 𝑦. We can consider the object group at 𝑥, 𝜋𝑋(𝑥), as the well-known fundamental group 𝜋1 (𝑋, 𝑥). In order to answer the question how these topologies can be generalized on the fundamental groupoids, the authors has introduced the Lasso topology on the fundamental groupoid of a locally path connected space in which makes it a topological groupoid. A topological groupoid is a groupoid 𝐺 together with topologies on 𝐺 and 𝐺0 such that the structure maps are continuous. R. Brown and G. Danesh-Naruie were the first and only ones to take this step. They have defined a topology on a quotient of the fundamental groupoid such that it became a topological groupoid when the given space 𝑋 is locally path connected and semilocally simply connected. Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space X in which it’s basis is known and by some assumptions, we can consider it as a generalization of the whisker topology on the fundamental group.

For a given topological space (𝑋, 𝜏), let [𝛼] ∈ 𝜋X(𝑥, 𝑦)wrwhw , 𝑥, 𝑦 ∈ 𝑋. If 𝑉, 𝑊 are open neighborhoods of 𝑥, y, respectively, one can define 𝑁([𝛼], 𝑉, 𝑊): = {[𝛽] ∈ 𝜋𝑋| 𝛽 ≃ 𝛾 ∗ 𝛼 ∗ 𝜆, 𝛾(𝐼) ⊆ 𝑉, 𝜆(𝐼) ⊆ 𝑊}, where 𝛾(1) = 𝛼(0) = 𝑥 and 𝛼(1) = 𝜆(0) = 𝑦. Theorem: The family {𝑁([𝛼], 𝑉, 𝑊);[𝛼] ∈ 𝜋𝑋(𝑥, 𝑦), 𝑥 ∈ 𝑉 ∈ 𝜏, 𝑦 ∈ 𝑊 ∈ 𝜏} forms a basis for a topology on fundamental groupoid. The topology that is generated by this basis, is called Whisker topology.

Based on the results that we presented in this paper: If 𝑋 is a small loop transfer space; The multiplication map 𝑚: 𝜋 𝑤ℎ𝑋 × 𝜋 𝑤ℎ𝑋 → 𝜋 𝑤ℎ𝑋 is continuous. The fundamental groupoid with the Whisker topology is a topological groupoid. The inherited topology from fundamental groupoid 𝜋 𝑤ℎ𝑋 on the object group 𝜋 𝑤ℎ𝑋(𝑥) equals to the Whisker topology on 𝜋1 (𝑋, 𝑥).

The first and main process capability indices (PCI), , was stated by Juran. The  process capability indices and then the  process capability indices have been formally stated by Chan et al. The  process capability indices is a combination of the  and  process capability indices and was introduced by Pearn et al. (2002). The  process capability indices are a tool for measuring process capability in production. So far, confidence intervals for the difference between process capability indices for the two processes have been obtained using the pivotal quantity method and also using the bootstrap resampling simulation method. In situations such as selecting a supplier and assessing process improvement, it is of interest to compare capability indices for two different processes or the same process before and after an adjustment. In this paper, first a generalized confidence interval for the  capability indices ratio of two different processes is presented and then this comparison is performed by performing a appropriate hypothesis test and providing an example. In addition, using simulating, the generalized confidence interval was compared with the Bootstrap confidence interval and the results showed that the generalized confidence interval works better. The  capability indices is based on the assumption that a given process can be represented by a normal probability model with mean  and standard deviation .

In this research, we introduce fundamental hypergroups and dependent hypergroups then we investigate a connection between hypergroup theory and nuclear physics and fundamental particles. We show that interaction process such as Lepton actions in fundamental particle physics and nuclear fusion Stars forms a fundamental hypergroup.

In this paper G is considered to be a finite group. We denote the set of elements order and the set of prime divisors of order G by 𝜋𝑒 (𝐺) 𝑎𝑛𝑑 𝜋(𝐺), respectively. The largest element order of G is denoted by 𝑘1(G), and the prime graph of G is denoted by Γ(𝐺), where two vertices u and v are adjacent if uv ∈ 𝜋𝑒 (𝐺). After classification of finite simple groups, the problem that came to the researchers’ attention was the problem of recognizing a group with a specific characteristic. Properties such as elements order, the set of elements with the same order, graphs, etc. In fact we say the group G by property M is recognizable, whenever by isomorphic G be the only group by property M. The other methods are group recognition by using the order of the group and the largest element order. In other words, we say the group G is recognizably by using the order of the G and the largest element order whenever there exists group H so that ∣ 𝐺 ∣=∣H∣ and 𝑘1 (𝐺) = 𝑘1(H) then 𝐺 ≅ H. It is known that some of groups are recognizable by this method. In this paper, we prove that the Stienberg group 𝐷4 3 (2 𝑛 ), where 2 4𝑛 − 2 2𝑛 + 1 is a prime number are recognizable by using the order of the group and the largest element order. In other words, we have the following main theorem. Main Theorem: Let G be a group with the Steinberg group 𝐷4 3 (2 𝑛 ), where2 4𝑛 − 2 2𝑛 + 1 is a prime number such that ∣ 𝐺 ∣=∣ 𝐷4 3 (2 𝑛 )∣ and 𝑘1 (𝐺) = 𝑘1( 𝐷4 3 (2 𝑛 )), then 𝐺 ≅ 𝐷4 3 (2 𝑛 ).

In this research we prove that Steinberg group 𝐷4 3 (2 𝑛 ), where 2 4𝑛 − 2 2𝑛 + 1 is a prime number by using the order of the group and the largest element order. In order to prove the main theorem, we used Lemmas 4.2, 7.2 of the reference [18].

In this section we prove the main result of this article. For simplicity the Steinberg simple group and prime number are denoted by D and p respectively. As mentioned in the previous section to prove the main result of this article we use the Lemma 4.2 of [18]. We prove p is an isolated vertex of prime graph. Using Lemma 4.2 we prove that G neither a Frobenius nor 2-Frobenius group. And for the case c this Lemma is satisfied. In other words, G has a normal series such that H and G/K and K/H are nonabelian simple groups. Moreover, H is a nilpotent group. Every odd components of prime is an odd component of the prime graph. In the next step, by using Lemma 7.2, since (5, ∣G∣)=1, we consider the groups of this Lemma. We also prove that isomorphism K/H≇ 𝐿2(q) , 𝐿3(q), 𝑈3(𝑞) , 𝐺2 (𝑞), 𝐺2 2 (𝑞), are a contradiction. Finally we have K/H ≅ 𝐷4 3 (2 𝑛 ). The proof be completed.

We conclude that in addition to a previously known criterion(test) for Steinberg groups recognition by their 2-sylow subgroups(ANTHONY HUGHES, CHARACTERIZATION OF 3D4 (q3 ), q = 2n BY ITS SYLOW 2-SUBGROUP, Proceedings of the Conference on Finite Groups,1976, Pages 103-105) and also the same order components (Guiyun Chen, Characterization of 3D4(q), Southeast Asian Bulletin of Mathematics, 25, pages 389– 401,2002). Next, in this paper we can recognize them by the order of the group and the largest element order of the group.

Determinantal ideal are one of important topics in Algebraic Geometry and Commutative Algebra. There are several examples of varieties as rational normal scrolls which their defining ideals are generated by minors of a matrix. In general, computing a Grobner bases for a special class of ideals help us to correspond a monomial ideal to the main ideals with the same invariants. In this paper, we compute a Grobner bases for an ideal generated by 2-minors of a 2×n matrix of monomials. Also height of determinantal ideal of a special class of monomial matrices is computed.

In this paper, direct numerical methods for solving a class of the fractional optimal control problems (FOCP) with different fractional derivative order and boundary conditions based on Genocchi hybrid functions are presented. For this purpose, first the importance of fractional calculus, definitions and required properties are provided. Then the hybrid functions including the combination of Genocchi polynomials with the block pulse basic functions, the advantages and properties of these polynomials are expressed. A required property has been proven. By using the new methods, the two fractional operators including the left Caputo fractional derivative and left Riemann-Liouville fractional integral of the Genocchi hybrid functions, are calculated directly and without approximation. Subsequently, some of the methods for solving fractional optimal control problems are presented in the form of classification of the direct methods. In the proposed direct methods, the fractional optimal control problem becomes a system of algebraic equations by discretizing state and control variables based on Genocchi hybrid functions with using fractional operators, Legendre-Gaussian formula for integral approximation and Lagrange multipliers. From the solution of the resulting system, the unknown coefficients of the state and control variables are obtained. We extend these methods with the ideas for the FOCP, including the final point. Then, the error bound of the function approximation are determined. Also, the convergence analysis of hybrid functions is investigated in the direct methods. At last, the efficiency and effectiveness of the proposed methods and comparison of the obtained results with those reported in the previous studies are discussed in the final section by solving some test problems.

In this paper, we construct a direct limit functor from the category of Bratteli diagrams to the category of dimension groups and we prove that it is an equivalence of categories. The notion of a classification functor was introduced by Elliott in 2010 for classification of separable C*-algebras, which is a functor from a (complicated) category to another (concrete) functor reducing the verification of isomorphism of two objects in the first category to verification of isomorphism of their images in the second category. Using this notion, we obtain several classification functors between the categories of dimension groups, C*-algebras, and Cantor minimal systems, leading to functorial formulations and generalizations of results of Giordano, Putnam, and Skau.

We classify orbit spaces of cohomogeneity two isometric actions on Riemannian manifolds of constant negative curvature.

In this paper, a system for detecting sleep deprivation of the driver was designed and constructed. Designed system is different from the systems that were previously implemented. There are many environmental factors that affect driver sleepiness, which can prevent driver sleepiness by measuring them and analyzing data. In this system, the relevant sensors are used to measure environmental parameters. An alert system is also used as a system output to alert the driver and his relatives if he / she detects a driver's sleep hazard before he / she is completely sleep-infected. In this method, fuzzy analysis is used to analyze the data obtained from environmental sensors. . The system is made up of sensors that minimize contact with the driver's body and thus do not interfere with driving, and works well at any time. One of the other features of this system is that, by checking the driver's environment, it will detect the driver's fatigue and sleepiness, depending on the conditions and before the person is fully drowsy

In this paper, modified hat functions and improved hat functions are proposed to solve stochastic Ito ̂-Volterra integral equations with multi stochastic terms. A linear system of equations are achieved by replacing the vector and matrix coefficients and operational matrices in the equation which is easy to solve with mathematical softwares. Also, under some conditions the error of these methods are o(h^3) and o(h^4 ) . The accuracy and reliability of these two methods are studied by solving and comparing the answers with block pulse functions and hat functions.

In this paper, we partition, enumerate and classify Krasner hyperrings and new classes of hyperrings (weak distributive, left inclusion distributive, right inclusion distributive, left near, right near) up to isomorphism of orders less than four and then determine their automorphism groups.

In this paper, we consider the classical and non-classical Lie symmetries of fractional HGF differential equations. Indeed, Lie symmetries are utilized to reduce the fractional nonlinear PDEs into the fractional nonlinear ODEs. Finally, exact solutions of the corresponding equations are extracted.

Due to the dynamic structure and nonlinear fluctuations of the stock market, it is difficult to accurately predict the trend of this market using the old methods. In this study, in order to improve the accuracy of predicting the index trend in different industries, we propose a new algorithm that combines algorithms fractal interpolation and support vector machine regression, abbreviated as fracsion algorithm. . For this purpose, after recognizing the fractal structure of industries using the Hurst exponent of each industry, we consider the value of the index in each fractal industry as the primary data to predict the trend of the index. Then, by modifying the fractal interpolation algorithm, we will generate new data, and finally, by calling the support vector regression algorithm on the obtained data, we will predict the index trend. The results of the implementation of the Hybrid fracsion algorithm and its comparison with two conventional methods, namely artificial neural network and support vector machine regression, indicate the superiority of the predictive accuracy of the proposed algorithm.