نشریه پژوهشهای ریاضی
سال نهم شماره 3 (پیاپی 26، پاییز 1402)

Let R be an arbitrary ring and N be a right R-module.  A right R-module M is called N-retractable if HomR(M,N')≠0, for any nonzero submodule N' of N. This is a generalization of the concept of retractable modules. The aim of this paper is to study of N-retractable modules, where N is an arbitrary right R-module. One of the most important results of this paper is the characterization of rings that have a module such that each module is retractable with respect to it. Also we show that the class of N-retractable modules is closed under direct sums and direct products.

Introduction In a vast sense, economy is a medium of products distribution and exchange of goods or services by different agents. Demand and supply issues for goods and services lead societies to plan market-based economics. The main goal of market-based economics is to balance demand and supply, known as the general equilibrium. Mathematical economy paves the way to verify the existence, uniqueness, and algorithms for computing the equilibria.  Differential geometries prefer to visualize the shape of the equilibrium manifold, i.e., the set of pairs of prices and endowments such that the aggregate excess demand function is equal to zero. Balasko had shown if the equilibrium price is unique for every economy, the price is constant, and the curvature is zero. Moreover, Loi and Matta generalized it in the form that for two commodities and an arbitrary number of agents, if the curvature of E(r) , which is equilibrium manifold with fixed total resources r , is zero, there is the uniqueness of equilibrium for every economy. The general equilibrium condition is also must be held in a flea market, in which all kinds of goods are in their depreciated form. By definition, depreciation is the decline in the value of an asset over time due to different factors such as usage, wear, and tear and obsolescence, which is a significant concept that must be considered when pricing an item for sale. One reason why depreciation must be considered is that, if it is not, businesses would have an unrealistic and distorted estimation of their assets. Consequently, their estimation of the depreciation period net profit in the book value (B.V) will be inaccurate. In turn, the amount of tax a company needs to pay may exceed the original amount as the losses are not considered accurately. Here, to calculate the incurred losses due to depreciation, we consider the current value of the original price paid for the item, considering the interest rate and different depreciation periods. This matter yields the current value of the item to be sold and, comparing it with the current value of its original price lets us understand whether selling the price is to the benefit of the seller or not.

In this study, a simple generalization of the Lehmann-Scheffe theorem is proposed in cases where UMVUEs exist but a sufficiently complete statistics does not exist. Also, another method is introduced based on the group action. In this method, UMVUE for the unknown parameter is found using  a commutative and associative binary operation.  Finally, the motivation for using the words "completeness" and "unbiasedness" is expressed in such a way that completeness and unbiasedness are not characteristic of a statistic or its parametric form, but a family of distributions of a statistic, and deleting even one point of the parameter space may change the completeness.

In this paper, the multi-traveling salesman problem as a two-objective optimization problem, are solved by using an integrated fuzzy approach. This approach, by defining the new concept of fuzzy domination, corresponds to each of the vectors of the objective functions of the problem a degree of Gaussian proximity, so it can be ranked and hence, we can compare the obtained Pareto solutions in a multi- objective optimization problem. More precisely, using this approach, the multi-objective optimization problem was considered as a single optimization problem. In this paper, by combining the concept of fuzzy domination and a meta-heuristic algorithm such as simulating annealing we study the multi-traveling salesman problem. To this end, by performing some different simulation, the performance of this proposed approach is evaluated. Numerical results indicate the effect of this approach in improving the quality of results and also reducing the computational time of problems.

Singular Spectrum Analysis (SSA) is a new powerful method in time series analysis. This non-parametric method due to its unique properties, such as there being nonecessity as to making assumptions about stationarity of time series and also about the normality of residuals, has caught the attention of many researchers in the field of Econometrics and time series analysis and its applications are increasingly getting wide spread. Also this method can be used for short time series. The main purpose of SSA method is to decompose time series into interpretable components such as trend, oscillatory component, and unstructured noise. The basis of SSA is singular value decomposition of the trajectory matrix built on the time series. In the basic SSA method the frequency of observations which used in the trajectory matrix is different and so there may be an error in reconstructing and forecasting the time series, especially at the beginning and end of the series. It occurs because the magnitude of eigenvalues, eigenvectors, and consequently, reconstruction and forecasting of future values of time series, is directly related to the trajectory matrix. The purpose of this paper is to improve the trajectory matrix of SSA method to increase the accuracy of the reconstructed time series and forecasting results, which is called singular spectrum decomposition (SSD). In this paper, SSA and SSD methods and their properties are briefly introduced and then the performance of SSD method over SSA method in time series reconstruction and forecasting for simulated and real data is discussed.

In this study‎, ‎an advanced meshfree technique is formulated to investigate the electromagnetic wave scattering problem from an infinitely transversally large anisotropic cylinder with arbitrary cross section‎. ‎A meshless local finite difference‎- ‎radial basis function is used to discrete the anisotropic interior (inside the scatterer) and isotropic exterior (outside the scatterer) problems‎. ‎The anisotropic and ordinary  Helmholtz operator‎ ‎inside and outside the scatterer are approximated by linear‎ ‎combinations of the generalized multiquadric radial basis functions‎. ‎The proposed numerical technique is an efficient instrument to deal with practical complex scatterers and also has a good advantage in treating the material discontinuity at the interface between two‎ ‎different media‎. ‎To confirm the accuracy and efficiency of this method‎, ‎in some numerical examples‎, ‎the radar cross section of some anisotropic scatterer with different cross section is calculated and compared with the answers obtained from analytical methods‎.

The rapid and continuous growth of the World Wide Web has made the process of extracting useful information with minimal volume of a large collection of documents a serious challenge these days. Summarizing documents are a very time-consuming and difficult task for humans, so it reveals the need for a powerful summarizing system to reduce the volume of texts and also to speed up access to useful information.  Recently, a summarization system based on the sparse representation approach has been presented, which tries to reconstruct each sentence in a sparse form by a linear combination of other sentences. In this approach, select a subset of sentences of the main text that contain important information about the text and send it to the output as a summary. It is also necessary to select the least number of text sentences that have the maximum reconstruction of other text sentences, which achieves this goal by using the sparse representation approach. This model consists of a penalty function based on the L2 norm to control sentence reconstruction and regularization. The reconstruction function based on the L2 norm causes all the words to have an equal role in reconstructing sentences, which may cause outlier words to change the summarization result. Therefore, to improve the quality of the summary obtained in this article, we rewrite the penalty function with the L2 norm. This causes a different amount of error to be allocated for each of the words in sentence reconstruction, which causes the sensitivity of the method to be reduced to outlier words. The implementation results show that the proposed method provides a quick and high-quality summary based on the ROUGE and measure-F criteria compared to the previous methods.

 Let a  be an ideal of Noetherian ring R  and M,N  be finitely generated R -modules. Recall that, for each i≥0 , i-th generalized local cohomology module M, N with respect to a  is defined byHaiM,N≔limn Hai(M/anM,N).Also, recall that  cda,M,N,  the cohomological dimension of R-modules M and N with respect to an ideal a  of a commutative Noetherian ring R  issupi∈N0: Hai M,N≠0.An important problem concerning local cohomology is determining the set of attached prime ideals of the top local cohomology modules. This problem has been studied by several authors. In this paper, we study attached prime ideals of top local cohomology modules.

 In this paper, we first obtain some subsets of the set of attached prime ideals of top local cohomology module. By using these, we obtain a result about finiteness of  top local cohomology modules.

  Let R  be a Noetherian ring and a  be an ideal of  R . Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞ . We will prove that i) p∈SuppRN:  cda,R/p=c, dimRp=c⊆AttR(HacN),ii)  AttR(Had+cM,N)⊆p∈SuppRN:  cda,M,R/p=d+c,iii) p∈SuppRN :cda,M,Rp=d+c,dimRp=c ⊆AttR(Had+cM,N).

 Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞.  The following conclusions were drawn from this research.  If  (R,m)   is a Noetherian local ring such that   Had+cM,N≠0  then Had+cM,N  is not of finite length. If R is a Noetherian domain, then under certain conditions we have AttR(Had+cM,N)=AttR(HacN) .

In the field of advertising‎, ‎there are always situations in which individuals or companies promote their products in order to find retrieval opportunities and attract customers in a competitive environment‎. ‎Several goals are followed in this paper‎. ‎First‎, ‎the historical development of applications of differential games in modeling strategic situations in competitive advertising is mentioned‎. ‎We then introduce the problem in a duopoly market under the influence of uncertainty in the framework of a stochastic differential game‎. ‎Finding the equilibrium strategy for this problem requires solving a nonlinear partial differential equations system also known as the Hamilton-Jacoby equation‎. ‎Another purpose of this paper is to propose an efficient and appropriate computational method for solving the Hamilton-Jacobi partial differential equations‎. ‎The proposed method for solving the problem is a combination of collocation methods by the derivative operator matrix based on Chelyshkov polynomials and policy iteration method‎. ‎The advantage of using the policy iteration method is that at each step‎, ‎instead of finding the solution to a nonlinear partial differential equation‎, ‎it is sufficient to solve a sequence of linear partial differential equations systems‎. ‎The convergence of the proposed method is provided in detail‎. ‎Finally‎, ‎we solve the corresponding Hamilton-Jacobi equations system by the proposed iterative algorithm‎.

In this paper we introduce the notion of generalized state coannihilators in state residuated lattices. We establish a connection between generalized state coannihilators and Galois connection. If A is a state residuated lattice, we show that the set of filters forms a Heyting algebra in which the relative pseudo-complement of G with respect to F is (F : G)ν. Also, we show that the set of state coannihilator filters form a Boolean lattice. Ultimately, we characterize generalized state coannihilators in terms of state prime filters and state minimal prime filters.

This paper studies the dynamical behavior of a discrete-time predator-prey analytically and numerically. The conditions and the critical coefficients for the transcritical, flip (period-doubling), and Neimark-Sacker are computed by using the center manifold and normal form technique. Besides, codimension-two bifurcations including strong resonances 1:2, 1:3, and 1:4  have been achieved. The numerical simulation and continuation method, not only confirm our analytical results but also reveals richer dynamics of the model, especially in the higher iteration.

‎Let $S$ be a commutative (not necessary unital) inverse semigroup with the set of idempotents $E$‎. Consider semigroup algebras $ell^1(S)$ and  $ell^1(E)$ and triangular Banach algebras ‎$mathcal{T}=begin{bmatrix}ell^1(S) &ell^1(S) /M_0&ell^1(S)end{bmatrix}$ and ‎$mathfrak{T}={begin{bmatrix}alpha &0&alphaend{bmatrix}: alpha in ell^1(E)]}$, where $M_0$ be the closed linear span of ${delta_{es}-delta_s‎: ‎ein E‎, ‎sin S}$. ‎Recently‎, the author of this paper along with Pourabbas shown that for every $nin N$, $(2n+1)$-weak module amenability of ‎$mathcal{T}} (as a ‎$mathfrak{T}$-module) and $(2n+1)$-weak module amenability of ‎$ell^1(S)} (as a ‎$ell^1(E)$-module), are equal. In this paper, we extend this result and prove that the result is also true for the even state (2n)-weak module amenability, in the non-unitary state of these algebras.

In this paper we show that if D is a valued F-central division algebra such that its residue division ring is a field, then each soluble-by-finite subgroup of its multiplicative group is soluble with derived length at most 3. We also prove that in some special cases the above bound for the length of the derived series is at most 2. Finally, using some examples we show that if the the residue division ring is not a field then the above resuklts are not true.

In this paper, the two-observational  percentile, percentile and maximum likelihood estimation of the probability density function of  Inverse Weibull random variable are studied. Finally, these estimates are compared using simulation studies and a real data.