فصلنامه مدل سازی پیشرفته ریاضی
سال دوازدهم شماره 4 (زمستان 1401)

In this article, we intend to use differential games to model Iran's relations with neighboring countries. According to the discussion of time continuity in the real world, the differential game model has been used to model the ongoing issues continuously over a period of time and provide more realistic results. In this article, we first introduce the differential game, then the game model and the Hamilton-Jacobi-Bellman equation are described. In the first method, the famous Cobb-Douglas function was used instead of the utility function, and in the second method, we used this function by making changes to the Berman function. In the first method, in general, the amount of military expenditure and the amount of military equipment are balanced in the situation of the Markovian strategy. But in the second method, by considering the sub-sets of each of the military cost and military equipment sets, we have obtained the values ​​of military costs and the amount of balance equipment for each sub-set in more detail

In this article, we examine the issue of pricing perpetual American option with a differential equation approach with free boundary properties. To describe the underlying asset dynamics in these options, we use the feature of jump-diffusion models under regime switching. In pricing these perpetual options, due to the possibility of early application, we need to solve the ordinary integro-differential equation with a free boundary. For this purpose, we write the equation created from this model first as a linear complementarity problems and then discretize by using the finite difference method. We use linear interpolation to approximate theintegral term. The discrete maximum principle is applied to the linear complementarityproblems to obtain the error estimates. We also illustrate some numerical results in orderto demonstrate and compare the accuracy of the method for our problem.

In Poisson sampling, each unit is selected independently of the other units with a certain inclusion probability, and the sample size n(s) is a random variable with large variation. Also, if there exists a trend between units of the population, it causes bias in the estimated population parameter. So Poisson-correlated sampling (CPS) and Poisson-correlated spatial sampling (SCPS) were introduced as alternative methods to Poisson sampling that reduces changes in the sample size and bias in the estimated population parameter by weighting strategies. In this paper, new strategies for choosing weights are introduced and it is shown by simulation that the new weighting strategies increase the efficiency of the estimated parameter compared to the earlier weighting strategies.

In this research, for a zero-dimensional space $X$, a Banach subalgebra $U(X)$ of $C^{*}(X,\mathbb{C})$ is introduced. It is shown that $U(X)$ is the uniform closure of the subalgebras $C^{F}(X,\mathbb{C})$ and $C^{*}_{c}(X,\mathbb{C})$ of the Banach algebra $C^{*}(X,\mathbb{C})$. Moreover a necessary and sufficient condition for the coincidence of $U(X)$ and $C^{*}(X,\mathbb{C})$ is given. It is shown that $U(X)$ consists exactly of all $f\in C^{*}(X,\mathbb{C})$ each of which has an extension to$\beta_{\circ}X$. Using this fact, an isometric isomorphism from $U(X)$ onto $C(\beta_{\circ}X,\mathbb{C})$ is defined. Finally, a description of the elements of $U(X)$ in terms of the inverse image of the closed subsets of $\mathbb{C}$ is given.

In this paper, after introducing the $m$-convexity by Toader, as an intermediate among the general convexity and star shaped property, we bring Hermite-Hadamard integral inequality on $(\alpha,m)$-convex function in the new form. Previous results about the Hermite-Hadamard inequality for $m$-convex functions are part of the results of our theorems. Illustrated examples of $(\alpha,m)$-convex and $m$-convex functions are also included in the article.

‎The square and real matricx $A$ is called a generalized row substochastic matrix‎, ‎if the sum of the absolute values of the entries in each row is less than or equal to one‎.‎Let $x,y\in \mathbb{R}^n$‎. ‎We say that $x$ is right $B$-majorized (resp‎. ‎left $B$-majorized) by $y$‎, ‎denoted by $x \prec _{rB} y$ ($x \prec _{lB} y$)‎, ‎if there exists a substochastic matrix $D$‎, ‎such that $x=yD$ (resp‎. ‎$x=Dy$)‎. ‎In this article‎, ‎we have found all the vectors such as $x$ that $x$ is right $B$-majorized (resp‎. ‎left $B$-majorized) by $y$‎, ‎for all row vector $y$ (resp‎. ‎column vector)‎. ‎Also‎, ‎we show $x \sim _{lB} y$ if and only if $\Vert x\Vert_\infty =\Vert y\Vert_\infty$ and prove $x \sim _{rB} y$ if and only if $\Vert x\Vert_1 =\Vert y\Vert_1$‎.‎We have also created conditions in which the left $B$-majorization is equivalent to the left majorization‎, ‎and created conditions in which the right $B$-majorization is equivalent to the right majorization‎.

Cells in all tissues of the body are constantly growing and dividinginto new cells. Abnormal proliferation of tissues outside the body leads tocancer. In all tissues of the body, a type of cell, called a stem cell, is foundthat has the ability to become specialized cells in the same tissue to be able tocompensate for damage in tissue disorders. In this paper, based on the existingmathematical model, the optimal control model is very effective for inhibitingthis growth in exchange for prescribing a specific drug (doxorubicin) is presented.In order to minimize the number of cancer cells over time, the cancer controlstrategy has been modeled as a problem from the theory of optimal controlduring the effect of a specific drug on non-cancerous stem cells in this model.To solve this problem and prescribe the optimal dose of the drug, first with thehelp of the maximum principle of Pontriagin and then the analytical solution ofthe first-order differential equations, the optimal solution has been determined.In order to provide the optimal dose of the drug to the patient, the proposedsolution is simulated numerically. This numerical implementation shows how byapplying this amount of drug with a specific dose, how the number of cancercells decreases over time, they will be.

The Ritz values of a matrix are the set of all the eigenvalues of the leading principal submatrices. In this paper, assuming that the set of Ritz values is given from the dimension of maximum three, we find a matrix such that the given set is its Ritz values. The conditions for the existing solution are also studied.

‎In ‎this ‎paper ‎we‎ introduce and study the three concepts of small endomorphic, small homomorphic and essential homomorphic modules using the tools of endomorphism and homomorphism which are known in module theory as important means of transmitting some algebraic properties. We have also examined some of the relationships between these three concepts as well as their relationship to different categories of modules.

‎In recent decades researchers introduced many numerical methods for solving partial differential equations. Some of these methods have limitations in solving problems with complex domains because of the need to construct meshes. Therefore, scientists developed a new set of numerical methods called meshless methods. In this paper, we introduce the direct meshless local Petrov-Galerkin method to the numerical study of the nonlinear two-dimensional Fisher equation. This method is based on the local weak form of the equation and uses the generalized moving least square method to approximate the unknown function. To show the efficiency and capability of the method, we report the numerical results in regular and irregular domains with a uniform and scattered distribution of nodes. Comparison of the obtained results with other methods indicates the accuracy and efficiency of this method.

In this paper, a new method for solving optimal control problems governedby nonlinear Volterra integral equations is presented. First by converting toa discretized form, the problem is considered as a quasi assignment problem and then an iterative method is applied to find approximate solutionfor discretized form of the integral equation. Next step using evolutionaryalgorithms, approximate solution of optimal control problems is obtained.An analysis for convergence of the proposed iterative method and its implementation for numerical examples are also given.

‎A hybrid censoring is a mixture of Type I and Type II censoring schemes‎, ‎which is divided into Type I and Type II hybrid censoring according to the criteria for terminating the test‎. ‎In this article‎, ‎considering Type II hybrid censoring scheme of Pareto distribution‎, ‎the optimal censoring scheme is determined‎. ‎To determine the optimal censoring scheme‎, ‎various factors can be considered‎, ‎the most important of which is the sampling cost criterion‎. ‎Therefore‎, ‎the optimal censoring scheme is determined so that the total cost of the test does not exceed a predetermined value‎. ‎To evaluate the obtained results‎, ‎numerical computations have been performed‎. ‎A real example is also expressed. Finally‎, ‎the conclusion of the article is presented‎.