Journal of Advanced Mathematical Modeling
Year:2024 | Volume:14 | Issue:4|Papers Count:11

In this paper, a system of elliptic equations with a perturbation in the system and also the nonlinear Neumann boundary conditions near a conical singular point are studied. By introducing the conical Sobolv space and using variational methods and Nehari manifold method, we will prove the existence of at least two positive solutions for this problem on the conical Sobolev space.

In this paper, we study the categorical structures of the concepts of ideal, grill, primal, and filter, which play a fundamental and important role in the study of topology spaces. Some researchers have shown that all these concepts are equivalent, regardless of the categorical viewpoint. Here, we define the homomorphisms and so the category of each of these concepts and then show that all of them are isomorphic, except for the category of filters. As an important result, it has been shown that these categories are topological, except for the category of filters. Consequently, they are complete and cocomplete, but they are not algebraic. Thus, many topological properties can be easily expressed and analyzed in terms of each of these concepts. Finally, some categorical structures of ideals are described.

Since most integral equations of the first kind are ill-posed, in this article we tried to present a method based on Legendre wavelets to solve some of them. The proposed method is that at first all the components of the equation are expressed in matrix form based on Legendre wavelet functions, then these matrices are replaced in the main equation. This method, along with the collocation method, adjusts the equation to a system of algebraic equations, which can be solved by usual numerical or analytical methods. To show the convergence of the method, some computable error bounds are obtained by preparing some theorems and lemmas. Moreover, the stability analysis, the existence, and uniqueness of the solution are investigated. Numerical results with comparisons are given to confirm the reliability of the proposed method. In practice, in order to check the stability, we calculated the condition number in the examples, and also the convergence rate was calculated and the convergence was displayed in graphs by the ratio test. The outcomes reveal that this method is very effective and more accurate than those of the literature. Also, in many cases, one can get the exact solution to the problem.

In this paper, we study the reaction-diffusion equation $\Delta u(x) = |u(x)|^{\gamma(x)-1} u(x) $ from a regularity point of view. This equation is used for modelling the distribution of a gas in an inhomogeneous porous catalyst. And the power $\gamma(x)$ can be a discontinuous function. In particular, we study the vanishing order of solution near the zero level set $\{u=0\}$.

In this paper ‎a bi-level game problem with L-R fuzzy coefficients is studied‎. ‎Using the concept of alpha-cut‎, ‎this problem is converted to a bi-level problem with nonfuzzy coefficients‎. ‎Three kinds of optimistic‎, ‎pessimistic and intermediate ‎solution‎ are introduced for the problem‎, ‎by considering the status of decision makers‎, ‎and in each case‎, ‎the bi-level problem is converted to a single-level problem using the Karush–Kuhn–Tucker optimality conditions‎. ‎Finally‎, ‎an interactive algorithm is proposed that obtains the appropriate solution for the decision maker‎. ‎At the end of the paper‎, ‎an economic example and a case study for the production of polyethylene bags are presented and solved by the proposed algorithm.

As an alternative to the proof given by Ansari and Bourdon [1], we present here a simple and self-contained proof that isometries are not supercyclic. As compared to their proof, which is based on a result that suggests that isometries always have nontrivial invariant subspaces [6], our proof is independent of this result and provides a more direct proof that isometries do not have supercyclic vectors.

In this ‎paper‎, we present a generalization of the evolute of a curve in the plane and study its geometry. Consider a curve in the plane, denoted by ‎‎$‎\gamma‎$‎, with curvature ‎$‎\kappa‎$‎. If ‎‎f‎‎ is a smooth real-valued function, we define the spatial curve ‎$‎\gamma_{f}‎$‎ in such a way that it ‎is the ‎generalization of the evolute of ‎$‎\gamma‎$‎. The process of obtaining this curve is through the introduction of an angular ‎surface‎. Theorem 2 shows that the singular points of ‎$‎\gamma_{f}‎$‎ correspond to the vertices of ‎$‎\gamma‎$‎ and are independent of the choice of the function ‎$‎f‎$‎. In such points, the generalized evolute has a cusp singularity if and only if the curve ‎$‎\gamma‎$‎ has a regular vertex at ‎$‎s=s_0‎$‎. Furthermore, we investigate the contact of the generalized evolute ‎$‎\gamma_f‎$‎ with a sphere and a plane. Additionally, by introducing a certain type of parallel curve, we study its geometric properties.

‎Let $ \mathcal{A} $ and $ \mathcal{B} $ be two algebras and $ \lambda‎, ‎\varphi $ and $ \psi $ be linear mappings from $ \mathcal{A} $ into $ \mathcal{B} $‎. ‎$ \lambda $ is said to be‎ ‎a pexider Lie $ (\varphi,\psi) $-derivation‎, ‎if $ \lambda([a_1,a_2])=[\varphi(a_1),a_2]‎ + ‎[a_1,\psi(a_2)]$ for all $ a_1,a_2 \in \mathcal{A} $‎, in which $[a_1,a_2]=a_1a_2‎ -‎a_2a_1$ is the Lie product of the elements $a_1,a_2 \in \mathcal{A}$‎.‎In this paper‎, ‎we introduce the concept of a pexider Lie $ \{\Phi_n,\Psi_n\} $-higher derivation as a sequence of linear mappings $ \{\Lambda_n\}_{n=0}^\infty $ from $\mathcal{A}$ into $\mathcal{B}$ satisfying the equation‎‎\begin{equation*}‎‎\Lambda_n([a_1,a_2])=\sum_{i+j=n}[\Phi_i(a_1),\Psi_j(a_2)]‎,‎\end{equation*}‎‎for all $ a_1,a_2 \in \mathcal{A} $ and all non-negative integers $ n $‎.‎Then we characterize it in terms of sequence of pexider Lie $ \{\varphi_n,\psi_n\} $-derivations $ \{\lambda_n\}_{n=1}^\infty $ from $\mathcal{A}$ into $\mathcal{B}$‎. Also, we show that there is a one-to-one correspondence between the set of all pexider Lie $ \{\Phi_n,\Psi_n\} $-higher derivations $ \{\Lambda_n\}_{n=0}^\infty $ and the set of all ‎sequences‎ $ \{\lambda_n\}_{n=1}^\infty $ of pexider Lie $ \{\varphi_n,\psi_n\} $-derivations.

Statistical significance determines whether the relationship between two or more variables is caused by factors other than chance and randomness. Statistical hypothesis testing is a method by which statistical significance is determined. By introducing fuzzy regression, several approaches were presented to estimate parameters of such the models. Now, most researches have conducted on the estimation method and little attention has been paid to the properties of estimators,, confidence intervals and significance tests. The main idea here is to investigate the significance of estimated parameters in an applied study with fuzzy-valued real data in the framework of fuzzy regression modeling using $m$-estimators. For this purpose, by accessing the water and sewage data source of Ahvaz city, the required variables were first introduced. Since $m$-estimators use an algorithm based on reweighted method, we are looking to determine the weight of the users in terms of the consumption pattern, that is, the weight of high consumption users and low consumption users is determined. Now, the company can first identify and classify the consumption patterns of each user based on the determined weights, and then deal with the stepped pricing of each cubic meter of water for each user. This idea requires providing a significant model of fuzzy regression. Since $m$-estimators of fuzzy regression model do not have a closed form, bootstrap is used to present the numerical indices of the estimators, confidence intervals and the significance test of the model. By analyzing the results and keeping significant parameters a suitable model was introduced.

In this article, we solve control problems with the conditions of parabolic partial differentialequations. In order to solve this type of mentioned problems, we use an iterative method based on basic polynomials and generating function. In this method, we first consider the solution of the problem based on the given partial differential equation, and then we approximate this solution based on the basic polynomials and the generating function (the function that applies to the initial conditions of the problem). which apply precisely in the given boundary conditions. After that, by placing this solution in the given sub-function and optimality conditions, we reach a system of nonlinear equations. By solving this device of unknown coefficients, we get the approximate answer. In fact, an approximation of the answer is made in a finitedimensional space. We have also shown the convergence of the proposed method for this type of problem.In order to check the effectiveness of the method, we have solved some examples using it and presented the obtained numerical results.

Let $\phi$ be a nonstandard involution on the set of all quaternion numbers and $\alpha$ be a quaternion such that $\phi(\alpha) = \alpha$. For any square quaternion matrix $A$, the numerical range of $A$ with respect to $\phi$ (shortly, the nonstandard quaternionic numerical range of $A$), is denoted by $W_\phi^{(\alpha)}(A)$. If $\alpha \neq 0$, then $W_\phi^{(\alpha)}(A)= \phi(\gamma) W_\phi^{(1)}(A) \gamma$ for some quaternion $\gamma$ with $\phi(\gamma) \gamma = \alpha$. So, the focus is on two particular nonstandard quaternionic numerical ranges $W_\phi^{(0)}(A)$ and $W_\phi^{(1)}(A)$. In this paper, a description of the intersection of $W_{\phi}^{(1)}(.)$ with a $2-$dimensional space is given, and then by using it, $W_{\phi}^{(1)}(.)$ is studied for $\phi-$hermitian quaternion matrices. After that, a class of quaternion matrices for which $W_\phi^{(0)}(.)$ and $W_\phi^{(1)}(.)$ are equal, is given. For further research, an open problem in the form of a conjecture is also given.