arXiv Math @arxiv_math@qoto.org

This study investigates the existence and uniqueness of solutions to Volterra integral equations with discontinuous kernels in both linear and nonlinear cases. The problem is two-dimensional, and the collocation method is employed to analyze the equations. The research aims to provide a comprehensive understanding of the solution properties of these integral equations, which are crucial in various mathematical and physical applications. By examining the existence and uniqueness of solutions, this study contributes to the development of numerical methods for solving Volterra integral equations with discontinuous kernels. The findings of this research have the potential to impact various fields, including physics, engineering, and economics, where integral equations play a significant role in modeling complex phenomena.

The Householder's method is a root-find algorithm which is a natural extension of the methods of Newton and Halley. The current paper mostly focuses on approximating the square root of a positive real number based on these methods. The resulting algorithms can be expressed using Chebyshev polynomials. An extension to the nth root is also proposed.

We examine space-filling curves, which are surjective continuous maps from $[0,1]$ to some higher-dimensional space, usually the unit square $[0,1]^2$. In particular, we define Peano's curve and Lebesgue's curve, and state some of their properties. We also discuss the Hahn-Mazurkiewicz theorem, which characterizes those subsets of $\mathbb{R}^n$ that are the image of a space-filling curve. Finally, we discuss real-world applications of Hilbert curves, in particular Google's $S2$ Cells.

We present if and only if conditions for a group homomorphism between spaces of smooth sections of Lie group bundles to be a weighted composition operator. These results provide new insights into a wide range of problems related to weighted composition operators. Specifically, we prove that the algebraic structure of the space of smooth sections of an algebra bundle, where the typical fiber is a finite dimensional simple algebra, completely determines the bundle structure. Furthermore, we derive a homomorphism version of the Shanks-Pursell theorem and identify a class of homomorphisms of multiplicative semigroups between spaces of smooth functions on finite dimensional manifolds, including all isomorphisms. Our approach is based on a method called range decreasing group homomorphism.

In this paper, we study hyperrigidity for $C^*$-algebras. We will show that hyperrigidity can be expressed solely in terms of representations, without the need to involve general unital completely positive maps.

Many applications like subseismic fault modeling, fractured reservoir modeling and interpretation/validation of fault connectivity involve the solution to an elliptic boundary value problem in a background medium perturbed by the presence of cracks that take the form of one or many pieces of surface (with boundary). When the background medium can be considered as homogeneous, boundary integral equations appear as a method of choice for the numerical solution to fractures problems. With such an approach, the problem is reformulated as a fully non-local equation posed at the surface of cracks. Discretization of boundary integral resulting in the so-called Boundary Element Method (BEM) leads to densely populated matrices due to the full non-locality of the operators under consideration. After the discretization process, geologists are faced with a system of equations that turns out difficult to solve numerically. Many empirical algorithms have been proposed by geologists to solve this system of equations. Unfortunately, none of them is guaranteed to converge in theory (in particular when faults (fractures) intersect each other forming a geometrically highly irregular structure). In practice, none of them appears to be either robust or efficient. We investigate another approach, referred to as interior point methods, for which convergence can be ensured (even if faults are too close). Interior point methods have proved their efficiency in a wide variety of domains, most notably for linear programming. Here, even though we do not have any optimization problem, we can adapt ideas from interior point methods for the numerical resolution of the system considered. The numerical results obtained demonstrate computational efficiency and accuracy, highlighting the robustness and effectiveness of the implemented methods.

In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function.

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients of $E$ of degree $d$. In this article, we survey some recent results on various invariants of $\mc Q$.

This paper studies a new class of linear-quadratic mean field games and teams problem, where the large-population system satisfies a class of $N$ weakly coupled linear backward stochastic differential equations (BSDEs), and $z_i$ (a part of solution of BSDE) enter the state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we derive a feedback form optimal strategy by introducing Riccati equations, stochastic differential equation (SDE) and BSDE. Finally, we provide a numerical example to simulate our results.

Rings in which the square of each unit lies in $1+Δ(R)$, are said to be $2$-$ΔU$, where $J(R)\subseteqΔ(R) =: \{r \in R | r + U(R) \subseteq U(R)\}$. The set $Δ(R)$ is the largest Jacobson radical subring of $R$ which is closed with respect to multiplication by units of $R$ and is studied in \cite{2}. The class of $2$-$ΔU$ rings consists several rings including $UJ$-rings, $2$-$UJ$ rings and $ΔU$-rings, and we observe that $ΔU$-rings are $UUC$. The structure of $2$-$ΔU$ rings is studied under various conditions. Moreover, the $2$-$ΔU$ property is studied under some algebraic constructions.

We refine the formulation of the Boolean satisfiability problem with $n$ Boolean variables in Clifford algebra ${\cal C}\ell(\mathbb{R}^{n,n})$ [3] and exploit this continuous setting to design a new unsatisfiability test. This algorithm is not combinatorial and proves unsatisfiability in polynomial time.

In this manuscript, corresponding to the weighted Ricci curvature, we discuss a class of Einstein Finsler metrics, which can be considered as a sequence of Einstein metrics on the general $CD(\infty,K)$ space. We investigate the non-Riemannian curvature conditions to ensure the equivalence of those metrics. With the bounds of the Ricci curvature, we can estimate the lower bounds of the S-curvature and the distortion. By arising a scalar curvature, we obtain an important formula relating the scalar curvature and the distortion. In some assumptions about some non-Riemannian curvatures, we get both the upper and lower bounds of the scalar curvature, the S-curvature and the distortion. We also discuss such bounds by using the lower bounds of the scalar curvature on the forward complete asymmetric $\infty$-Einstein Finsler manifold. The results in this manuscript will further lead us to study geometric analysis problems in general Finsler geometry.

This paper deals with a stochastic optimal feedback control problem for the controlled stochastic partial differential equations. More precisely, we establish the existence of stochastic optimal feedback control for the controlled stochastic partial differential equations with fully monotone coefficients by a minimizing sequence for the control problem. Using the Faedo-Galerkin approximations, the uniform estimates and the tightness in some appropriate space for the Faedo-Galerkin approximating solution can be obtain to prove the well-posedness of the controlled stochastic partial differential equations with fully monotone coefficients. The results obtained in the present paper may be applied to various types of controlled stochastic partial differential equations, such as the controlled stochastic convection diffusion equation.

In the following text for Khalimsky $n-$dimensional space $\mathcal{K}^n$ we show self--map $f:\mathcal{K}^n\to\mathcal{K}^n$ has closed graph if and only if there exist integers $λ_1,\ldots,λ_n$ such that $f$ is a constant map with value $(2λ_1,\cdots,2λ_n)$. We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.

In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.

The property of $*$-cleanness in group rings has been studied for some groups considering the classical involution, given by $g^*=g^{-1}$. A group is called an SLC-group if its quotient by its center is isomorphic to the Klein group; these groups are equipped with its own canonical involution, which usually does not coincide with the classical one. In this paper we study the $*$-cleanness of $RG$ when $G$ is an SLC-group, considering $*$ as its canonical involution. In that context, we prove that if $RG$ is $*$-clean then $G$ is the direct product of $Q_8$ and an abelian group with some extra properties and we find a converse for some specific cases, generalizing a result by Gao, Chen and Li for $Q_8$.

The purpose of this paper is to provide conditions for the existence or non existence of non trivial Einstein multiply warped products, specially of generalised Kasner type; as well as to show estimates of the Einstein parameter that condition the existence of such metrics.