Acta Mathematica Vietnamica最新文献

In this paper, we investigate local derivations and local generalized derivations on path algebras associated with finite acyclic quivers. We show that every local derivation on a path algebra is a derivation, and every local generalized derivation on a path algebra is a generalized derivation. Also, we apply main results on several related maps to local derivations. The established results generalize several ones on some known algebras such as incidence algebras.

We study families of porous medium equations with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the L^p norms.

We derive an expression for the collision-induced amplitude dynamics in a fast collision between two (N + 1) −dimensional spatiotemporal solitons in saturable nonlinear media with weak perturbations in a spatial dimension of N, where N ≥ 1. The perturbed spatiotemporal soliton evolution is under a framework of the coupled saturable (N + 1 + 1) −dimensional nonlinear Schrödinger equations in the presence of weakly nonlinear loss and delayed Raman response. The perturbation approach is based on an extended perturbation technique for analyzing the collision-induced dynamics of one-dimensional temporal solitons and two-dimensional solitons. The accuracy of our theoretical calculations is validated by numerical simulations of the interaction of two 3D spatiotemporal solitons, also known as two light bullets, of the coupled nonlinear Schrödinger equations in the presence of delayed Raman response and cubic loss.

We establish several identities and inequalities of Hardy type and Sobolev type with Dunkl weights. We also investigate the sharp constants and optimal functions of the Hardy-Sobolev inequality with Dunkl weights.

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

We consider gradient estimates for H¹ solutions of linear elliptic systems in divergence form (partial _{alpha }(A_{ij}^{alpha beta } partial _{beta } u^{j}) = 0). It is known that the Dini continuity of coefficient matrix (A = (A_{ij}^{alpha beta }) ) is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L² mean oscillation ω_A,2 of A satisfies

where C_∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If X_A,2 = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying X_A,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when (nabla u in L^{infty } cap VMO).

We give a proof of the Lieb–Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma.

This paper is concerned with primal and dual second-order optimality conditions for the second-order strict efficiency of nonsmooth vector equilibrium problem with set, cone and equality conditions. First, we propose some second-order constraint qualifications via the second-order tangent sets. Second, we establish necessary optimality conditions of order two in terms of second-order contingent derivatives and second-order Shi sets for a second-order strict local Pareto minima to such problem under suitable assumptions on the second-order constraint qualifications. An application of the result for the twice Fréchet differentiable functions for the second-order local strict efficiency of that problem is also presented. Some illustrative examples are also provided for our findings.

Endpoint weak-type bounds and non-endpoint strong type bounds are obtained for the spherical maximal function in the setting of Choquet spaces with respect to certain Hausdorff contents or Sobolev capacities.

Partially saturated flow in a porous medium is typically modeled by the Richards equation, which is nonlinear, parabolic and possibly degenerated. This paper presents domain decomposition-based numerical schemes for the Richards equation, in which different time steps can be used in different subdomains. Two global-in-time domain decomposition methods are derived in mixed formulations: the first method is based on the physical transmission conditions and the second method is based on equivalent Robin transmission conditions. For each method, we use substructuring techniques to rewrite the original problem as a nonlinear problem defined on the space-time interfaces between the subdomains. Such a space-time interface problem is linearized using Newton’s method and then solved iteratively by GMRES; each GMRES iteration involves parallel solution of time-dependent problems in the subdomains. Numerical experiments in two dimensions are carried out to verify and compare the convergence and accuracy of the proposed methods with local time stepping.