A long-time dynamic for granular materials arising in the hypoplastic theory of Kolymbas type is investigated. It assumed that hardness allows exponential degradation, which leads to densification material states. The governing system a rate-independent strain under stress control described by implicit differential equations. Its analytical solution arbitrary inhomogeneous coefficients construc...

We prove that a certain finite difference scheme converges to the weak solution of the Cauchy problem on a finite interval with periodic boundary conditions for the Camassa– Holm equation ut−uxxt +3uux−2uxuxx−uuxxx = 0 with initial data u|t=0 = u0 ∈ H1([0, 1]). Here it is assumed that u0 − u′′ 0 ≥ 0 and in this case, the solution is unique, globally defined, and energy preserving.

We consider the product of a finite number of non-Hermitian random matrices with i.i.d. centered entries of growing size. We assume that the entries have a finite moment of order bigger than two. We show that the empirical spectral distribution of the properly normalized product converges, almost surely, to a non-random, rotationally invariant distribution with compact support in the complex pl...

In this paper, we prove existence results for a Predator-prey system in a polluted environment. The existence result is proved by the Schauder fixed-point theorem. Moreover, we construct a combined finite volume finite element scheme to our model, we establish existence of discrete solutions to this scheme, and show that it converges to a weak solution. The convergence proof is based on derivin...

We describe a nonlinear finite element technique to approximate the solutions of stationary Hamilton–Jacobi equations in two space dimensions using continuous finite elements of arbitrary degree. The method consists of minimizing a functional containing the L1-norm of the Hamiltonian plus a discrete entropy. It is shown that the approximate sequence converges to the unique viscosity solution un...

This paper studies action-based reinforcement learning in finite perfectioninformation games. Restrictions on the valuation updating rule that guarantee that the play eventually converges to a subgame-perfect Nash equilibrium (SPNE) are identified. These conditions are mild enough to contain interesting and plausible learning behavior. We provide two examples of such updating rule that suggest ...

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the disc...

In this paper, we consider the Darboux problem for a (1+1)dimensional cubic nonlinear Klein-Gordon equation with an external source. Stable finite difference scheme is constructed on a four-point stencil, which does not require additional iterations for passing from one level to another. It is proved, that the finite difference scheme converges with the rate O(h2), when the exact solution belon...

We describe a nonlinear finite element technique to approximate the solutions of stationary Hamilton-Jacobi equations in two space dimensions using continuous finite elements of arbitrary degree. The method consists of minimizing a functional containing the L-norm of the Hamiltonian plus a discrete entropy. It is shown that the approximate sequence converges to the unique viscosity solution und...