In the numerical solution of stochastic diierential equations (SDEs) such appearances as sudden, large uctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. Fo...

in this paper for a given prescribed ritz values that satisfy in the some special conditions, we nd a symmetric nonnegative matrix, such that the given set be its ritz values.

We prove instantaneous blow-up of nonnegative solutions for a class of semilinear hyperbolic inequalities with first order terms and singular coefficients. The present approach relies on a suitable choice of the functions used to test the differential inequalities.

Nonnegative solutions are established for nonlinear integral equations where the nonlinearity G(t, y) may be singular at y = 0. Existence is established using the Leray–Schauder alternative. c © 2007 Elsevier Ltd. All rights reserved.

For given nonnegative integers k, s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is ut = ∇·(u∇(−∆)−1/2u). For definiteness, the problem is posed in {x ∈ RN , t ∈ R} with nonnegative initial data u(x, 0) that are integrable and decay at infinity. Previous papers have established the existence of mass-preserving, nonnegative ...

We prove local integral (entropy) estimates for nonnegative solutions of the fourth order degenerate parabolic equation u t + div(u n ru) = 0 in space dimensions two and three. These estimates enable us to show that solutions have nite speed of propagation if n 2 (1 8 ; 2) and that the support cannot shrink if the growth exponent n is larger than 3=2. In addition, we prove decay estimates for s...

The number ν d (n) of linearly independed homogeneous invariants of degree n for the ternary form of degree d is calculated. The folloving formula is hold is number of nonnegative integer solutions of the system of equations

Given a nondecreasing nonlinearity f , we prove uniqueness of large solutions to the equation (1) below, in the following two cases: the domain is the ball or the domain has nonnegative mean curvature and the nonlinearity is asymptotically convex.

We establish a Fujita-type theorem for the blow-up of nonnegative solutions to a certain class of parabolic inequalities in the Heisenberg group. Our proof is based on a duality argument.