Existence and uniqueness of solutions of reaction–convection equations with non-Lipschitz nonlinearity

In this work we study single balance law _ut+∇⋅Φ(u)=f(u)$u_t+\mathrm{\nabla }\cdot \Phi (u)=f(u)$ with bounded initial value, and find that there may exist maximal and minimal solutions, if f(u)$f(u)$ is not Lipschitz continuous at u=0$u=0$. We also show that comparison principle is valid for such solutions, and the solutions may blow up or not under certain conditions. It is determined by the strength of source supply, as well as the competition between the source and flux.     

Verification by Stochastic Perron's Method in stochastic exit time control problems

We apply the Stochastic Perron Method, created by Bayraktar and Sîrbu, to a stochastic exit time control problem. Our main assumption is the validity of the Strong Comparison Result for the related Hamilton–Jacobi–Bellman (HJB) equation. Without relying on Bellman's optimality principle we prove that inside the domain the value function is continuous and coincides with a viscosity solution of the Dirichlet boundary value problem for the HJB equation.     

A characterization of spaces of constant curvature by minimum covering radius of triangles

In this paper, we prove that if in a Riemannian manifold, the minimum covering radius of a point triple of small diameter depends only on the geodesic distances between the points, then the manifold must be of constant curvature. This implies that if in a complete connected Riemannian manifold, the volume of the intersection of three small geodesic balls of equal radii depends only on the distances between the centers and the radius, then it is one of the simply connected spaces of constant curvature. This generalizes an earlier result of the first author and D. Kunszenti-Kovács (2010).     

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.     

Stability and boundedness criteria of nonlinear impulsive systems employing perturbing Lyapunov functions

This paper develops a new comparison principle for nonlinear impulsive differential systems, then the stability, practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of impulsive systems under much weaker assumptions. The reported novel results complement the existing results. It may provide a greater prospect for solving problems which exhibit impulsive effects.     

Terrestrial Bradley aberration cannot be observed

Because terrestrial sources radiate in all directions, collimation cannot produce a beam fixed in direction relative to absolute space. Therefore, terrestrial aberration, even though present, cannot be observed; and terrestrial aberration cannot be used to measure the absolute velocity of the closed laboratory (contradicting a prior claim in this journal). In other words, the large parallax of terrestrial sources frustrates the observation of aberration.     

A NEW LAPLACIAN COMPARISON THEOREM AND THE ESTIMATE OF EIGENVALUES

This paper establishes a new Laplacian comparison theorem which is specially useful to the manifolds of nonpositive curvature.It leads naturally to the corresponding heat kernel comparison and eigenvalue comparison theorems. Furthermore, a lower estimate of L^2-spectrum of an n-dimensional non-compact complete Cartan-Hadamard manifold is given by (n-1)k/4,provided its Ricci curvature ≤-(n-1)k(k=const.≥0).     

p(x)-Laplace方程的强极大值原理

本文给出了如下形式的p(x)-Laplace方程 -div(|u|~(p(x)-2)u)+d(x)|u|~q(x)-2)u=0,x∈Ω的一个强极大值原理,其中p(x),q(x),d(x)满足一定的条件。     

Reflected Backward SDEs with Two Barriers Under Monotonicity and General Increasing Conditions

In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on y, with Lipschitz condition on z.