We extend the proof of Reifenberg’s Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set E for the existence of a bi-Lipschitz parameterization of E by a d-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers β1(x, r). In particular, it applies in the locally Ahlfors-regular ...

Positive results are proved here about the ability of numerical simulations to reproduce the exponentialmean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step si...

It is well known that under the Wolfe–Powell inexact line search, global convergence of nonlinear conjugate gradient method always requires Lipschitz continuous condition for nonconvex functions. In this paper, we find unnecessary proving particular algorithms if its searching direction has well-known sufficient descent property and trust region feature. Thus, family proposed by Yuan et al. (Nu...

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the W 1,p (1 < p < ∞) estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. T...

*Correspondence: [email protected] 1College of Information Sciences and Technology, Donghua University, Shanghai, China 2School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing, China Full list of author information is available at the end of the article Abstract In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed st...

In many practical decision-making problems it happens that functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. In this paper, a global optimization problem is considered where both the goal function f(x) and its gradient f (x) are black-box functions. It is supposed that f (x) satisfies the Lipschitz condition over the search hy...

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo simulation and other numerical integrations for estimating conditional expec...