Generalized flows and singular ODEs on differentiable manifolds

Lipschitz perturbations of differentiable implicit functions

Let $y=f(x)$ be a continuously differentiable implicit function solving the equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show that if $F_\eps$ is a Lipschitz function such that the Lipschitz constant of $F_\eps-F$ goes to 0 as $\eps\to 0$ then the equation $F_\eps(x,y)=0$ has a Lipschitz solution $y=f_\eps(x)$ such that the Lipschitz constant of $f_\eps-f$ goes to 0 as $\eps\to 0$ either. As an application we evaluate the length of time intervals where the right hand parts of some nonautonomous discontinuous systems of ODEs are continuously differentiable with res…

可微奇异值分解

Differentiable Singular Value Decomposition

针对一般复矩阵SVD导数计算在精度与可扩展性上的局限，本文提出两种基于伴随方法和反向自动微分的高效求导算法，经验证精度达5至7位有效数字，并在大型湍流数据集POD中展示了良好的可扩展性。

Large algebras of singular functions vanishing on prescribed sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.

Ordinary differential equations with singular coefficients: an intrinsic formulation with applications to the Euler-Bernoulli beam equation

We study a class of linear ordinary differential equations (ODE)s with distributional coefficients. These equations are defined using an {\it intrinsic} multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with non-intersecting singular supports [L. Hörmander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983]. We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and dis…