New concepts related to approximating a Lipschitz function between Banach spaces by a ne functions are introduced Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X Y for which any Lipschitz function from X to Y can be so approx imated is obtained This is applied to the study of Lipschitz and uniform quotient mapp...

We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival...

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c0 and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical `p-spaces into c0 and give other applications. We prove that if a Banach space embeds almost isometrical...

We consider a rate independent evolution quasivariational inequality in a Hilbert space X with closed convex constraints having nonempty interior. We prove that there exists a unique solution which is Lipschitz dependent on the data, if the dependence of the Minkowski functional on the solution is Lipschitzian with a small constant and if also the gradient of the square of the Minkowski functio...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [8]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Bana...

New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The nonsmooth nature of PWM system...

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truthtable reducibility [6]. This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solo...