Let Kn be the Lorentz/second-order cone in IR. For any function f from IR to IR, one can define a corresponding vector-valued function f soc (x) on IR by applying f to the spectral values of the spectral decomposition of x ∈ IR with respect to Kn. It was shown by J.-S. Chen, X. Chen and P. Tseng that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity,...

The theory of generic differentiability of convex functions on Banach spaces is by now a well-explored part of infinite-dimensional geometry. All the attempts to solve this kind of problem have in common, as a working hypothesis, one special feature of the finite-dimensional case. Namely, convex functions are always considered to be defined on convex sets with nonempty interior. But typically, ...

This paper focuses on the differentiability properties of the control-to-state mapping for entropy solutions to a scalar hyperbolic conservation law on R with respect to the switching times of an on/off control. As an example we consider the traffic density on a unidirectional road described by the LWR-model that is controlled by a traffic light switching between red and green at certain times....

1. In this paper we will give necessary and sufficient conditions for a measurable function to be equivalent to one which is differentiable a. e. on a set E. The condition is in terms of Marcinkiewicz type integrals which have also been recently the main objects in problems of differentiability. The reader is especially referred to the important paper on differentiability by E. M. Stein and A. ...

We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x . This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions. 2010 Mathematics Subject Classification 03E15; 03F60 (p...

We prove that every perfect torsion theory for a ring R is differential (in the sense of [2]). In this case, we construct the extension of a derivation of a right R-module M to a derivation of the module of quotients of M . Then, we prove that the Lambek and Goldie torsion theories for any R are differential.