Abstract In this chapter, we discuss the higher order differentiability properties of Σ g when lies in $$\mathcal {C}^r\cap \mathcal {D}^p\cap {K}^{\max \{p,r\}}$$ C r ∩ D p K max</mm...

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.

For a bounded domain Ω ⊂ Rn , n > 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system −∆u + u · ∇u +∇p = f , div u = k, u|∂Ω = g with u ∈ L q , q > n, and very general data classes for f , k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular s...

This paper considers some path pro~erties of real separable Gaussian processes ~ with parameter set an arbitrary interval. The following results are established, among others. At every fixed point the paths of ~ are continuous, or differentiable, with probability zero or one. If ~ is measurable, then with probability one its paths have essentially the same points of continuity and differentiabi...

The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from (vectorial) Tresca friction problem found in contact mechanics. weak formulation considered leads variational inequality second kind depending on perturbation parameter. unique solution this then c...

In the paper compressible, stationary Navier-Stokes (N-S) equations are considered. The model is well-posed, there exist weak solutions in bounded domains, subject to inhomogeneous boundary conditions. The shape sensitivity analysis is performed for N-S boundary value problems, in the framework of small perturbations of the so-called approximate solutions. The approximate solutions are determin...