The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansion...

We study Lagrangian systems on a closed manifold M . We link the differentiability of Mather’s β -function with the topological complexity of the complement of the Aubry set. As a consequence, when dimM ≤ 3, the differentiability of the β -function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of two conjectur...

In this paper we study differentiability properties of the map T 7→ φ(T ), where φ is a given function in the disk-algebra and T ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of doubl...

In this paper, solution of fuzzy differential equation under general differentiability is obtained by genetic programming (GP). The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution to this problem is qualitatively better. An illustrative numerical example is presented for the proposed method. Keywords—Fuzzy differential ...

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ω1 which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal α > 0, the set of Turing indices of C[0, 1] functions that are differentiable with rank at most ...

We discuss in this paper local uniqueness, continuity and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use two types of techniques. One approach consists in formulating variational inequalities in a form of optimization problems, based on regularized gap functions, and applying a general theory of perturbation analys...

The differentiability properties of the metric projection Pc on a closed convex set C in Hilbert space are characterized in terms of the smoothness type of the boundary of C. Our approach is based on using variational type second derivatives as a sufficiently flexible tool to describe the boundary struc ture of the set C with regard to the differentiability of Pc. We extend results by R.B. Holm...