Let I ⊆ R be an interval and let k : I2 → C be a reproducing kernel on I . We show that if k(x, y) is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if k is a positive definite kernel on I (possibly unbounded) w...

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(...

A new formula for continuum percolation on the Euclidean space R (d ≥ 2), which is analogous to Russo’s formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite c...

One dimensional optimization on non-Archimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimization. This is due to the total disconnectedness of the given non-Archimedean field in the order topology, ...

We find an equivalent condition for a real function to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a V BG 1 2 function, which plays an analogous rôle for the second order differentiability as the classical notion of a V BG∗ function for the first order differentiability. In fact, for a function f : [a, b] → R, being Lebesgue equivalent...

In this paper, we investigate the existence and characterizations of the Fréchet derivatives of the solution to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique the factorization of the difference of the far-field pattern for two different scatterers introduced by Kress and Païvarinta to establish Fréchet differentiabil...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1,61, 389) with so much ingenuity...and other equations occurring in the literature of mathematics, do not di...