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...

The theory of scale relativity [14] is an attempt to study the consequences of giving up the hypothesis of space-time differentiability. One can show [14] [15] that a continuous but nondifferentiable space-time is necessarily fractal. Here the word fractal [12] is taken in a general meaning, as defining a set, object or space that shows structures at all scales, or on a wide range of scales. Mo...

We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as black-box optimization,...

We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to ...

A simple and deep standard mathematical theorem asserts the existence, for any oneparameter differentiable group, of an additive parameter, such as the angle for rotations and the rapidity parameter for Lorentz transformations. The importance of this theorem for the applications of group theory in physics is stressed, and an elementary proof is given. The theorem then is applied to the construc...

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...

Speech enhancement employing deep neural networks (DNNs) for denoising is called noise suppression (DNS). The DNS trained with mean squared error (MSE) losses cannot guarantee good perceptual quality. Perceptual evaluation of speech quality (PESQ) a widely used metric evaluating However, the original PESQ algorithm non-differentiable, therefore, directly be as optimization criterion gradient-ba...