Related papers: A note on Leibniz rule for difference quotient

A short note on bounds on distance to variety of a point in terms of the Taylor coefficients at the point.

We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new "conceptual" proof to equidistribution results…

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…

It is well known that the Leibniz rule for the integer derivative of order one does not hold for the fractional derivative case when the fractional order lies between 0 and 1. Thus it poses a great difficulty in the calculation of…

We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…

In this paper, we study the fundamental properties of Leibniz rings. Special attention is given to the structure of Leibniz rings whose additive group is "small". The results obtained illustrate a significant difference between the classes…

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…

This note is the follow up to a paper by M. Waldschmidt.

This note describes a way of obtaining e that differs from the standard one. It could be used as an alternate way of showing how the value of e is obtained. No attempt is made to show the existence of the limit in the definition of e that…

These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

We establish a large deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, the large deviation principle is derived for super-Brownian…

In this work we present a novel proof of the Saalschutz formula by using the theory of discrete fractional calculus. The proofs of some results within this theory, namely, the fractional power rule and the fractional Leibniz rule are…

In this work, a generalization of the well known Bernoulli inequality is obtained by using the theory of discrete fractional calculus. As far as we know our approach is novel.

This note is designed to show some classes of differential-difference equations admitting Lax representation which generalize evolutionary equations known in the literature.

We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel--Freidlin theorem, but under the considerably…

New cases of the multiplicity conjecture are considered.

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

These are the notes from my courses on the arithmetic of quadratic forms.

We survey the classical results of the Dirichlet Approximation Theorem.

In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.