Related papers: Arithmetic Euler Top
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…
The theory of differential equations has an arithmetic analogue in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear…
Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental…
We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us…
This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by…
We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The…
We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat…
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations…
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati…
Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the…
The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is…
In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have…
In this paper we show that there are applications that transform the movement of a pendulum into movements in $\mathbb{R}^3$. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top…
Different fractional difference types of Euler-Lagrange equations are obtained within Riemann and Caputo by making use of different versions of integration by part forumlas in fractional difference calculus. An example is presented to…
In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give…
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…
Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations…