Related papers: Arithmetic Laplacians
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 is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois…
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 study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some…
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…
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,…
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…
The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlev\'{e}, etc.) were previously…
In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition,…
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 prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
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…
Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…
Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and…
New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We…
The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the…
The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are…
We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the…