Related papers: Arithmetic Laplacians

In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex…

In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum…

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an…

In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where…

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns…

We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to…

Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $\delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable…

Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining integer-order derivatives. Although the two approaches lead (under…

For smooth bounded open sets in euclidean space, we construct corresponding contractive linear extension operators for the space of continuous functions which preserve regularity of functions in the domain of the Robin Laplacian. We also…

This paper is concerned with the construction of discrete counterparts of the Laplace-Beltrami operator on Riemannian manifolds that can be effectively used in the numerical solution of partial differential equations. Since existing…

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach…

We present and prove a version of the elliptic regularity theorem for partial differential equations involving fractional Riemann-Liouville derivatives. In this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the forcing…

In a first part, we give a method for solving a family of fuchsian systems of operators of pseudo-derivations associated to a family of homographies with two parameters which unify and generalize the differential, the difference and the…

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…

This paper develops a framework of algebra whereby every Diophantine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Newtonian triangles. We then apply this framework…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{equation*} \mathcal L =y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right), \qquad\alpha_1, \alpha_2 \in\mathbb R…

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

In this paper, we investigate critical quasilinear elliptic partial differential equations on a complete Riemannian manifold with nonnegative Ricci curvature. By exploiting a new and sharp nonlinear Kato inequality and establishing some…

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle ($L^2(\mathcal C)$) and in $l_2(\mathbb Z)$, which are related to each other by means of the Fourier transform and the…