Related papers: Arithmetic Laplacians
A super-Laplacian is a set of differential operators in superspace whose highest-dimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differential operators of arbitrary finite degree…
We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function.…
We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit…
We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of…
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation…
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main…
For a pair $(E,P)$ of an elliptic curve $E/\mathbb{Q}$ and a nontorsion point $P\in E(\mathbb{Q})$, the sequence of \emph{elliptic Fermat numbers} is defined by taking quotients of terms in the corresponding elliptic divisibility sequence…
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all…
Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
In a series of recent scientific contributions the role of bosonic and fermionic ladder operators in a macroscopic realm has been investigated. Creation, annihilation and number operators have been used in very different contexts, all…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish…
For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…
We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this…
We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operators which are closely related to symmetry…
The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators…
We study the regularity of the solution to an obstacle problem for a class of integro-differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional…
It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that…
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus…