Related papers: On Div-Curl for Higher Order
In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual…
We characterize the finite codimension sub-K-algebras of K[[t]] as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension…
We utilize group-theoretical methods to develop a matrix representation of differential operators that act on tensors of any rank. In particular, we concentrate on the matrix formulation of the curl operator. A self-adjoint matrix of the…
We relate non integer powers ${\mathcal L}^{s}$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal L$ in a real separable Hilbert space $\mathcal H$ with a certain differential operator of order $2\lceil{s}\rceil$,…
We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are…
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as…
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…
This paper presents some new propositions related to the fractional order $h$-difference operators, for the case of general quadratic forms and for the polynomial type, which allow proving the stability of fractional order $h$-difference…
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…
In this paper differential operators on various moduli spaces (e.g. of holomorphic vector bundles) are described in a canonical way in terms of the geometry of a certain distinguished completion of an appropriate configuration space.
We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.
In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental {\pi}…
Let $O$ be an order of odd discriminant $D$ in an imaginary quadratic field $K$. Let $Cl(O)$ be the group of proper $O$-ideals and $Cl(O)[2]$ the kernel of multiplication by $2$ in $Cl(O)$. We describe explicitly the group $Cl(O)[2]$. In…
We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of…
In this paper we have considered higher order two dimensional coupled system of non-linear ordinary differential equations. We have given necessary and sufficient conditions on the non-linear functions such that the solutions pair oscilla
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
In this article, we give a sequence of operators for producing an approximation result. The relation between the rate of approximation of sequence operators including Dunkl variant of exponential function with first and second-order modulus…
We prove H\"older regularity results for a class of nonlinear elliptic integro-differential operators with integration kernels whose ellipticity bounds are strongly directionally dependent. These results extend those in [9] and are also…