Related papers: Operator calculus - the exterior differential comp…
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star,…
The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the…
The Dirac-Dolbeault operator for a compact K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows to express the values of the sections of the…
The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a…
Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal…
The basic results for nonlinear operators are given. These results include nonlinear versions of classical uniform boundedness theorem and Hahn-Banach theorem. Furthermore, the mappings from a metrizable space into another normed space can…
As shown in earlier work, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended…
In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\S)\to…
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely…
By means of fixed point index theory for multi-valued maps, we provide an analogue of the classical Birkhoff--Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general…
We give new examples of linear differential operators of order $k=2m+1$ (any given odd integer) that are invariant under the isometries of $\mathbb R^n$ and satisfy so-called $L^1$-duality estimates and div/curl inequalities.
We construct potentials for the exterior derivative, in particular, for the gradient, the curl, and the divergence operators, over domains with shellable triangulations. Notably, the class of shellable triangulations includes local patches…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the…
An L2 theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the…
This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, which is a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther [Bull.…
We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior…
We develop a very general operator-valued functional calculus for operators with an $H^{\infty}-$calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an $H^{\infty}$calculus. Using…
We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for…
Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general…