Related papers: Generalized vector potential and Trace Theorem for…

Recently, several extensions of massive vector theory in curved space-time have been proposed in many literatures. In this paper, we consider the most general vector-tensor theories that contain up to two derivatives with respect to metric…

It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories…

While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many…

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is…

The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…

Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, and let $E\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets…

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite…

Helmholtz decomposition theorem for vector fields is presented usually with too strong restrictions on the fields. Based on the work of Blumenthal of 1905 it is shown that the decomposition of vector fields is not only possible for…

We present proofs for the existence of distributional potentials $F\in{\mathcal D}'(\Omega)$ for distributional vector fields $G\in{\mathcal D}'(\Omega)^n$, i.e. $\operatorname{grad} F=G$, where $\Omega$ is an open subset of ${\mathbb…

A generalised Weber function is given by $\w_N(z) = \eta(z/N)/\eta(z)$, where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\w_N^e$ evaluated…

We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few…

In ZM theory the direction of time has a non-zero projection onto space and this projection corresponds to the local velocity relative to the observer. Classical trajectories can be obtained by following the local direction of time. The…

We give an integral expression for the vector potential of a time-independent, steady azimuthal current density. Our derivation is substantially simpler and somewhat more general than others given in the literature. As an illustration, we…

We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators,…

We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\bR^n\times \bR^m\to\bR^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\bR^n\times \bR^m\to \bR^n\times \bR^m$ such…

A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with…

The article presents an alternative approach to the definition of vector electrodynamic potential and its properties. It is shown that generally it has vortical and potential components. The system of differential equations of generalized…

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers…

The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a `big set' $X$…

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$…