Related papers: Minkowski valuations in a 2-dimensional complex ve…
All SL($n$) contravariant vector valuations on polytopes in $\mathbb R^n$ are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique such valuation for $n\geq3$. In dimension…
A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.
A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one…
Minkowski valuations provide a systematic framework for quantifying different aspects of morphology. In this paper we apply vector- and tensor-valued Minkowski valuations to neuronal cells from the cat's retina in order to describe their…
This article explores \Z_2-graded L_\infinity algebra structures on a 2|1-dimensional vector space. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the…
In this paper, we obtain the classification theorem for three-dimensional complete space-like $\lambda$-translators $x:M^{3} \rightarrow \mathbb R^{4}_{1}$ with constant norm of the second fundamental form and constant $f_{4}$ in the…
A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which…
We establish a full classification of degree $2$ codimension one distributions on $\mathbb{P}^3$ according to invariants of their tangent sheaves.
$Sp(2M)$ invariant field equations in the space ${\cal M}_M$ with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ${\cal M}_M$ which include usual $4d$ Minkowski space as a…
Higher-rank Minkowski valuations are efficient means for describing the geometry and connectivity of spatial patterns. We show how to extend the framework of the scalar Minkowski valuations to vector- and tensor-valued measures. The…
All $\textrm{SL}(n)$ contravariant matrix-valued valuations on polytopes in $\mathbb{R}^n$ are completely classified without any continuity assumptions. Moreover, the symmetry assumption of matrices is removed. The general Lutwak-Yang-Zhang…
We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of…
We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel…
In this paper, the 2-category $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ on (some version of) Kapranov and Voevodsky's 2-category of (complex) 2-vector spaces…
In this paper we introduce the systematic study of invariant functions and equivariant mappings defined on Minkowski space under the action of the Lorentz group. We adapt some known results from the orthogonal group acting on the Euclidean…
The bicovariant differential calculus on the four-dimensional kappa-Poincare group and the corresponding Lie-algebra like structure are described. The deifferential calculus on the n-dimensional kappa-Minkowski space covariant under the…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
All continuous, SL(n) covariant valuations on Orlicz spaces are completely classified without any symmetric assumptions. It is shown that the moment matrix is the only such valuation if n\geq3, while a new functional shows up in dimension…
This paper is aimed to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of {1,3m}, {1,2,3m}, {1,4m} and {1,2,4m}-inverses are given in order to…
All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.