Related papers: Representative elementary volume via averaged scal…
This paper introduces Least Volume (LV)--a simple yet effective regularization method inspired by geometric intuition--that reduces the number of latent dimensions required by an autoencoder without prior knowledge of the dataset's…
We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$ for…
The orthogonal projection of a 4-cube onto a uniform random 3-subspace in R^4 is a convex 3-polyhedron P with 14 vertices almost surely. Three numerical characteristics of P -- volume, surface area and mean width -- are studied. These…
We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard…
This work demonstrates that the popular arithmetic mean conformation tensor frequently used in the analysis of turbulent viscoelastic flows is not a good representative of the ensemble. Alternative means based on recent developments in the…
We study sets of $\delta$ tubes in $\mathbb{R}^3$, with the property that not too many tubes can be contained inside a common convex set $V$. We show that the union of tubes from such a set must have almost maximal volume. As a consequence,…
A nonempty closed convex set in ${\mathbb R}^n$, not containing the origin, is called a pseudo-cone if with every $x$ it also contains $\lambda x$ for $x\ge 1$. We consider pseudo-cones with a given recession cone $C$, called…
The group-theoretic method for constructing symmetric isometric embeddings is used to describe all possible four-dimensional surfaces in flat $(1,9)$-dimensional space, whose induced metric is static and spherically symmetric. For such…
Neural radiance fields (NeRF) rely on volume rendering to synthesize novel views. Volume rendering requires evaluating an integral along each ray, which is numerically approximated with a finite sum that corresponds to the exact integral…
The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting…
We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local…
A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on…
We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under two kinds of discrete subgroups of $O(3)$ of order four. We also characterize the convex bodies with the minimal volume product…
In practical implementations of density-functional theory, the only term where an orbital description is needed is the kinetic one. Even this term in principle depends on the density only, but its explicit form is unknown. We provide a…
In Minkowski geometry the unit ball is a compact convex body $K$ containing the origin in its interior. The boundary of the body is formed by the unit vectors. We also have a so-called Minkowski functional to measure the length of vectors.…
We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a…
\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…
Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner…
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is…