Related papers: How do difference bodies in complex vector spaces …
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
We present a quantitative measure of physical complexity, based on the amount of information required to build a given physical structure through self-assembly. Our procedure can be adapted to any given geometry, and thus to any given type…
The language and methods of algebraic topology, particularly homotopy theory, have been extensively used in the study of the identification, the classification and the evolution of defects. Topological methods provide the means for the…
In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants…
We study the effects on length spaces imposed by quadratic inequalities on the six distances between the points in every quadruple.
We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof…
Positive geometries encode the physics of scattering amplitudes in flat space-time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables,…
We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes)…
We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a…
Space-Time in general relativity is a dynamical entity because it is subject to the Einstein field equations. The space-time metric provides different geometrical structures: conformal, volume, projective and linear connection. A deep…
The space of Minkowski valuations on an m-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is…
This paper is devoted to a detailed investigation of the perturbed pendulum-like motions of a heavy rigid body about a fixed point. Canonical variables that allow one to simplify the analysis of homoclinic and heteroclinic orbits are…
W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to $d$-dimensional balls. In this paper, we extend their result to any $d$-dimensional convex body with a smooth boundary and finite order…
Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…
Mixing, and coherence are fundamental issues at the heart of understanding transport in fluid dynamics and other non-autonomous dynamical systems. Recently, the notion of coherence has come to a more rigorous footing, and particularly…
Quantum many-body systems exhibit a rich and diverse range of exotic behaviours, owing to their underlying non-classical structure. These systems present a deep structure beyond those that can be captured by measures of correlation and…
It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete…