Related papers: Projective Geometry and PDE Prolongation
For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1} $ and the fibres are locally…
We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical…
In this paper, we present a mathematical model for the angular projection of a rectangular arrangement of points in a grid. This simple, yet interesting problem, has both a scholarly value and applications for data extraction techniques to…
We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras.…
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
The aim of this text is to extend the theory of generalized ordinary differential equations to the setting of metric spaces. We present existence and uniqueness theorems that significantly improve previous results even when restricted back…
Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections and mappings. Projective geometry identifies a line with a single point, like the perspective…
We deal with generalizations of the Fundamental Theorem of Projective Geometry to other related geometries (of dimension $\geq 3$) and non bijective maps. We consider locally projective geometries and locally affino-projective geometries…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
This is an introduction (in German) to projective geometry by the late Heinz Lueneburg. Projective spaces are treated as lattices with particular properties, and finite geometries receive special attention. The final chapters deal with…
Projective metrics on vector spaces over finite fields, introduced by Gabidulin and Simonis in 1997, generalize classical metrics in coding theory like the Hamming metric, rank metric, and combinatorial metrics. While these specific metrics…
Shape-constrained inference has wide applicability in bioassay, medicine, economics, risk assessment, and many other fields. Although there has been a large amount of work on monotone-constrained univariate curve estimation, multivariate…
In this small paper we bring together various open problems on geometric multidimensional continued fractions.
Differentiable rendering is a technique to connect 3D scenes with corresponding 2D images. Since it is differentiable, processes during image formation can be learned. Previous approaches to differentiable rendering focus on mesh-based…
This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter…
This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…
In extension problems of partial graph drawings one is given an incomplete drawing of an input graph $G$ and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of…
We study the graded derivation-based noncommutative differential geometry of the $Z_2$-graded algebra ${\bf M}(n| m)$ of complex $(n+m)\times(n+m)$-matrices with the ``usual block matrix grading'' (for $n\neq m$). Beside the…
We revisit the problem of local persistence in directed percolation, reporting improved estimates of the persistence exponent in 1+1 dimensions, discovering strong corrections to scaling in higher dimensions, and investigating the mean…
The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so-called discretization assumptions allow to discretize the search space of…