Related papers: The Fundamental Theorems of Affine and Projective …

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to…

In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in…

In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a line arrangement and a reducible curve with a line component. We give some applications to this result.

In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local…

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These…

Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics,…

A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b$, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n$. The affine maps over $\ff=\rr$ or $\cc$ are…

We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly…

In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number…

Let $(\Omega,{\mathcal F},P)$ be a probability space and $L^0({\mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(\Omega,{\mathcal F},P)$. A left module $M$ over the algebra $L^0({\mathcal…

We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine B\"acklund theorem. We describe some geometrical conditions…

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study…

I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a…

We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman…

We prove that finite sets of mutual neighbor points in an affine scheme admit affine combinations, preserved by any map. Furthermore, such combination has a value which is neighbor point of all the original points.

A partial affine plane of order $n$ is a point-line incidence structure with $n^2$ points and $n$ points on each line, such that every two lines meet in at most one point. In this paper, we show that a partial affine plane of order $n$, $n$…

A standard theorem in nonsmooth analysis states that a piecewise affine function $F:\mathbb R^n\rightarrow\mathbb R^n$ is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero…

Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of ordered affine geometry. There are 22 axioms for the ordered affine geometry, of which the…

A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional…

We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.