Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe fundamental theorem of affine geometry in regular $L^0$-modules
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $L^0({mathcal F})$(briefly, an $L^0({mathcal F})$-module) is said to be regular if $x=y$ for any given two elements $x$ and $y$ in $M$ such that there exists a countable partition ${A_n,nin mathbb N}$ of $Omega$ to $mathcal F$ such that ${tilde I}_{A_n}cdot x={tilde I}_{A_n}cdot y$ for each $nin mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({mathcal F})$-modules: let $V$ and $V^prime$ be two regular $L^0({mathcal F})$-modules such that $V$ contains a free $L^0({mathcal F})$-submodule of rank $2$, if $T:Vto V^prime$ is stable and invertible and maps each $L^0$-line segment onto an $L^0$-line segment, then $T$ must be $L^0$-affine.
rate research
Read More
Let $(E, lVert . rVert)$ be a two-dimensional real normed space with unit sphere $S = {x in E, lVert x rVert = 1}$. The main result of this paper is the following: Consider an affine regular hexagon with vertex set $H = {pm v_1, pm v_2, pm v_3} subseteq S$ inscribed to $S$. Then we have $$min_i max_{x in S}{lVert x - v_i rVert + lVert x + v_i rVert} leq 3.$$ From this result we obtain $$min_{y in S} max_{x in S}{lVert x - y rVert + lVert x + y rVert} leq 3,$$ and equality if and only if $S$ is a parallelogram or an affine regular hexagon.
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.
Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the (DFS)-space ${mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[gamma]$ in $L^1([0,1],{mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and smooth in $[gamma]$. As tools for the proof, we develop several new results concerning $L^p$-regularity of infinite-dimensional Lie groups, for $1leq pleq infty$, which will be useful also for the discussion of other classes of groups. Moreover, we obtain new results concerning the continuity and complex analyticity of non-linear mappings on open subsets of locally convex direct limits.
Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound. Our proof uses a relaxed notion of motion planning algorithm that instead of contractibility yields a lower bound for the $mathbb{Z}/2$-coindex of a space.
The minus partial order is already known for sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces. Recently, this partial order has been studied on Rickart rings. In this paper, we extend the concept of the minus relation to the module theoretic setting and prove that this relation is a partial order when the module is regular. Moreover, various characterizations of the minus partial order in regular modules are presented and some well-known results are also generalized.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
