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Register a new userThe Fundamental Theorems of Affine and Projective Geometry Revisited
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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.
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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.
The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.
We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${mathbb P}^2$ or ${mathbb P}^1times {mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaokas generic semipositivity theorem.
This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartans theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing.
Affine jump-diffusions constitute a large class of continuous-time stochastic models that are particularly popular in finance and economics due to their analytical tractability. Methods for parameter estimation for such processes require ergodicity in order establish consistency and asymptotic normality of the associated estimators. In this paper, we develop stochastic stability conditions for affine jump-diffusions, thereby providing the needed large-sample theoretical support for estimating such processes. We establish ergodicity for such models by imposing a `strong mean reversion condition and a mild condition on the distribution of the jumps, i.e. the finiteness of a logarithmic moment. Exponential ergodicity holds if the jumps have a finite moment of a positive order. In addition, we prove strong laws of large numbers and functional central limit theorems for additive functionals for this class of models.
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