Some equivalence theorems in affine hypersurface theory

1992 ◽  
Vol 113 (3) ◽  
pp. 245-254 ◽  
Author(s):  
Barbara Opozda
Keyword(s):  
Affine Hypersurface ◽  
Equivalence Theorems

Absence of Envy among “Neighbors” Can Be Enough

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chiara Donnini ◽  
Marialaura Pesce
Keyword(s):  
Mixed Markets ◽  
Entire Society ◽  
Equivalence Theorems ◽  
Definition Of ◽  
Equal Income ◽  
Walrasian Equilibria

AbstractWe assume that the set of agents is decomposed into several classes containing individuals related each other in some way, for example groups of neighbors. We propose a new definition of fairness by requiring efficiency and envy-freeness only within each group. We identify conditions under which absence of envy among “neighbors” is enough to ensure fairness in the entire society. We also show that equal-income Walrasian equilibria are the only fair allocations according to our notion, deriving as corollaries the equivalence theorems of Zhou (1992) and Cato (2010). The analysis is conducted in atomless economies as well as in mixed markets.

scholarly journals Almost vanishing polynomials and an application to the Hough transform

2014 ◽  
Vol 13 (08) ◽  
pp. 1450057 ◽  
Author(s):  
Maria-Laura Torrente ◽  
Mauro C. Beltrametti
Keyword(s):  
Hough Transform ◽  
Affine Space ◽  
Bounded Region ◽  
Pattern Recognition Technique ◽  
Local Study ◽  
Affine Hypersurface ◽  
Automated Recognition ◽  
Real Affine ◽  
Necessary And Sufficient ◽  
Recognition Technique

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

scholarly journals A classification of isotropic affine hyperspheres

2016 ◽  
Vol 27 (09) ◽  
pp. 1650074 ◽  
Author(s):  
Marilena Moruz ◽  
Luc Vrancken
Keyword(s):  
Acta Math ◽  
Complete Classification ◽  
Positive Definite ◽  
Affine Hypersurface ◽  
Affine Hypersphere ◽  
Difference Tensor ◽  
Affine Spheres

We study affine hypersurfaces [Formula: see text], which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955–1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918–962.] We first show that the dimension of an isotropic affine hypersurface is either [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Next, we assume that [Formula: see text] is an affine hypersphere and we obtain for each of the possible dimensions a complete classification.

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