Harmonicity of gradient mappings of level surfaces in a real affine space

1995 ◽  
Vol 56 (2) ◽  
pp. 177-184 ◽  
Author(s):  
Hirohiko Shima
Keyword(s):  
Affine Space ◽  
Real Affine

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 Fonctions régulues

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Goulwen Fichou ◽  
Johannes Huisman ◽  
Frédéric Mangolte ◽  
Jean-Philippe Monnier
Keyword(s):  
Affine Space ◽  
Rational Functions ◽  
Continuous Extension ◽  
Prime Ideals ◽  
Geometrical Characterization ◽  
The Real ◽  
Constructible Sets ◽  
Real Affine ◽  
Zero Locus

AbstractNous étudions l’anneau des fonctions rationnelles qui se prolongent par continuité surWe study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.

On the crystallography of infinite Coxeter groups

1977 ◽  
Vol 82 (1) ◽  
pp. 13-24 ◽  
Author(s):  
George Maxwell
Keyword(s):  
Affine Space ◽  
Coxeter Groups ◽  
Root Systems ◽  
Space Groups ◽  
Coxeter Graph ◽  
Finite Dimensional ◽  
Multipartite Graphs ◽  
Real Affine ◽  
Point Groups ◽  
Complete Multipartite

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

The Space Groups of Two Dimensional Minkowski Space

1978 ◽  
Vol 30 (5) ◽  
pp. 1103-1120
Author(s):  
George Maxwell
Keyword(s):  
Vector Space ◽  
Minkowski Space ◽  
Bilinear Form ◽  
Orthogonal Group ◽  
Affine Space ◽  
Symmetric Bilinear Form ◽  
Two Dimensional ◽  
Space Groups ◽  
Dimensional Minkowski Space ◽  
Real Affine

LetEbe an n-dimensional real affine space,Vits vector space of translations andA(E)the affine group ofE.Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature(n —1, 1), O(V) its orthogonal group andS(V)its group of similarities.

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