The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Semi-affine graphs. It is profitable to treat the so-called Coxeter-Dynkin diagrams as graphs. A classification of finite graphs with an adjacency matrix having 2 as the largest eigenvalue is made in a paper of John Smith [JHS]. It is in a combinatorial context and no reference is made to Coxe...

The goal of this paper is to define a new class of objects which we call triple affine Artin groups and to relate them with Cherednik’s double affine Hecke algebras. This has as immediate consequences new and simple descriptions of double affine Weyl and Artin groups, the double affine Hecke algebras as well as the corresponding elliptic objects. We also recover in an transparent and elementary...

We classify the cohomology spaces H(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on g, hence a canonical...

Affine arithmetic (AA) is widely used in range analysis in word-length optimization of hardware designs. To reduce the uncertainty in the AA and achieve efficient and accurate range analysis of multiplication, this paper presents a novel refined affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE). The affine form of multiplication is divided into two part...

We prove that the Lam-Shimozono “down operator” on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of “near rectangles” in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of ...

Interest point detection is essential process for many computer vision applications, which must provide invariant points to several image variations, such as, rotation, zoom, blur, illumination variation and change of viewpoints. Harris-Affine detector is considered as one of the most effective interest point detectors, although it still presents vulnerability to some image. This paper proposes...

Using a classical result on algebraic invariants of the unimodular group, we present in this paper some basic geometric affine invariant quantities, and we use them to construct some distinctive descriptors for object detection. Although full affine invariance cannot be guaranteed due to noncommutativity of camera blur with affine maps and the domain problem (that is, the difficulty of finding ...

An affine invariant point on the class of convex bodies Kn in R, endowed with the Hausdorff metric, is a continuous map from Kn to R which is invariant under one-to-one affine transformations A on R, that is, p ` A(K) ́ = A ` p(K) ́ . We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K) = p(K) for every K ∈ Kn, where K denotes the polar of K with...

This paper presents a novel approach for detecting affine invariant interest points. Our method can deal with significant affine transformations including large scale changes. Such transformations introduce significant changes in the point location as well as in the scale and the shape of the neighbourhood of an interest point. Our approach allows to solve for these problems simultaneously. It ...