Quotient

About: Quotient is a research topic. Over the lifetime, 9450 publications have been published within this topic receiving 106649 citations. The topic is also known as: ratio.

Shape manifolds, procrustean metrics, and complex projective spaces

Abstract: The shape-space l. k m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S k ~ 2 ; when m ^ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space I* ca n be identified with a version of CP*~ 2 . Of special importance are the shape-measures induced on CP k ~ 2 by any assigned diffuse law of distribution for the k vertices. We determine several such shape-measures, we resolve some of the technical problems associated with the graphic presentation and statistical analysis of empirical shape distributions, and among applications we discuss the relevance of these ideas to testing for the presence of non-accidental multiple alignments in collections of (i) neolithic stone monuments and (ii) quasars. Finally the recently introduced Ambartzumian density is examined from the present point of view, its norming constant is found, and its connexion with random Crofton polygons is established.

A simple unpredictable pseudo random number generator

Abstract: Two closely-related pseudo-random sequence generators are presented: The ${1 / P}$generator, with input P a prime, outputs the quotient digits obtained on dividing 1 by P. The $x^2 \bmod N$generato...

Stable real cohomology of arithmetic groups

Abstract: Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism $$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),$$ (1) where I G q denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j Γ q is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j Γ q is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, “logarithmic growth,” at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima’s arguments remain valid in the non-compact case for square integrable forms.

Tilting theory and cluster combinatorics

Abstract: We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.