Congruence and Similarity Classification of Real Hyperquadrics
ثبت نشده
چکیده
We shall begin by recalling some material from quadrics1.pdf, which gave the classification of hyperquadrics in R and C up to affine equivalence: CONGRUENCE OR ISOMETRY CLASSIFICATION PROBLEM. Given two affine hyperquadrics Σ1 and Σ2 in R , is there a congruence or isometry T from R to itself mapping Σ1 onto Σ2? There are two possible versions of the question. An arbitrary isometry T from R to itself is an affine transformation of the form T(x) = P x + q, where P is some orthogonal matrix (i.e, P = P and q is some vector); in some writings, the term “congruence” refers to an arbitrary isometry, while in others it refers to an isometry for which det P = +1. In this document we shall be interested mainly in the less restrictive (first) option.
منابع مشابه
On the use of Heronian means in a similarity classifier
This paper introduces new similarity classifiers using the Heronian mean, and the generalized Heronian mean operators. We examine the use of these operators at the aggregation step within the similarity classifier. The similarity classifier was earlier studied with other operators, in particular with an arithmetic mean, generalized mean, OWA operators, and many more. The two classifiers here ar...
متن کاملObstructions to Embeddability into Hyperquadrics and Explicit Examples
We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codime...
متن کامل3 0 Ju n 20 09 Classification of commutative algebras and tube realizations of hyperquadrics
In this paper we classify up to affine equivalence all local tube realizations of real hyper-quadrics in C n. We show that this problem can be reduced to the classification, up to isomorphism, of commutative nilpotent real and complex algebras. We also develop some structure theory for com-mutative nilpotent algebras over arbitrary fields of characteristic zero.
متن کاملThe Classification Problem for Graphs and Lattices Is Wild
We prove that the classification problem for graphs and several types of algebraic lattices (distributive, congruence and modular) up to isomorphism contains the classification problem for pairs of matrices up to simultaneous similarity.
متن کاملMeasuring the Similarity of Trajectories Using Fuzzy Theory
In recent years, with the advancement of positioning systems, access to a large amount of movement data is provided. Among the methods of discovering knowledge from this type of data is to measure the similarity of trajectories resulting from the movement of objects. Similarity measurement has also been used in other data mining methods such as classification and clustering and is currently, an...
متن کامل