Showing papers on "Equivalence class published in 1994"

Variation of Geometric Invariant Theory Quotients

Abstract: Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several fundamental questions naturally arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space; and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.

On the estimation of join result sizes

Abstract: Good estimates of join result sizes are critical for query optimization in relational database management systems. We address the problem of incrementally obtaining accurate and consistent estimates of join result sizes. We have invented a new rule for choosing join selectivities for estimating join result sizes. The rule is part of a new unified algorithm called Algorithm ELS (Equivalence and Largest Selectivity). Prior to computing any result sizes, equivalence classes are determined for the join columns. The algorithm also takes into account the effect of local predicates on table and column cardinalities. These computations allow the correct selectivity values for each eligible join predicate to be computed. We show that the algorithm is correct and gives better estimates than current estimation algorithms.

Abstract Model Checking of Infinite Specifications

Abstract: A new method for analyzing specifications in languages like Z and VDM is proposed. Theorems are checked automatically by exhaustive search of the state space. An abstraction over the actual states can be defined that reduces an infinite state space to a finite number of equivalence classes, allowing it to be searched exhaustively by treating each class as a single abstract state. A prototype has been built that has verified some small theorems from the literature.

Recursive and r.e. quotient Boolean algebras

Abstract: We prove a converse to one of the theorems from [F], giving a description in terms of Turing complexity of sets which can be coded into recursive and r.e. quotient Boolean algebras.

Avoiding Non-Termination when Learning Logical Programs: A Case Study with FOIL and FOCL

Abstract: Many systems that learn logic programs from examples adopt θ-subsumption as model of generalization and refer to Plotkin's framework in order to define their search space. However, they seldom take into account the fact that the lattice defined by Plotkin is a set of equivalence classes rather than simple clauses. This may lead to non-terminating learning processes, since the search gets stuck within an equivalence class, which contains an infinite number of clauses.

On exchangeability-based equivalence relations induced by strongly Robinson and, in particular, by quadripolar Robinson dissimilarity matrices

Abstract: Let I denote a totally ordered set of n ≥ 1 elements. It is notationally convenient to identify I with the set of the first n integers or, on occasion, with the row vector (1,…,n). In the former case the order on I is embodied in the natural order on the reals, and in the latter case in the ordering amongst the elements of a vector

Spectral asymptotics of foliated manifolds

Abstract: In this paper we study the spectral distribution functions for the analytic and combinatorial leafwise Laplacians on a foliated manifold admitting a transverse invariant measure. We show (Theorem 4.1) that the dilational equivalence classes near zero of the two spectral distribution functions are the same. An immediate consequence is that this dilational equivalence class is independent of the metric (in the analytic case) and the bounded triangulation (in the combinatorial case) used to define it. Our main result (Theorem 2.6) is that this dilational equivalence class is invariant under a measure preserving leafwise homotopy equivalence. This result is equivalent to the invariance of the dilational equivalence class near infinity of the trace of the leafwise heat kernels. Our results are motivated by those of [Ef-Sh], [G-Sh] and [El] concerning the equivariant homotopy invariance of the asymptotic behaviour of the spectral distribution function for Riemannian manifolds with a free isometric action of a discrete group. We extend many of the ideas of these papers to our situation. The main techniques used in the proof of the homotopy invariance, however, are those developed in [H-L2] (where we prove the leafwise homotopy invariance of the foliation betti numbers), the simplicial techniques of [D] and [W], and those of [H-L1].

On 2{PFA}s and the Hadamard quotient of formal power series

Abstract: We present the main results on 2PFA’s and on the Hadamard quotient of formal power series, the connection between the two notions being a result stating that the event dened by a 2PFA is the Hadamard quotient of two rational power series.

The role of morse-Lyapunov functions in the design of nonlinear global feedback dynamics

Abstract: The main objective of this chapter has been to present a totally different approach to the controllability question for nonlinear systems. This approach, by first specifying an equivalence class of desirable control dynamics that we hope to achieve, makes the controllability problem easier to address. For most points in state space it is seen that the controllability problem is trivially verified (relative to some arbitrary Morse function). The remaining points belong to some obstructing set, whose topological features at least are frequently known. The way we solve the problem of bypassing the obstruction is, we believe, a natural generalization of the linear case, interpreted geometrically, and not as a condition involving Lie brackets of vector fields. It is hoped that, by understanding the topology of the problem better, it will be possible to derive existence conditions for Morse-Lyapunov functions that are more general than the ones derived here.

Identifying unrecognizable regular languages by queries

Abstract: We describe a new technique useful in identifying a subclass of regular trace languages (defined on a free partially commutative monoid). We extend an algorithm defined by Dana Angluin in 1987 for DFA's and using equivalence and membership queries. In trace languages the words are equivalence classes of strings, and we show how to extract, from a given class, a string that can drive the original learning algorithm. In this way we can identify a class of regular trace languages which includes languages which are not recognizable by any automaton.

Aplicaciones cerradas y espacios en los que componentes y cuasicomponentes coinciden

Abstract: This interesting and well-written survey article is devoted to the class P of topological spaces in which components and quasi-components coincide. This class includes the compact Hausdorff spaces and the locally connected spaces. It also includes every subset of the real line but not every subset of the plane. This class is closed under homotopy type, but the authors state that "it does not seem to be possible to give easily-stated conditions'' for membership in P . They do give some sufficient conditions using the fact that, to any topological space X , one can associate the quotient space ΔX in which each quasi-component is identified to a point (they show that this association is categorically natural). These conditions include the assumption that the quotient map is closed. For example, they show that, if X is normal and ΔX is zero-dimensional, then X∈P . Variations of this include the result that, if ΔX is zero-dimensional and the quasi-components are compact, then X∈P , and the result that, if X is locally compact Lindelof and Hausdorff, then X∈P . No proofs are given. Can the fact that P is closed under homotopy equivalence be improved by allowing a more arbitrary homotopy index set (or not using a product structure at all)? What is an example of a space X whose quotient map is closed but X∉P ?

Identifying Regular Languages over Partially-Commutative Monoids

Abstract: We define a new technique useful in identifying a subclass of regular languages defined on a free partially commutative monoid (regular trace languages), using equivalence and membership queries. Our algorithm extends an algorithm defined by Dana Angluin in 1987 to learn DFA's. The words of a trace language can be seen as equivalence classes of strings. We show how to extract, from a given equivalence class, a string of an unknown underlying regular language. These strings can drive the original learning algorithm which identify a regular string language that defines also the target trace language. In this way the algorithm applies also to classes of unrecognizable regular trace languages and, as a corollary, to a class of unrecognizable string languages. We also discuss bounds on the number of examples needed to identify the target language and on the time required to process them.

Mutations in finite groups

Abstract: One generally studies the different types of algebraic structures from the equivalence relation “being isomorphic”, thus establishing a series of invariants and canonical models for each equivalence class under the mentioned relation. The main goal of this paper is to study how we can minimally quantify, by means of certain parameters, the degree of non-isomorphy between two given groups of the same order, i.e., to study the degree of invariance between two distinct equivalence classes under the relation “being isomorphic” in the category of finite groups. The search of an efficient estimator of the qualification “non-isomorphic groups”, which allow us to know if they are “hardly or nearly” isomorphic, has led us to define the concept of mutation, which, formalized in the category of internal Ω-algebras, is widely studied in this paper in the category of finite groups. We present the concept and the results relative to mutations, according with the three stages their study has taken us chronologically. First of all, from the comparative analysis of pairs of similar structures having underlying sets of the same cardinality, i.e., groups of the same small order, 4,6,8, etc., it arises the concept of mutation, as being a bijection maximally satisfying the homomorphy condition, in order to minimize the number of times one has to mutate the group law to get an isomorphism from the given map. It is also worth noting that the concept of mutation furthermore lets us rapidly check whether two given distinct presentation correspond to the same group. Once introduced the concept of mutation between non-isomorphic groups and dealt with its properties we obtain in a second stage the evolutive chains in the sets

Object reconstruction using the cooperation of 3D segments and 3D facets information

Abstract: Given a series of trinocular images of an object, we have developed a method for building 3D Facets and 3D segments model of the object. From each triplet, a partial description of the object, called 3D View, is extracted. From the set of all extracted 3D Facets, a strategy for guiding the object reconstruction, based on a statistical method is developed. A 3D Matching Builder computes matchings between the 3D Primitives of consecutive 3D Views. Guided by the strategy, a Superstructure gathers all the matching informations given by the 3D Matching Builder in a set of equivalence classes. For each equivalence class of Superstructure, a representative is derived. The 3D Primitives model is finally computed merging informations of 3D Facet representatives and 3D Segment representatives. This method implemented in Smalltalk80, has been applied on a series of stereoscopic real images triplets; some results are provided at the end of the paper. >