Showing papers on "Equivalence class published in 1999"

Exceptional quotient singularities

Abstract: A singularity is said to be exceptional (in the sense of V. Shokurov), if for any log canonical boundary, there is at most one exceptional divisor of discrepancy -1. This notion is important for the inductive treatment of log canonical singularities. The exceptional singularities of dimension 2 are known: they belong to types E 6 , E 7 , E 8 after Brieskorn. In our previous paper, it was proved that the quotient singularity defined by Klein's simple group in its 3-dimensional representation is exceptional. In the present paper, the classification of all the three-dimensional exceptional quotient singularities is obtained. The main lemma states that the quotient of the affine 3-space by a finite group is exceptional if an only if the group has no semiinvariants of degree 3 or less. It is also proved that for any positive e, there are only finitely many e-log terminal exceptional 3-dimensional quotient singularities.

Formal Rough Concept Analysis

Abstract: In this paper, we present a novel approach for approximating concepts in the framework of formal concept analysis. Two main problems are investigated. The first, given a set A of objects (or a set B of features), we want to find a formal concept that approximates A (or B). The second, given a pair (A,B), where A is a set of objects and B is a set of features, the objective is to find formal concepts that approximate (A,B). The techniques developed in this paper use ideas from rough set theory. The approach we present is different and more general than existing approaches.

Index structures for path expressions

Abstract: In recent years there has been an increased interest in managing data that does not conform to traditional data models, like the relational or object oriented model. In this paper we propose a novel, general index structure for semistructured databases, called template index, or T-index. It improves over the previous approaches in several ways. First, T-indexes allow us to trade space for generality. Second, T-indexes can be efficiently constructed. Our technique consists in grouping database objects into equivalence classes containing objects that are indistinguishable w.r.t to a class of paths defined by a path template as described above. A T-index is built from these equivalence classes, by constructing a non-deterministic automaton whose states represent the equivalence classes and whose transitions correspond to edges between objects in those classes.

Poles and zeros at infinity of linear time-varying systems

Abstract: The notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time-varying systems. This study is based on the notion of a "newborn systems which is, in a mathematical point of view, a graded module extension over the noncommutative ring of differential operators. It is proved to be a relevant generalization to the time-varying case of the equivalence class, for the so-called "restricted equivalence" of Rosenbrock's polynomial matrix descriptions. The authors' approach is intrinsic and unifies the definitions previously given in the literature in the time-invariant case.

Tactual equivalence class formation and tactual-to-visual cross-modal transfer

Abstract: Five adults, two who were deaf-blind and three who were hearing-sighted but blindfolded for the experiment, learned the conditional discriminations AB and BC that consisted of complex tactual stimuli. Class-consistent responding to symmetry, transitivity, and equivalence probes demonstrated the emergence of tactual equivalence classes by four of the subjects. These data confirm that visual or auditory stimuli are not needed for the induction of equivalence classes. They also suggest that equivalence class procedures might facilitate the formation of communicational skills by individuals who are deaf and blind. Following the formation of the tactual equivalence classes by the two hearing-sighted adults, they immediately showed tactual-to-visual cross-modal generalization of equivalence relations. These results suggest that prior equivalence class formation facilitate the reliable emergence of cross-modal transfer.

Formal Verification of Designs with Complex Control by Symbolic Simulation

Abstract: A new approach for the automatic equivalence checking of behavioral or structural descriptions of designs with complex control is presented. The verification tool combines symbolic simulation with a hierarchy of equivalence checking methods, including decision-diagram based techniques, with increasing accuracy in order to optimize overall verification time without giving false negatives. The equivalence checker is able to cope with different numbers of control steps and different implementational details in the two descriptions to be compared.

An Algorithm for Finding Equivalence Relations from Tables with Non-Deterministic Information

Abstract: Rough sets theory depending upon DIS(Deterministic Information System) is now becoming a mathematical foundation of soft computing. Here, we pick up NIS(Non-deterministic Information System) which is more general system than DIS and we try to develop the rough sets theory depending upon NIS. We first give a definition of definability for every object set X, then we propose an algorithm for checking it. To find an adequate equivalence relation from NIS for X is the most important part in this algorithm, which is like a resolution. According to this algorithm, we implemented some programs by prolog language on the workstation.

The reduced semiclassical description method

Abstract: We develop a new version of the semiclassical analysis of a system of bound states in centrally symmetrical potentials. The set of potentials is in a 1∶1 correspondence with a certain set of pairs of functions of the orbital momentum. The first of these functions determines the usual WKB quantization condition and groups the potentials into equivalence classes. Its Mellin transform demonstrates similar behavior for the typical potentials, which allows describing the equivalence class with a small number of parameters. We can chose these parameters as the asymptotically exact estimates of the number of states. We obtain an equation that allows classifying states in a self-consistent atomic potential without knowing the explicit form of the potential. The second of these functions distinguishes the potentials within an equivalence class and also gives the first correction to the quantization condition.

The intersection of some classical equivalence classes of matrices

Abstract: Let A be an n X ra complex matrix. Let Sim (A) denote the similarity equivalence class of A, Conj(.4) denote the conjunctivity equivalence class of .4, UEquiv(.4) denote the unitaryequivalence equivalence class of .4, and 2/{{A) denote the unitary similarity equivalence class of A. Each of these equivalence classes has been studied for some time and is generally wellunderstood. In particular, canonical forms have been given for each equivalence class. Since the intersection of any two equivalence classes of .4 is again an equivalence class of .4, we consider two such intersections: CS(.4) = Sim(.4) fl Conj(.4) and UES(.4) = Sim(A) n UEquiv(.4). Though it is natural to first think that each of these is simply U{A), for each .4. we show by examples that this is not the case. We then try to classify which matrices .4 have CS(.4) = U{A). For matrices having CS(.4) ^ 1({A), we try to count the number of disjoint unitary similarity classes contained in CS(.4). Though the problem is not completely solved for CS(.4). we reduce the problem to non-singular, non-co-Hermitian matrices .4. A similar analysis is performed for UES(.4), and a (less simple) reduction of the problem is also achieved.

Equivalence classes of clone circuits for physical-design benchmarking

Abstract: To provide a better understanding of physical design algorithms and the underlying circuit architecture they are targeting, we need to exercise the algorithms and architectures with many benchmark circuits. The lack of existing benchmarks makes us consider the automatic generation of netlists. In this paper, we formally define the equivalence class of circuit "clones" of a given seed circuits, based upon physical properties of the seed circuit's netlist graph. Using these equivalence classes of circuits, a given seed circuit can be used to generate many similar circuits. A more finely grained statistical analysis of algorithm behaviour can then be obtained from using the multiple benchmarks than would be available from using the seed benchmark alone.

Braid invariants and instability of periodic solutions of time-periodic $2$-dimensional ODE's

Abstract: We present a topological approach to the problem of the existence of unstable periodic solutions for 2-dimensional, time-periodic ordinary differential equations. This approach makes use of the braid invariant, which is one of the topological invariants for periodic solutions exploiting a concept in the low-dimensional topology. Using the braid invariant, an equivalence relation on the set of periodic solutions is defined. We prove that any equivalence class consisting of at least two solutions must contain an unstable one, except one particular equivalence class. Also, it is shown that more than half of the equivalence classes contain unstable solutions.

Rationals and the modular group

Abstract: The modular group k is the quotient group PSL2(Z) = SL,(Z)/{I we refer to these as reduced strings. We explained this free product structure in terms of the action of the modular group on the irrationals. In this note we describe the action on the rationals; this can be viewed as a way of describing the inverse of the Euclidean algorithm. The group SL2(Z) acts via linear transformations on R2 as column vectors and this gives an action of .A' via linear fractional transformations on the projective line P'(R), the real numbers together with m. We may also view P'(R) as the slopes of non-zero vectors, that is, the equivalence classes of R2 -(:)induced by non-zero scalar multiplication; the equivalence class of the vector e = ( i ) is , ,

Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Interpretation

Abstract: In the preceding paper, 26 gauge-invariant variables were defined that characterize an almost-Bianchi type I or almost-Bianchi type V universe filled with a nonbarotropic perfect fluid. One can think of these basic variables, denoted collectively by D, as having at least two aspects. First, D gives an explicit (i.e., analytical) representation of the equivalence class of perturbations. In fact, this equivalence class is uniquely determined from D and vice versa. Second, any gauge-invariant quantity with respect to a Bianchi type I or Bianchi type V background model is obtainable linearly from D through purely algebraic and differential operations. Among many other things, the above properties of D facilitate new insights into the question of why a standard and well-known formulation based on the Stewart–Walker lemma does not enable one to describe the equivalence classes of perturbations and to find gauge-invariant variables independent of D. If we pose an analogous question in regard to almost-Robertson–Walker universe models, a different but none-the-less instructive answer is obtained. In this case, the Stewart–Walker lemma provides a complete framework both for constructing all gauge-invariant variables and for determining the equivalence classes of perturbations. Because of the sometimes confusing statements in the literature, nontrivial comparisons with other work on linear perturbations in anisotropic background models are made. We also present what we believe are some interesting applications of our ideas to the subject of quantum field theory in curved space–time.

Modular Invariants, Operator Algebra and Quotient Singularities

Abstract: The workshop was led by Prof. Miles Reid (Warwick) and Prof. David E Evans (Cardiff) from the UK, and Prof. Yasuyuki Kawahigashi (Tokyo Univ.) and Prof. Iku Nakamura (Hokkaido) from Japan. It set itself the task of exploring connections between different approaches to the algebra and geometry of finite subgroups of SU(2) and SU(3), including the many analogies between II_1 and III_1 subfactors in operator algebras, modular invariant partition functions in conformal field theory, finite subgroups of SU(n) and the resolution of their orbifolds. The SU(2) cases are governed by an ADE classification, and combinatorics generalising the ADE classification (for SU(3), say) appear in many different areas of recent research. The main objective of this workshop is to probe for connections to explain the empirically observed similarities between the results.

A Counting Equivalence Classes Method to Prove Negative Results

Abstract: In this paper we review the techniques developed to prove limitation on the real time recognition ability of Iterative Array and One-way Cellular Automata. The different proofs are based on the same principle which uses equivalence relations and counting arguments.