scholarly journals Multiplication of weak equivalence classes may be discontinuous

2019 ◽  
Vol 372 (11) ◽  
pp. 8091-8106
Author(s):  
Anton Bernshteyn
Keyword(s):  
Equivalence Classes ◽  
Weak Equivalence

scholarly journals Equivalence classes of functions on finite sets

1982 ◽  
Vol 5 (4) ◽  
pp. 745-762
Author(s):  
Chong-Yun Chao ◽  
Caroline I. Deisher
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Permutation Groups ◽  
Weak Equivalence ◽  
Finite Sets ◽  
The Family ◽  
Polya's Theorem ◽  
Finite Set ◽  
Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

The Symmetries of Genus One Handlebodies

1991 ◽  
Vol 43 (2) ◽  
pp. 371-404 ◽  
Author(s):  
John Kalliongis ◽  
Andy Miller
Keyword(s):  
Finite Groups ◽  
Focal Point ◽  
Complete Classification ◽  
Equivalence Classes ◽  
Weak Equivalence ◽  
Finite Group Actions ◽  
3 Dimensional ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Genus One

The symmetries of manifolds are a focal point of study in low-dimensional topology and yet, outside of some totally asymmetrical 3- and 4-manifolds, there are very few cases in which a complete classification has been attained. In this work we provide such a classification for symmetries of the orientable and nonorientable 3-dimensional handlebodies of genus one. Our classification includes a description, up to isomorphism, of all of the finite groups which can arise as symmetries on these manifolds, as well as an enumeration of the different ways in which they can arise. To be specific, we will classify the equivalence, weak equivalence and strong equivalence classes of (effective) finite group actions on the genus one handlebodies.

scholarly journals Topology and convexity in the space of actions modulo weak equivalence

2017 ◽  
Vol 38 (7) ◽  
pp. 2508-2536 ◽  
Author(s):  
PETER BURTON
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Convex Combination ◽  
Compact Convex ◽  
Type Theorem ◽  
Extreme Points ◽  
Equivalence Classes ◽  
Compact Convex Subset ◽  
Weak Equivalence ◽  
Convex Structure

We analyze the structure of the quotient $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$. This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is compatible with the topology, and as a consequence deduce that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ for amenable $\unicode[STIX]{x1D6E4}$ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ can be represented as a compact convex subset of a Banach space if and only if $\unicode[STIX]{x1D6E4}$ is amenable. In the case of general $\unicode[STIX]{x1D6E4}$ we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ are dense in this space. We also consider the space $\text{A}_{{\sim}_{s}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group $\mathbb{F}_{N}$, we show that if one restricts to the compact convex set $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})\subseteq \text{A}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$ consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$.

scholarly journals Weak containment of measure-preserving group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2681-2733 ◽  
Author(s):  
PETER J. BURTON ◽  
ALEXANDER S. KECHRIS
Keyword(s):  
Global Structure ◽  
Equivalence Classes ◽  
The Other ◽  
Unitary Representations ◽  
Weak Equivalence ◽  
Interesting Fact ◽  
Weak Containment ◽  
Probability Spaces ◽  
Measure Preserving Actions ◽  
Standard Probability

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

scholarly journals Explicit computation of some families of Hurwitz numbers, II

2020 ◽  
Vol 20 (4) ◽  
pp. 483-498
Author(s):  
Carlo Petronio
Keyword(s):  
Target Surface ◽  
Equivalence Classes ◽  
Weak Equivalence ◽  
Source Surface ◽  
Explicit Computation ◽  
Combinatorial Method ◽  
Branched Covers ◽  
Hurwitz Numbers ◽  
Branching Points

AbstractWe continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2, …, 2], [2h + 1, 3, 2, …, 2], $\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di’s.

scholarly journals MET: a Java package for fast molecule equivalence testing

2020 ◽  
Vol 12 (1) ◽  
Author(s):  
Jördis-Ann Schüler ◽  
Steffen Rechner ◽  
Matthias Müller-Hannemann
Keyword(s):  
Chemical Properties ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Equivalence Classes ◽  
Equivalence Testing ◽  
Equivalence Test ◽  
Graph Isomorphism Problem ◽  
2D Structure ◽  
Node Labels ◽  
Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

On equivariant disconnected rational homotopy theory

2010 ◽  
Vol 17 (2) ◽  
pp. 229-240
Author(s):  
Marek Golasiński
Keyword(s):  
Homotopy Theory ◽  
Weak Equivalence ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Hamiltonian Group ◽  
Simplicial Set ◽  
De Rham

Abstract An equivariant disconnected Sullivan–de Rham equivalence is developed using Kan's result on diagram categories. Given a finite Hamiltonian group G, let X be a G-simplicial set. It is shown that the associated system of algebras indexed by the category 𝒪(G) of a canonical orbit can be “approximated” (up to a weak equivalence) by such a system ℳ X with the properties required by nonequivariant minimal algebras.

Parallel and Distributed Processability of Objects

1989 ◽  
Vol 12 (3) ◽  
pp. 317-356
Author(s):  
David C. Rine
Keyword(s):  
Software Engineering ◽  
Software Design ◽  
Equivalence Classes ◽  
Minimum Amount ◽  
Software Components ◽  
Limited Extent ◽  
Design Environment ◽  
Distributed Software

Partitioning and allocating of software components are two important parts of software design in distributed software engineering. This paper presents two general algorithms that can, to a limited extent, be used as tools to assist in partitioning software components represented as objects in a distributed software design environment. One algorithm produces a partition (equivalence classes) of the objects, and a second algorithm allows a minimum amount of redundancy. Only binary relationships of actions (use or non-use) are considered in this paper.

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