The elementary-equivalence classes of clopen algebras of P-spaces

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.

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Parallel and Distributed Processability of Objects

Fundamenta Informaticae ◽

10.3233/fi-1989-12304 ◽

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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A dichotomy for bounded displacement equivalence of Delone sets

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.60 ◽

2021 ◽

pp. 1-18

Author(s):

YOTAM SMILANSKY ◽

YAAR SOLOMON

Keyword(s):

Compact Space ◽

Equivalence Classes ◽

Natural Topology ◽

Fell Topology ◽

Local Complexity ◽

Substitution Tilings ◽

Local Topology ◽

Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

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A Note on Equivalence Classes of Directed Acyclic Independence Graphs

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003004 ◽

1993 ◽

Vol 7 (3) ◽

pp. 409-412 ◽

Cited By ~ 1

Author(s):

David Madigan

Keyword(s):

Model Selection ◽

Expert Systems ◽

Equivalence Classes ◽

Influence Diagrams ◽

Single Class ◽

Selection Procedures ◽

Probabilistic Expert Systems ◽

Recent Developments ◽

Independence Graphs ◽

Markov Properties

Directed acyclic independence graphs (DAIGs) play an important role in recent developments in probabilistic expert systems and influence diagrams (Chyu [1]). The purpose of this note is to show that DAIGs can usefully be grouped into equivalence classes where the members of a single class share identical Markov properties. These equivalence classes can be identified via a simple graphical criterion. This result is particularly relevant to model selection procedures for DAIGs (see, e.g., Cooper and Herskovits [2] and Madigan and Raftery [4]) because it reduces the problem of searching among possible orientations of a given graph to that of searching among the equivalence classes.

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A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006312 ◽

2008 ◽

Vol 17 (05) ◽

pp. 579-599 ◽

Cited By ~ 14

Author(s):

MARIA RITA CASALI ◽

PAOLA CRISTOFORI

Keyword(s):

Automatic Generation ◽

Computational Approach ◽

Equivalence Classes ◽

Colored Graphs ◽

One To One ◽

Genus Two ◽

Jsj Decompositions

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

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Uncountable models and infinitary elementary extensions

Journal of Symbolic Logic ◽

10.2307/2273044 ◽

1973 ◽

Vol 38 (3) ◽

pp. 460-470 ◽

Cited By ~ 4

Author(s):

John Gregory

Keyword(s):

Countable Subset ◽

Elementary Extension ◽

Admissible Set ◽

Natural Numbers ◽

Elementary Equivalence ◽

Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

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