scholarly journals On cardinalities of k-abelian equivalence classes

2017 ◽  
Vol 658 ◽  
pp. 190-204 ◽  
Author(s):  
Juhani Karhumäki ◽  
Svetlana Puzynina ◽  
Michaël Rao ◽  
Markus A. Whiteland
Keyword(s):  
Equivalence Classes ◽  
Abelian Equivalence

scholarly journals The binomial equivalence classes of finite words

2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1397
Author(s):  
Marie Lejeune ◽  
Michel Rigo ◽  
Matthieu Rosenfeld
Keyword(s):  
Nilpotent Group ◽  
Equivalence Class ◽  
Free Monoid ◽  
Equivalence Classes ◽  
Single Element ◽  
Free Nilpotent Group ◽  
Context Free ◽  
Abelian Equivalence

Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

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.

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.

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

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.

scholarly journals A Note on Equivalence Classes of Directed Acyclic Independence Graphs

1993 ◽  
Vol 7 (3) ◽  
pp. 409-412 ◽  
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.

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

2008 ◽  
Vol 17 (05) ◽  
pp. 579-599 ◽  
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.

scholarly journals Five-point zero-divisor graphs determined by equivalence classes

2011 ◽  
Vol 4 (1) ◽  
pp. 53-64
Author(s):  
Florida Levidiotis ◽  
Sandra Spiroff
Keyword(s):  
Zero Divisor ◽  
Equivalence Classes
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