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Equivalence Classes (equivalence + class)

Distribution by Scientific Domains
Mathematics and Statistics41%
Chemistry25%

Selected Abstracts

Between ends and fibers

JOURNAL OF GRAPH THEORY, Issue 2 2007
C. Paul Bonnington
Abstract Let , be an infinite, locally finite, connected graph with distance function ,. Given a ray P in , and a constant C , 1, a vertex-sequence is said to be regulated by C if, for all n,,, never precedes xn on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 1964, 125,137) defined two rays to be end-equivalent if they are joined by infinitely many pairwise-disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 1993, 477,484) defined rays P and Q to be b-equivalent if there exist sequences and VQ regulated by some constant C , 1 such that for all n,,; he named the resulting equivalence classes b-fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f-equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b -equivalence while, under specified conditions, are no coarser than end-equivalence. Indeed, for every , there exists an "end-defining function" that is unbounded and sublinear and such that implies that P and Q are end-equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of "initially metric" rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end-equivalent. In the case that , contains translatable rays we give some sufficient conditions for every f -equivalence class to contain uncountably many g -equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost-transitive planar map are never bundle- equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125,153, 2007 [source]

Sharp bounds for the number of 3-independent partitions and the chromaticity of bipartite graphs

JOURNAL OF GRAPH THEORY, Issue 1 2001
F. M. Dong
Abstract Given a graph G and an integer k,,,1, let ,(G,,k) denote the number of k -independent partitions of G. Let ,,,s(p,q) (resp., ,,2,s(p,q)) denote the family of connected (resp., 2-connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p,,,q,,,2. This paper first gives a sharp upper bound for ,(G,3), where G ,,,,,,s(p,q) and 0,,,s,,,(p,,,1)(q,,,1) (resp., G ,,,,,2,s(p,q) and 0,,,s,,,p,+,q,,,4). These bounds are then used to show that if G ,,,,,,s(p,q) (resp., G ,,,,,2,s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ,,,si(p+i,q,i) where max and si,=,s,,,i(p,q+i) (resp., a subset of ,,2,s(p,q), where either 0,,,s,,,q,,,1, or s,,,2q,,,3 and p,,,q,+,4). By applying these results, we show finally that any 2-connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q,,,1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48,77, 2001 [source]

New Concepts in Evolutionary Search for Boolean Functions in Cryptology

COMPUTATIONAL INTELLIGENCE, Issue 3 2004
William Millan
In symmetric cryptology the resistance to attacks depends critically on the nonlinearity properties of the Boolean functions describing cipher components like Substitution boxes (S-boxes). Some of the most effective methods known to generate functions that satisfy multiple criteria are based on evolutionary heuristics. In this paper, we improve on these algorithms by employing an adaptive strategy. Additionally, using recent improvements in the understanding of these combinatorial structures, we discover essential properties of the graph formed by affine equivalence classes of Boolean functions, which offers several advantages as a conceptual model for multiobjective seeking evolutionary heuristics. Finally, we propose the first major global cooperative effort to discover new bounds for cryptographic properties of Boolean functions. [source]

Quasilocal defects in regular planar networks: Categorization for molecular cones

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 4-5 2003
D. J. Klein
Abstract Graphical networks are cast into structural equivalence classes, with special focus on ones related to two-dimensional regular translationally symmetric nets (or lattices). A quasilocal defect in a regular net is defined as consisting of a finite subnet surrounded outside this region by an infinitely extended network of which arbitrary, simply connected regions are isomorphic to those of the regular undefected net. The global equivalence classes for such quasilocal defects are identified by a "circum-matching" characteristic. One or more such classes are identified as a "turn" number, or equivalently as a discrete "combinatorial curvature" ,, which associates closely to the geometric Gaussian curvature of "physically reasonable" embeddings of the net in Euclidean space. Then for positive ,, geometric cones result; for , = 0, the network is flat overall; and for negative ,, fluted or crenalated cones result. As , or q varies through its discrete range, the number of defect classes varies between 1 and , and repeats with a period depending on the parent regular net. For the square-planar net, the numbers of defect classes at succeeding turn numbers (q) starting at q = 0 are ,, 2, 3, 2, repeating with a period of 4. For the hexagonal and triangular nets, the numbers of classes at suceeding q starting at q = 0 are ,, 1, 2, 2, 2, 1, repeating with a period of 6. A further refinement of the classes of quasilocal defects breaks these classes up into "irrotational" subclasses, as are relevant for multiwall cones. The subclasses are identified via a "quasispin" characteristic, which is conveniently manipulatable for the categorization of multiwall cones. Besides the development of the overall comprehensive topo-combinatoric categorization scheme for quasilocal defects, some consequences are briefly indicated, and combining rules for the characteristics of pairs of such defects are briefly considered. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003 [source]

Between ends and fibers

JOURNAL OF GRAPH THEORY, Issue 2 2007
C. Paul Bonnington
Abstract Let , be an infinite, locally finite, connected graph with distance function ,. Given a ray P in , and a constant C , 1, a vertex-sequence is said to be regulated by C if, for all n,,, never precedes xn on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 1964, 125,137) defined two rays to be end-equivalent if they are joined by infinitely many pairwise-disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 1993, 477,484) defined rays P and Q to be b-equivalent if there exist sequences and VQ regulated by some constant C , 1 such that for all n,,; he named the resulting equivalence classes b-fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f-equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b -equivalence while, under specified conditions, are no coarser than end-equivalence. Indeed, for every , there exists an "end-defining function" that is unbounded and sublinear and such that implies that P and Q are end-equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of "initially metric" rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end-equivalent. In the case that , contains translatable rays we give some sufficient conditions for every f -equivalence class to contain uncountably many g -equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost-transitive planar map are never bundle- equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125,153, 2007 [source]

Differentiable structure of the set of coaxial stress,strain tensors,

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2009
Josep Clotet
Abstract In order to study stress,strain tensors, we consider their representations as pairs of symmetric 3 × 3-matrices and the space of such pairs of matrices partitioned into equivalence classes corresponding to change of bases. We see that these equivalence classes are differentiable submanifolds; in fact, orbits under the action of a Lie group. We compute their dimension and obtain miniversal deformations. Finally, we prove that the space of coaxial stress,strain tensors is a finite union of differentiable submanifolds. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Russell, His Paradoxes, and Cantor's Theorem: Part I

PHILOSOPHY COMPASS (ELECTRONIC), Issue 1 2010
Kevin C. Klement
In these articles, I describe Cantor's power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell's work. These include Russell's paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor's theorem, its proof, how it can be used to manufacture paradoxes, Frege's diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. [source]

Russell, His Paradoxes, and Cantor's Theorem: Part II

PHILOSOPHY COMPASS (ELECTRONIC), Issue 1 2010
Kevin C. Klement
Sequel to Part I. In these articles, I describe Cantor's power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell's work. These include Russell's paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell's own various attempts to solve these paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ,logical fictions' or ,logical constructions' instead. [source]

Zones and sublattices of integral lattices

ACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2004
A. Janner
Methods are presented for an analysis of zones and sublattices of integral lattices, whose relevance is revealed by sharp peaks in the frequency distribution of hexagonal and tetragonal lattices, as a function of the axial ratio . Starting from a few examples, zone symmetries, lattice,sublattice relations and integral scaling transformations are derived for hexagonal lattices with axial ratios , , and 1 (the isometric case) and for the related and tetragonal lattices. Sublattices and zones connected by linear rational transformations lead to rational equivalence classes of integral lattices. For properties like the axial ratio and the point-group symmetry (lattice holohedry), rational equivalence can be extended so that also metric tensors differing by an integral factor become equivalent. These two types of equivalence classes are determined for the lattices mentioned above. [source]

Training and testing music skills in a boy with autism using a matching-to-sample format

BEHAVIORAL INTERVENTIONS, Issue 2 2010
Erik Arntzen
A 16-year old boy with autism was taught music skills using a matching to sample procedure. He was trained and subsequently tested for the formation of four 4-member classes, including different visual music stimuli, and Norwegian and Vietnamese labels for different major and minor chords. Four different stimuli sets were trained both in one-to-many (OTM) and many-to-one (MTO) training structures. Further, we explored if the reaction times to comparison stimuli increased from training to testing. Results showed that the participant formed equivalence classes with music relations. Furthermore, there were small differences only between OTM and MTO with respect to stimulus equivalence responding. The reaction times to comparison stimuli increased from training to testing, and were most pronounced for the equivalence trials. Copyright © 2010 John Wiley & Sons, Ltd. [source]