scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals ON THE DEFINITION OF L2-BETTI NUMBERS OF EQUIVALENCE RELATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 383-396
Author(s):  
SERGEY NESHVEYEV ◽  
SIMEN RUSTAD
Keyword(s):  
Betti Numbers ◽  
Equivalence Relations ◽  
Definition Of

We show that the L2-Betti numbers of equivalence relations defined by R. Sauer coincide with those defined by D. Gaboriau.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals The absorption theorem for affable equivalence relations

2008 ◽  
Vol 28 (5) ◽  
pp. 1509-1531 ◽  
Author(s):  
THIERRY GIORDANO ◽  
HIROKI MATUI ◽  
IAN F. PUTNAM ◽  
CHRISTIAN F. SKAU
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Closed Subset ◽  
Cantor Set ◽  
Equivalence Relations ◽  
Orbit Structure ◽  
Orbit Equivalent ◽  
Finite Set ◽  
Significant Generalization

AbstractWe prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

scholarly journals Functional Metagenomics of a Biostimulated Petroleum-Contaminated Soil Reveals an Extraordinary Diversity of Extradiol Dioxygenases

2016 ◽  
Vol 82 (8) ◽  
pp. 2467-2478 ◽  
Author(s):  
Laura Terrón-González ◽  
Guadalupe Martín-Cabello ◽  
Manuel Ferrer ◽  
Eduardo Santero
Keyword(s):  
Contaminated Soil ◽  
Phylogenetic Analyses ◽  
Metagenomic Library ◽  
Type I ◽  
Type Ii ◽  
Extradiol Dioxygenase ◽  
Metagenomic Dna ◽  
Petroleum Contaminated Soil ◽  
Definition Of ◽  
Extradiol Dioxygenases

ABSTRACTA metagenomic library of a petroleum-contaminated soil was constructed in a fosmid vector that allowed heterologous expression of metagenomic DNA. The library, consisting of 6.5 Gb of metagenomic DNA, was screened for extradiol dioxygenase (Edo) activity using catechol and 2,3-dihydroxybiphenyl as the substrates. Fifty-eight independent clones encoding extradiol dioxygenase activity were identified. Forty-one different Edo-encoding genes were identified. The population of Edo genes was not dominated by a particular gene or by highly similar genes; rather, the genes had an even distribution and high diversity. Phylogenetic analyses revealed that most of the genes could not be ascribed to previously defined subfamilies of Edos. Rather, the Edo genes led to the definition of 10 new subfamilies of type I Edos. Phylogenetic analysis of type II enzymes defined 7 families, 2 of which harbored the type II Edos that were found in this work. Particularly striking was the diversity found in family I.3 Edos; 15 out of the 17 sequences assigned to this family belonged to 7 newly defined subfamilies. A strong bias was found that depended on the substrate used for the screening: catechol mainly led to the detection of Edos belonging to the I.2 family, while 2,3-dihydroxybiphenyl led to the detection of most other Edos. Members of the I.2 family showed a clear substrate preference for monocyclic substrates, while those from the I.3 family showed a broader substrate range and high activity toward 2,3-dihydroxybiphenyl. This metagenomic analysis has substantially increased our knowledge of the existing biodiversity of Edos.

Proposed Recommended Nutrient Densities for Moderately Malnourished Children

2009 ◽  
Vol 30 (3_suppl3) ◽  
pp. S267-S342 ◽  
Author(s):  
Michael H. Golden
Keyword(s):  
Dietary Intake ◽  
Severe Malnutrition ◽  
Type I ◽  
Type Ii ◽  
Healthy Individuals ◽  
Malnourished Children ◽  
Factorial Approach ◽  
Accelerated Growth ◽  
Moderate Malnutrition ◽  
Definition Of

Recommended Nutrient Intakes (RNIs) are set for healthy individuals living in clean environments. There are no generally accepted RNIs for those with moderate malnutrition, wasting, and stunting, who live in poor environments. Two sets of recommendations are made for the dietary intake of 30 essential nutrients in children with moderate malnutrition who require accelerated growth to regain normality: first, for those moderately malnourished children who will receive specially formulated foods and diets; and second, for those who are to take mixtures of locally available foods over a longer term to treat or prevent moderate stunting and wasting. Because of the change in definition of severe malnutrition, much of the older literature is pertinent to the moderately wasted or stunted child. A factorial approach has been used in deriving the recommendations for both functional, protective nutrients (type I) and growth nutrients (type II).

BLOCKSPIN AND MULTIGRID FOR STAGGERED FERMIONS IN NON-ABELIAN GAUGE FIELDS

1992 ◽  
Vol 03 (01) ◽  
pp. 121-147 ◽  
Author(s):  
T. KALKREUTER ◽  
G. MACK ◽  
M. SPEH
Keyword(s):  
Gauge Theories ◽  
Free Action ◽  
Real Space ◽  
Gauge Fields ◽  
Group Approach ◽  
Abelian Gauge ◽  
Staggered Fermions ◽  
Real Space Renormalization Group ◽  
Renormalization Group Approach ◽  
Definition Of

We discuss blockspins for staggered fermions, i. e. averaging and interpolation procedures which are needed in a real space renormalization group approach to gauge theories with staggered fermions and in a multigrid approach to the computation of gauge covariant propagators. The discussion starts from the requirement that the symmetries of the free action should be preserved by the blocking procedure in the limit of a pure gauge. A definition of an averaging kernel as a solution of a gauge covariant eigenvalue equation is proposed, and the properties of a corresponding interpolation kernel are examined in the light of general criteria for good choices of blockspins. Some results of multigrid computations of bosonic propagators in an SU(2) gauge field in 4 dimensions are also presented.

scholarly journals Lattice Theory of Generalized Partitions

1959 ◽  
Vol 11 ◽  
pp. 97-106 ◽  
Author(s):  
Juris Hartmanis
Keyword(s):  
Equivalence Relation ◽  
Lattice Theory ◽  
Equivalence Relations ◽  
Unified Theory ◽  
Fixed Set ◽  
Higher Types ◽  
Definition Of ◽  
Partition Lattices

In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.

scholarly journals Inner amenability, property Gamma, McDuff factors and stable equivalence relations

2017 ◽  
Vol 38 (7) ◽  
pp. 2618-2624 ◽  
Author(s):  
TOBE DEPREZ ◽  
STEFAAN VAES
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Countable Group ◽  
Acta Math ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Von Neumann ◽  
Stable Equivalence ◽  
Inner Amenability

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

scholarly journals Examples of Calabi-Yau threefolds with small Hodge numbers

2021 ◽  
Vol 2081 (1) ◽  
pp. 012030
Author(s):  
A O Shishanin
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Gauge Group ◽  
Complete Intersection ◽  
Euler Characteristic ◽  
Discrete Group ◽  
Free Action ◽  
Discrete Groups ◽  
Hodge Numbers ◽  
Fixed Point Sets

Abstract We observe some suitable examples of Calabi-Yau threefolds for heterotic superstring compactifications. It is reasonable to seek CY threefolds with Euler characteristic equals ±6 because of generation’s number. Hosotani mechanism for violations of the gauge group by the Wilson loops requires such CY space has a non-trivial fundamental group. These spaces can be obtained by factoring the complete intersection Calabi-Yau spaces by the free action of some discrete group. Also we shortly discuss cases when discrete groups act with fixed point sets.

scholarly journals Cladistic hypotheses as degree of equivalence relational structures: implications for three-item statements

2021 ◽  
Author(s):  
Valentin Rineau ◽  
Stéphane Prin
Keyword(s):  
Phylogenetic Trees ◽  
Phylogenetic Reconstruction ◽  
Equivalence Relations ◽  
Strong Connection ◽  
Relational Structures ◽  
Degree Of Equivalence ◽  
Axiomatic Definition ◽  
Finite Set ◽  
Definition Of ◽  
Supertree Methods

AbstractThree-item statements, as minimal informative rooted binary phylogenetic trees on three items, are the minimal units of cladistic information. Their importance for phylogenetic reconstruction, consensus and supertree methods relies on both (i) the fact that any cladistic tree can always be decomposed into a set of three-item statements, and (ii) the possibility, at least under some conditions, to build a new cladistic tree by combining all or part of the three-item statements deduced from several prior cladistic trees. In order to formalise such procedures, several k-adic rules of inference, i.e., rules that allow us to deduce at least one new three-item statement from exactly k other ones, have been identified. However, no axiomatic background has been proposed, and it remains unknown if a particular k-adic rule of inference can be reduced to more basic rules. In order to solve this problem, we propose here to define three-item statements in terms of degree of equivalence relations. Given both the axiomatic definition of the latter and their strong connection to hierarchical classifications, we establish a list of the most basic properties for three-item statements. With such an approach, we show that it is possible to combine five three-item statements from basic rules although they are not combinable only from dyadic rules. Such a result suggests that all higher k-adic rules are well reducible to a finite set of simpler rules.

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