scholarly journals Coarse structures on groups defined by conjugations

2021 ◽  
Vol 32 (1) ◽  
pp. 65-75
Author(s):  
I. Protasov ◽  
◽  
K. Protasova ◽  
Keyword(s):  
Asymptotic Properties ◽  
Infinite Group ◽  
Locally Finite ◽  
Coarse Structure ◽  
Dedekind Group ◽  
Algebraic Properties ◽  
Coarse Structures ◽  
Coarse Space

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

scholarly journals Coarse cohomology with twisted coefficients

2020 ◽  
Vol 70 (6) ◽  
pp. 1413-1444
Author(s):  
Elisa Hartmann
Keyword(s):  
Sheaf Cohomology ◽  
Grothendieck Topology ◽  
Coarse Structure ◽  
Relative Cohomology ◽  
Number Of Ends ◽  
Constant Coefficients ◽  
Coarse Space ◽  
Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

scholarly journals Extremal balleans

2019 ◽  
Vol 20 (1) ◽  
pp. 297
Author(s):  
Igor Protasov
Keyword(s):  
Coarse Structure ◽  
Unbounded Subset ◽  
Coarse Space ◽  
Extremal Properties

<p>A ballean (or coarse space) is a set endowed with a coarse structure. A ballean X is called normal if any two asymptotically disjoint subsets of X are asymptotically separated.  We say that a ballean X is ultra-normal (extremely normal) if any two unbounded subsets of X are not asymptotically disjoint (every unbounded subset of X is large).   Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold.   A normal ballean is ultranormal if and only if the Higson′s corona of X is a singleton.   A discrete ballean X is ultranormal if and only if X is maximal.  We construct a series of concrete balleans with extremal properties.</p>

scholarly journals On a notion of entropy in coarse geometry

2019 ◽  
Vol 7 (1) ◽  
pp. 48-68
Author(s):  
Nicolò Zava
Keyword(s):  
Large Scale ◽  
Topological Spaces ◽  
Coarse Geometry ◽  
Locally Finite ◽  
Basic Properties ◽  
Algebraic Entropy ◽  
Probability Spaces ◽  
Scale Properties ◽  
Coarse Space

AbstractThe notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.

scholarly journals On Optimal and Asymptotic Properties of a Fuzzy L2 Estimator

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1956
Author(s):  
Jin Hee Yoon ◽  
Przemyslaw Grzegorzewski
Keyword(s):  
Regression Model ◽  
Asymptotic Properties ◽  
Asymptotic Relative Efficiency ◽  
Confidence Regions ◽  
Fuzzy Regression ◽  
Monte Carlo Simulation Study ◽  
Fuzzy Input ◽  
Fuzzy Regression Model ◽  
Algebraic Properties ◽  
Best Linear Unbiased

A fuzzy least squares estimator in the multiple with fuzzy-input–fuzzy-output linear regression model is considered. The paper provides a formula for the L2 estimator of the fuzzy regression model. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. Using the proposed operations, the formula for the variance, provided and this paper, proves that the estimators have several important optimal properties and asymptotic properties: they are Best Linear Unbiased Estimator (BLUE), asymptotic normality and strong consistency. The confidence regions of the coefficient parameters and the asymptotic relative efficiency (ARE) are also discussed. In addition, several examples are provided including a Monte Carlo simulation study showing the validity of the proposed theorems.

ENDS OF CUSP-UNIFORM GROUPS OF LOCALLY CONNECTED CONTINUA — I

2005 ◽  
Vol 15 (04) ◽  
pp. 765-798 ◽  
Author(s):  
DAN P. GURALNIK
Keyword(s):  
Hyperbolic Group ◽  
Infinite Group ◽  
Graph Of Groups ◽  
Locally Finite ◽  
Cut Point ◽  
Word Hyperbolic Group ◽  
Complete Knowledge ◽  
Simplicial Tree ◽  
Locally Connected Continuum ◽  
Uniform Group

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

scholarly journals An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 534
Author(s):  
Lino G. Garza ◽  
Luis E. Garza ◽  
Edmundo J. Huertas
Keyword(s):  
Difference Equation ◽  
Recurrence Relation ◽  
Positive Measure ◽  
Asymptotic Properties ◽  
Nonlinear Difference Equation ◽  
Sobolev Type ◽  
Real Numbers ◽  
Recurrence Coefficients ◽  
Algebraic Properties ◽  
Three Term Recurrence Relation

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.

Free coarse groups

2019 ◽  
Vol 22 (4) ◽  
pp. 775-782
Author(s):  
Igor Protasov ◽  
Ksenia Protasova
Keyword(s):  
Topological Groups ◽  
Coarse Structure ◽  
Variety Of Groups ◽  
And Inversion ◽  
Coarse Space

AbstractA coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let {(X,\mathcal{E})} be a coarse space, and let {\mathfrak{M}} be a variety of groups different from the variety of singletons. We prove that there is a coarse group {F_{\mathfrak{M}}(X,\mathcal{E})\in\mathfrak{M}} such that {(X,\mathcal{E})} is a subspace of {F_{\mathfrak{M}}(X,\mathcal{E})}, X generates {F_{\mathfrak{M}}(X,\mathcal{E})} and every coarse mapping {(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}, where {G\in\mathfrak{M}}, {(G,\mathcal{E}^{\prime})} is a coarse group, can be extended to coarse homomorphism {F_{\mathfrak{M}}(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}. If {\mathfrak{M}} is the variety of all groups, the groups {F_{\mathfrak{M}}(X,\mathcal{E})} are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

scholarly journals Finitary approximations of coarse structures

2021 ◽  
Vol 55 (1) ◽  
pp. 33-36
Author(s):  
I. V. Protasov
Keyword(s):  
Natural Number ◽  
Countable Base ◽  
Coarse Structure ◽  
Coarse Structures

A coarse structure $ \mathcal{E}$ on a set $X$ is called finitary if, for each entourage $E\in \mathcal{E}$, there exists a natural number $n$ such that $ E[x]< n $ for each $x\in X$. By a finitary approximation of a coarse structure $ \mathcal{E}^\prime$, we mean any finitary coarse structure $ \mathcal{E}$ such that $ \mathcal{E}\subseteq \mathcal{E}^\prime$.If $\mathcal{E}^\prime$ has a countable base and $E[x]$ is finite for each $x\in X$ then $ \mathcal{E}^\prime$has a cellular finitary approximation $ \mathcal{E}$ such that the relations of linkness on subsets of $( X,\mathcal{E}^\prime)$ and $( X, \mathcal{E})$ coincide.This answers Question 6 from [8]: the class of cellular coarse spaces is not stable under linkness. We define and apply the strongest finitary approximation of a coarse structure.

scholarly journals Selectors and orderings of coarse spaces

2021 ◽  
Vol 18 (1) ◽  
pp. 71-79
Author(s):  
Igor Protasov
Keyword(s):  
Linear Orders ◽  
Coarse Structure ◽  
Coarse Space

Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.

scholarly journals Closeness and linkness in balleans

2020 ◽  
Vol 53 (1) ◽  
pp. 100-108
Author(s):  
I.V. Protasov ◽  
K. Protasova
Keyword(s):  
General Question ◽  
Coarse Structure ◽  
Coarse Space

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.

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