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 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>

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 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 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.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

scholarly journals FINDING DESCENDING SEQUENCES THROUGH ILL-FOUNDED LINEAR ORDERS

2021 ◽  
pp. 1-39
Author(s):  
JUN LE GOH ◽  
ARNO PAULY ◽  
MANLIO VALENTI
Keyword(s):  
Linear Orders

scholarly journals Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

2021 ◽  
Vol 26 (2) ◽  
pp. 44
Author(s):  
Eric Chung ◽  
Hyea-Hyun Kim ◽  
Ming-Fai Lam ◽  
Lina Zhao
Keyword(s):  
Deep Neural Network ◽  
Elliptic Problems ◽  
Learning Approach ◽  
High Contrast ◽  
Spectral Problems ◽  
Significant Challenge ◽  
Machine Learning Approach ◽  
Balancing Domain Decomposition ◽  
Computationally Expensive ◽  
Coarse Space

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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