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.

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 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 $(X, \mathcal{E})$.

Identifying Community Interests in International Law

2018 ◽  
Author(s):  
Rüdiger Wolfrum
Keyword(s):  
International Law ◽  
Dispute Settlement ◽  
International Community ◽  
Available P ◽  
General Question ◽  
Global Commons ◽  
Settlement Systems ◽  
Community Interest ◽  
National Jurisdiction ◽  
Advisory Opinions

This chapter explores the general question of how to establish that the regulation of a certain matter constitutes a matter of community-wide concern, which is the necessary step for the recognition of community obligation. The hypothesis is that such a qualification must, first, be well founded factually and, secondly, accepted as such in a legal or political legitimizing process. On this basis, the chapter suggests that the governance of spaces beyond national jurisdiction constitutes a community interest and has to be guided by the interests of the international community. Exploring this question with respect to key common spaces and particular issues, the chapter notes the difficulty of most of the dispute settlement systems, which, being bilateral, are not fully adequate to address questions related to the management of global commons as well as for the protection of the environment. To avoid this difficulty, the chapter suggests greater reliance on advisory opinions where available.

scholarly journals A New Methodological Approach to Correlate Protective and Microscopic Properties by Soft X-ray Microscopy and Solid State NMR Spectroscopy: The Case of Cusa’s Stone

2021 ◽  
Vol 11 (13) ◽  
pp. 5767
Author(s):  
Veronica Ciaramitaro ◽  
Alberto Spinella ◽  
Francesco Armetta ◽  
Roberto Scaffaro ◽  
Emmanuel Fortunato Gulino ◽  
...  
Keyword(s):  
Solid State ◽  
Nmr Spectroscopy ◽  
Solid State Nmr ◽  
Methodological Approach ◽  
Polymer Chains ◽  
General Question ◽  
Protective Agent ◽  
Vapour Permeability ◽  
X Ray ◽  
Solid State Nmr Spectroscopy

Hydrophobic treatment is one of the most important interventions usually carried out for the conservation of stone artefacts and monuments. The study here reported aims to answer a general question about how two polymers confer different protective performance. Two fluorinated-based polymer formulates applied on samples of Cusa’s stone confer a different level of water repellence and water vapour permeability. The observed protection action is here explained on the basis of chemico-physical interactions. The distribution of the polymer in the pore network was investigated using scanning electron microscopy and X-ray microscopy. The interactions between the stone substrate and the protective agents were investigated by means of solid state NMR spectroscopy. The ss-NMR findings reveal no significant changes in the chemical neighbourhood of the observed nuclei of each protective agent when applied onto the stone surface and provide information on the changes in the organization and dynamics of the studied systems, as well as on the mobility of polymer chains. This allowed us to explain the different macroscopic behaviours provided by each protective agent to the stone substrate.

The pace of “the good life”: Connecting past, present, and future in the context of a housing affordability crisis

2021 ◽  
pp. 0961463X2098781
Author(s):  
Petr Kubala ◽  
Tomáš Hoření Samec
Keyword(s):  
Life Course ◽  
Good Life ◽  
Social Inequalities ◽  
Housing Affordability ◽  
General Question ◽  
The Good Life ◽  
The Czech Republic ◽  
Ideal Types ◽  
Depth Interviews ◽  
The Ideal

This article focuses on the topic of the young adult’s cleft habitus influenced by a housing affordability crisis in the Czech Republic and examines how this situation affects the young adult’s relation to the imagination of a temporally structured life course and synchronization of life spheres (housing, family, and work). This article is based on qualitative in-depth interviews conducted in the four cities most affected by the house and rent price increase. The general question addresses if and how social inequalities, sharpened by the current housing affordability crisis, affect the process of narrative life course coherence creation (the connection of past, present, and future) in relation to an orientation toward a vision of “the good life.” We furthermore complement the already existing ideal types of the young adult’s relation toward time— confident continuity and cautious contingency—with two other two types— cautious continuity and total contingency—defined on the basis of our data. We argue that the ability of young adults to envision a coherent future is related to the feeling of secured housing and that the idea of the good life is depicted to a large extent through the ideal of homeownership, although the precarity of the housing market makes homeownership harder to reach for those from unprivileged backgrounds.

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.

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