Diamonds associated with adic spaces

2020 ◽  
pp. 74-89
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Topological Space ◽  
Topological Information ◽  
Residue Fields

This chapter focuses on diamonds associated with adic spaces. The goal is to construct a functor which forgets the structure morphism to Spa Zp, but retains topological information. The chapter studies how much information is lost when applying this construction. The intuition is that only topological information is kept. A morphism of adic spaces is a universal homeomorphism if all pullbacks are homeomorphisms. As in the case of schemes, in characteristic 0 the map f is a universal homeomorphism if and only if it is a homeomorphism and induces isomorphisms on completed residue fields. In keeping with the intuition, universal homeomorphisms induce isomorphisms of diamonds. The chapter then considers the underlying topological space of diamonds, as well as the étale site of diamonds.

Basic Topological Notions and their Relation to BIM

2010 ◽  
pp. 451-472 ◽  
Author(s):  
Norbert Paul
Keyword(s):  
Topological Space ◽  
Information Model ◽  
Topological Model ◽  
Topological Information ◽  
Building Information Model ◽  
Building Element ◽  
Connectivity Information ◽  
Building Information ◽  
Global Connectivity ◽  
Building Sets

Each building sets up a topological space in the mathematical sense. Therefore every Building Information Model (BIM) has to store topological information. Such information can be found, for example, in the IFC (Liebich et al. 2005). The volume modelling part of the IFC uses a so-called ‘IfcTopologyResource’ which is a topological model on the local scope of each single building element. At a global scope, the ‘IfcRelConnects’ class and its subclasses are used for the connectivity of the building parts. This chapter presents a generalizing concept which handles both “local” and “global” connectivity information in a common way and provides means to mutually relate them.

Combinatorial Design of Molecule using Activity-Linked Substructural Topological Information as Applied to Antitubercular Compounds

2018 ◽  
Vol 15 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Chandan Raychaudhury ◽  
Md. Imbesat Hassan Rizvi ◽  
Debnath Pal
Keyword(s):  
Biological Activities ◽  
Combinatorial Design ◽  
Combinatorial Structure ◽  
Potential Drug ◽  
Structure Generation ◽  
Topological Information ◽  
Active Compounds ◽  
Antitubercular Drugs ◽  
Topological Distance ◽  
Drug Molecules

Background: Generating a large number of compounds using combinatorial methods increases the possibility of finding novel bioactive compounds. Although some combinatorial structure generation algorithms are available, any method for generating structures from activity-linked substructural topological information is not yet reported. Objective: To develop a method using graph-theoretical techniques for generating structures of antitubercular compounds combinatorially from activity-linked substructural topological information, predict activity and prioritize and screen potential drug candidates. </P><P> Methods: Activity related vertices are identified from datasets composed of both active and inactive or, differently active compounds and structures are generated combinatorially using the topological distance distribution associated with those vertices. Biological activities are predicted using topological distance based vertex indices and a rule based method. Generated structures are prioritized using a newly defined Molecular Priority Score (MPS). Results: Studies considering a series of Acid Alkyl Ester (AAE) compounds and three known antitubercular drugs show that active compounds can be generated from substructural information of other active compounds for all these classes of compounds. Activity predictions show high level of success rate and a number of highly active AAE compounds produced high MPS score indicating that MPS score may help prioritize and screen potential drug molecules. A possible relation of this work with scaffold hopping and inverse Quantitative Structure-Activity Relationship (iQSAR) problem has also been discussed. The proposed method seems to hold promise for discovering novel therapeutic candidates for combating Tuberculosis and may be useful for discovering novel drug molecules for the treatment of other diseases as well.

The Spatial Dimensions of Social Networks

2020 ◽  
pp. 367-383
Author(s):  
Zachary P. Neal
Keyword(s):  
Social Networks ◽  
Topological Space ◽  
Network Science ◽  
Spatial Dimensions

The first law of geography holds that everything is related to everything else, but near things are more related than distant things, where distance refers to topographical space. If a first law of network science exists, it would similarly hold that everything is related to everything else, but near things are more related than distant things, but where distance refers to topological space. Frequently these two laws collide, together holding that everything is related to everything else, but topographically and topologically near things are more related than topographically and topologically distant things. The focus of the spatial study of social networks lies in exploring a series of questions embedded in this combined law of geography and networks. This chapter explores the questions that have been asked and the answers that have been offered at the intersection of geography and networks.

scholarly journals Data-driven biological network alignment that uses topological, sequence, and functional information

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Shawn Gu ◽  
Tijana Milenković
Keyword(s):  
Prediction Accuracy ◽  
Biological Network ◽  
Network Alignment ◽  
Data Driven ◽  
Sequence Information ◽  
Functional Prediction ◽  
Research Knowledge ◽  
Topological Information ◽  
Functional Knowledge ◽  
Related Proteins

Abstract Background Network alignment (NA) can transfer functional knowledge between species’ conserved biological network regions. Traditional NA assumes that it is topological similarity (isomorphic-like matching) between network regions that corresponds to the regions’ functional relatedness. However, we recently found that functionally unrelated proteins are as topologically similar as functionally related proteins. So, we redefined NA as a data-driven method called TARA, which learns from network and protein functional data what kind of topological relatedness (rather than similarity) between proteins corresponds to their functional relatedness. TARA used topological information (within each network) but not sequence information (between proteins across networks). Yet, TARA yielded higher protein functional prediction accuracy than existing NA methods, even those that used both topological and sequence information. Results Here, we propose TARA++ that is also data-driven, like TARA and unlike other existing methods, but that uses across-network sequence information on top of within-network topological information, unlike TARA. To deal with the within-and-across-network analysis, we adapt social network embedding to the problem of biological NA. TARA++ outperforms protein functional prediction accuracy of existing methods. Conclusions As such, combining research knowledge from different domains is promising. Overall, improvements in protein functional prediction have biomedical implications, for example allowing researchers to better understand how cancer progresses or how humans age.

On a topological construction of Juhasz and Shelah

1992 ◽  
Vol 57 (1) ◽  
pp. 166-171
Author(s):  
Dan Velleman
Keyword(s):  
Topological Space ◽  
Additional Structure ◽  
Homogeneous Set ◽  
Stationary Set ◽  
Topological Construction

In [2], Juhasz and Shelah use a forcing argument to show that it is consistent with GCH that there is a 0-dimensional T2 topological space X of cardinality ℵ3 such that every partition of the triples of X into countably many pieces has a nondiscrete (in the topology) homogeneous set. In this paper we will show how to construct such a space using a simplified (ω2, 1)-morass with certain additional structure added to it. The additional structure will be a slight strengthening of a built-in ◊ sequence, analogous to the strengthening of ordinary ◊k to ◊S for a stationary set S ⊆ k.Suppose 〈〈θα∣ ∝ ≤ ω2〉, 〈∝β∣α < β ≤ ω2〉〉 is a neat simplified (ω2, 1)-morass (see [3]). Let ℒ be a language with countably many symbols of all types, and suppose that for each α < ω2, α is an ℒ-structure with universe θα. The sequence 〈α∣α < ω2 is called a built-in ◊ sequence for the morass if for every ℒ-structure with universe ω3 there is some α < ω2 and some f ∈αω2 such that f(α) ≺ , where f(α) is the ℒ-structure isomorphic to α under the isomorphism f. We can strengthen this slightly by assuming that α is only defined for α ∈ S, for some stationary set S ⊆ ω2. We will then say that is a built-in ◊ sequence on levels in S if for every ℒ-structure with universe ω3 there is some α ∈ S and some f ∈ αω2 such that f(α) ≺ .

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

scholarly journals Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1781
Author(s):  
Samer Al Ghour
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Soft Sets ◽  
New Concepts ◽  
Bitopological Spaces ◽  
The Relationship ◽  
Soft Topological Space

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

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