Topological Representation of Precontact Algebras

2006 ◽  
pp. 1-16 ◽  
Author(s):  
Georgi Dimov ◽  
Dimiter Vakarelov
Keyword(s):  
Topological Representation

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Logic-based topological representation of vague moving regions: computational models for well-behaved GIS solutions

2012 ◽  
Vol 57 (2) ◽  
pp. 259-281 ◽  
Author(s):  
Mohammad H. Vahidnia ◽  
Ali A. Alesheikh ◽  
Mohammad R. Malek
Keyword(s):  
Computational Models ◽  
Topological Representation

scholarly journals Overlapping Representation of Basic Tastes in the Human Gustatory Cortex

2021 ◽  
Author(s):  
Du Zhang ◽  
Xiaoxiao Wang ◽  
Yanming Wang ◽  
Benedictor Alexander Nguchu ◽  
Zhoufang Jiang ◽  
...  
Keyword(s):  
Fundamental Property ◽  
Insular Cortex ◽  
Taste Preference ◽  
Topological Representation ◽  
Activation Patterns ◽  
Gustatory Cortex ◽  
Taste Response ◽  
Neural Activities ◽  
Sensory Cortices ◽  
Multivariate Pattern

The topological representation is a fundamental property of human primary sensory cortices. The human gustatory cortex (GC) responds to the five basic tastes: bitter, salty, sweet, umami, and sour. However, the topological representation of the human gustatory cortex remains controversial. Through functional magnetic resonance imaging(fMRI) measurements of human responses to the five basic tastes, the current study aimed to delineate the taste representations within the GC. During the scanning, the volunteers tasted solutions of the five basic tastes, then washed their mouths with the tasteless solution. The solutions were then sucked from the volunteers' mouths, eliminating the action of swallowing. The results showed that the bilateral mid-insula activated most during the taste task, and the active areas were mainly in the precentral and central insular sulcus. However, the regions responding to the five basic tastes are substantially overlapped, and the analysis of contrasts between each taste response and the averaged response to the remaining tastes does not report any significant results. Furthermore, in the gustatory insular cortex, the multivariate pattern analysis (MVPA) was unable to distinguish the activation patterns of the basic tastes, suggesting the possibility of weakly clustered distribution of the taste-preference neural activities in the human insular cortex. In conclusion, the presented results suggest overlapping representations of the basic tastes in the human gustatory insular cortex.

A Topological Representation of Information: A Heuristic Study

2008 ◽  
Vol 8 (20) ◽  
pp. 3743-3747 ◽  
Author(s):  
A. Haouas ◽  
B. Djebbar ◽  
R. Mekki
Keyword(s):  
Topological Representation ◽  
Heuristic Study

scholarly journals Molecular Cavity Topological Representation for Pattern Analysis: A NLP Analogy-Based Word2Vec Method

2019 ◽  
Vol 20 (23) ◽  
pp. 6019 ◽  
Author(s):  
Dongliang Guo ◽  
Qiaoqiao Wang ◽  
Meng Liang ◽  
Wei Liu ◽  
Junlan Nie
Keyword(s):  
Molecular Dynamics ◽  
Language Processing ◽  
Pattern Analysis ◽  
Real Data ◽  
Molecular Function ◽  
Topological Representation ◽  
Data Set ◽  
Novel Method ◽  
Topological Characteristics

Cavity analysis in molecular dynamics is important for understanding molecular function. However, analyzing the dynamic pattern of molecular cavities remains a difficult task. In this paper, we propose a novel method to topologically represent molecular cavities by vectorization. First, a characterization of cavities is established through Word2Vec model, based on an analogy between the cavities and natural language processing (NLP) terms. Then, we use some techniques such as dimension reduction and clustering to conduct an exploratory analysis of the vectorized molecular cavity. On a real data set, we demonstrate that our approach is applicable to maintain the topological characteristics of the cavity and can find the change patterns from a large number of cavities.

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