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A Spatial Logic for a Simplicial Complex Model [article]

Michele Loreti, Michela Quadrini
2021 arXiv   pre-print
In this work, we propose a spatial logic interpreted on simplicial complexes.  ...  We discuss how the satisfaction of logical formulas can be verified by a correct and complete model checking algorithm, which is linear to the dimension of the simplicial complex and logical formula.  ...  Therefore, σ i and σ j are spatial adjacent. Spatial logics for Simplicial Complex In this section, we introduce a Spatial Logic for Simplicial Complex (SLSC).  ... 
arXiv:2105.08708v2 fatcat:xnh6dvrgknewhg2frdzlevsg4u

Spatial Topology and its Structural Analysis based on the Concept of Simplicial Complex

Bin Jiang, Itzhak Omer
2007 Transactions on GIS  
This paper introduces a model that identifies spatial relationships for a structural analysis based on the concept of simplicial complex.  ...  The model relies on the simplical complex for structural representation and analysis.  ...  The model aims to represent spatial relationships of various objects as a simplicial complex.  ... 
doi:10.1111/j.1467-9671.2007.01073.x fatcat:diwcsovybfdlbdmrterybbmw6q

Topology Based Object Tracking

Padraig Corcoran
2019 Mathematical and Computational Applications  
A generalisation of this model for tracking spatially close objects lying in an ambient metric space is also proposed.  ...  A model for tracking objects whose topological properties change over time is proposed.  ...  [14] proposed models for tracking topological changes in spatial phenomena using a geosensor network.  ... 
doi:10.3390/mca24030084 fatcat:aknvj3sivvgpnklyzzizvkh7gy

Topological schemas of cognitive maps and spatial learning in the hippocampus [article]

A. Babichev, S. Cheng, Yu. Dabaghian
2016 arXiv   pre-print
In this paper we suggest computational frameworks for integrating the spiking signals of individual cells into a spatial map, which we call schemas.  ...  Spatial navigation in mammals is based on building a mental representation of their environment---a cognitive map.  ...  Spatial learning A key property of our approach, crucial for modeling spatial learning, is that schemas are dynamic objects.  ... 
arXiv:1509.00171v2 fatcat:67xvm7xfdvgcdhcpjwudygve2a

Modular programming of computing media using spatial types, for artificial physics [article]

Frederic Gruau
2019 arXiv   pre-print
The simplicial proximity between bits is used to define operations on fields, thus implementing " spatial type". Expression combining operations can be translated in logical circuits.  ...  This high level of complexity forced us to work out an efficient and expressive scheme for programming the medium, the goal of this paper it to present it.  ...  Acknowledgment We thank Luidnel Maignan for his fruitful comments.  ... 
arXiv:1904.05787v1 fatcat:qjavzqpwonc33jeimiqjndapsm

Modelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes

Padraig Corcoran, Christopher B. Jones
2017 IEEE Access  
This paper presents a model of swarm behaviour that encodes the spatial-temporal characteristics of topological features such as holes and connected components.  ...  Validation of the proposed model is performed using a real data set corresponding to a swarm of fish.  ...  For a given simplicial complex there exists a number of methods for computing the corresponding Betti numbers [38] , [8] . C.  ... 
doi:10.1109/access.2017.2749319 fatcat:hxlerrg3zjaanmxdnkk3ky3xyq

The why, how, and when of representations for complex systems [article]

Leo Torres, Ann S. Blevins, Danielle S. Bassett, Tina Eliassi-Rad
2020 arXiv   pre-print
observed data (i.e. graphs, simplicial complexes, and hypergraphs), and relevant computational methods for each formalism.  ...  We hope this work can serve as an opportunity of reflection for experienced complexity scientists, as well as an introductory resource for new researchers.  ...  As with the previous dependency types, failure to account for a spatial dependency can greatly bias our models and results. Consider an outbreak of a contagious disease.  ... 
arXiv:2006.02870v1 fatcat:6rp6iechmbdzdpdolgprr2vh7e

Topological Schemas of Cognitive Maps and Spatial Learning

Andrey Babichev, Sen Cheng, Yuri A. Dabaghian
2016 Frontiers in Computational Neuroscience  
In this paper we suggest computational frameworks for integrating the spiking signals of individual cells into a spatial map, which we call schemas.  ...  Spatial navigation in mammals is based on building a mental representation of their environment-a cognitive map.  ...  Spatial Learning A key property of our approach, crucial for modeling spatial learning, is that schemas are dynamic objects.  ... 
doi:10.3389/fncom.2016.00018 pmid:27014045 pmcid:PMC4781836 fatcat:vbwipledyvdsvkk635dcspruta

Separating Topological Noise from Features using Persistent Entropy [article]

Nieves Atienza, Rocio Gonzalez-Diaz, Matteo Rucco
2016 arXiv   pre-print
In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.  ...  This can be obtained constructing a filtration of the simplicial complex which is a collection of simplicial complexes {K(t)|t ∈ R} of K such that K(t) ⊂ K(s) for t < s and there exists tmax ∈ R such that  ...  This structure satisfies the definition of a simplicial complex.  ... 
arXiv:1605.02885v1 fatcat:nak6tlw4mvho7pkonukqcysqyy

Ozone

Xin Wei Sha, Michael Fortin, Navid Navab, Timothy Sutton
2010 Proceedings of the international conference on Multimedia - MM '10  
A potential-energy engine evolves superposed states over simplicial complexes modeling the topological space of metaphorical states.  ...  This paper describes Ozone, a new media choreography system based on layered, continuous physical models, designed for building a diverse range of interactive spaces that coordinate arbitrary streams of  ...  The trajectory followed by a player's state is a path in the simplicial complex underlying the model.  ... 
doi:10.1145/1873951.1874221 dblp:conf/mm/WeiFNS10 fatcat:vosim4ozh5brxe57je4gacgsfa

Logic and the 2-Simplicial Transformer [article]

James Clift, Dmitry Doryn, Daniel Murfet, James Wallbridge
2019 arXiv   pre-print
We show that this architecture is a useful inductive bias for logical reasoning in the context of deep reinforcement learning.  ...  We introduce the 2-simplicial Transformer, an extension of the Transformer which includes a form of higher-dimensional attention generalising the dot-product attention, and uses this attention to update  ...  However, since the simplicial agent has a more complex model, each environment step takes longer to execute and the gradient descent steps are slower.  ... 
arXiv:1909.00668v1 fatcat:2heqijpptfc5nfas3rjnzz73sm

Community structures in simplicial complexes: an application to wildlife corridor designing in Central India – Eastern Ghats landscape complex, India [article]

Saurabh Shanu, Shashankaditya Upadhyay, Arijit Roy, Raghunandan Chundawat, Sudeepto Bhattacharya
2020 arXiv   pre-print
a network of wildlife corridors for Tigers (Panthera Tigris Tigris) in Central India Eastern Ghats landscape complex.  ...  A simplicial complex network based on the network centralities calculated in the landscape matrix presents a tiger corridor network in the landscape complex that is proposed to correspond better to reality  ...  Hence, to model such complex systems we adopt a perspective using the theory of complex networks for modelling.  ... 
arXiv:2008.02537v1 fatcat:7fd27tlozrcjrjsz5ftcbn2jmu

From topological analyses to functional modeling: the case of hippocampus [article]

Yuri Dabaghian
2019 arXiv   pre-print
The resulting functional model provides a unifying framework for integrating spiking information at different timescales and understanding the course of spatial learning at different levels of spatiotemporal  ...  In particular, the model allows quantifying contributions of various physiological phenomena---brain waves, synaptic strengths, synaptic architectures, etc., into spatial cognition.  ...  A model of spatial learning.  ... 
arXiv:1909.07536v1 fatcat:oktdmg5be5atxeyu52g5ewxeyi

A formal definition of binary topological relationships [chapter]

Max J. Egenhofer
1989 Lecture Notes in Computer Science  
A model of topological relations is presented which is based upon fundamental concepts of algebraic topology in combination with set theory.  ...  Spatial query languages, for example, must offer terms for spatial relationships; spatial database management systems need algorithms to determine relationships.  ...  The spatial objects considered within the context of this paper are restricted to the following subset: All objects are represented as simplicial complexes, a model for spatial objects which will be discussed  ... 
doi:10.1007/3-540-51295-0_148 fatcat:sauay7ihwnb4bnx3dc6rftheri

A Compact Topological DBMS Data Structure For 3D Topography [chapter]

Friso Penninga, Peter van Oosterom
2007 Lecture Notes in Geoinformation and Cartography  
Based on earlier results (i.e. the full decomposition of space, the use of a TEN structure and applying Poincaré simplicial homology as mathematical foundation) a simplicial complex-based TEN structure  ...  The objective is to develop a data structure that is capable of handling large data volumes and offers support for querying, analysis and validation.  ...  As can be seen in Figure 8 , the Euler-Poincaré formula holds for all simplicial complexes, including simplicial complexes that consist of simplexes of different dimensions.  ... 
doi:10.1007/978-3-540-72385-1_27 fatcat:3tswnfta4fhytfy7w36aufm2tq
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