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A Certified Module to Study Digital Images with the Kenzo System [chapter]

Jónathan Heras, Vico Pascual, Julio Rubio
2012 Lecture Notes in Computer Science  
The proof is carried out using ACL2, a system for proving properties of programs written in (a subset of) Common Lisp.  ...  The description of the main function in charge of this task is shown here: simplicial-complex-generator ls: From a list of simplexes, ls, this function generates the associated simplicial complex, that  ...  For each p we obtain two triangles which are two facets of the simplicial complex associated with I.  ... 
doi:10.1007/978-3-642-27549-4_15 fatcat:ynkiskuvrvapne5ogt72torv2e

Knowledge and simplicial complexes [article]

Hans van Ditmarsch, Eric Goubault, Jeremy Ledent, Sergio Rajsbaum
2020 arXiv   pre-print
Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in  ...  We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in prior work by Goubault and others.  ...  In this work we further explore and survey what the logic of knowledge can contribute to the description of simplicial complexes and their dynamic evolution.  ... 
arXiv:2002.08863v1 fatcat:h2wtjfp4orc4fnbdjbni2ske5a

Page 4406 of Mathematical Reviews Vol. , Issue 90H [page]

1990 Mathematical Reviews  
Formulas relating HW;(p) and SW;(p) are known (and mentioned in the paper) for i = 1, 2, 3, but not for i = 4. For the proofs the author uses simplicial algebraic topology in Grothendieck sites.  ...  Second, the author states that his complex I(r) is the same as the complex that Lichtenbaum and Milne would call I'(r)[—2r].  ... 

Higher Čech Theory

Tibor Beke
2004 K-theory  
We introduce a notion of 'cover of level n' for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram  ...  Our approach is purely simplicial and combinatorial. Mathematics Subject Classifications (2000): 18G55, 55N30  ...  Duskin for several email messages and generous simplicial guidance; this work could not have started without his definitions and insights.  ... 
doi:10.1007/s10977-004-0840-0 fatcat:47pb6w6tlvghdbenrdpnqwqndi

Formalization of a normalization theorem in simplicial topology

Laureano Lambán, Francisco-Jesús Martín–Mateos, Julio Rubio, José-Luis Ruiz–Reina
2012 Annals of Mathematics and Artificial Intelligence  
simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex.  ...  In this paper we present a complete formalization of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a  ...  Acknowledgements We thank the anonymous referees for their careful revision and useful feedback.  ... 
doi:10.1007/s10472-011-9274-6 fatcat:rk4mq532xnbudfetu5rdbr2qau

Geometric Model Checking of Continuous Space [article]

Nick Bezhanishvili and Vincenzo Ciancia and David Gabelaia and Gianluca Grilletti and Diego Latella and Mieke Massink
2022 arXiv   pre-print
The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being  ...  Topological Spatial Model Checking is a recent paradigm where model checking techniques are developed for the topological interpretation of Modal Logic.  ...  We use the terminology cells in this way for the purposes of this paper; there is no relation between such cells and the so-called cell complexes of algebraic topology.  ... 
arXiv:2105.06194v2 fatcat:w6sqisekjndgrnhrsydoaokftq

Incidence Simplicial Matrices Formalized in Coq/SSReflect [chapter]

Jónathan Heras, María Poza, Maxime Dénès, Laurence Rideau
2011 Lecture Notes in Computer Science  
Simplicial complexes are at the heart of Computational Algebraic Topology, since they give a concrete, combinatorial description of otherwise rather abstract objects which makes many important topological  ...  In this paper we present a formalization in the COQ theorem prover of simplicial complexes and their incidence matrices as well as the main theorem that gives meaning to the definition of homology groups  ...  Simplicial complexes provide a purely combinatorial description of topological spaces which admit a triangulation.  ... 
doi:10.1007/978-3-642-22673-1_3 fatcat:2xjkvd3frzc3fjk3phtlwbmwwq

Cosimplicial C-infinity rings and the de Rham complex of Euclidean space [article]

Herman Stel
2013 arXiv   pre-print
We also analyse the notion of R-module (following Quillen) for a (co-)simplicial C-infinity ring R.  ...  We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring.  ...  I give explanation of this terminology in the corresponding section.  ... 
arXiv:1310.7407v1 fatcat:gckdfjfkfrckbhzwnkezdfwns4

Stacky fans and tropical moduli in polymake [article]

Dominic Bunnett, Michael Joswig, Julian Pfeifle
2021 arXiv   pre-print
For instance, the next logical step would be to analyse the cells of type (020) and (111) in M pl 3 . D.Bunnett and M.  ...  For instance, the second barycentric subdivision of a finite symmetric ∆-complex yields a finite simplicial complex on which the group action is regular [4, Definition II.1.2 and Exercise III.5], which  ... 
arXiv:2101.07316v2 fatcat:zpv6l7ju5fc75c75laqtlk2kn4

A simplicial complex model of dynamic epistemic logic for fault-tolerant distributed computing [article]

Eric Goubault, Sergio Rajsbaum
2017 arXiv   pre-print
Then we use dynamic epistemic logic to study how the simplicial complex epistemic model changes after the agents communicate with each other.  ...  For each state of the Kripke model there is a facet in the complex, with one vertex per agent.  ...  Sergio Rajsbaum would like to acknowledge the Ecole Polytechnique for financial support through the 2016-2017 Visiting Scholar Program.  ... 
arXiv:1703.11005v2 fatcat:2j5mgeumrrgq5hqqh3ujovuzwu

Enriched categories and models for spaces of evolving states

Timothy Porter
2008 Theoretical Computer Science  
We model both the 'space' and the directed paths by a simplicially enriched category, and show how to adapt some classical constructions to produce a differential graded enrichment.  ...  That structure is, however, non-commutative and computational techniques for handling it are more complex than for, say, simplicial vector spaces or chain complexes.  ...  The terminology 'chain complex (of vector spaces)' is usually considered to be synonymous with 'non-negatively graded dgvs', whilst a cochain complex is a 'non-positively graded dgvs'.  ... 
doi:10.1016/j.tcs.2008.06.029 fatcat:toiz5ntwcrgo5hnsym6zyqv4du

Shellability is NP-Complete

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner, Marc Herbstritt
2018 International Symposium on Computational Geometry  
For d ≥ 3, the problem is already NP-hard for pure d-dimensional simplicial complexes that are cones (hence contractible).  ...  We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete.  ...  This is a purely combinatorial description of a simplicial complex and a natural input model for computational questions. 9 A (finite) geometric simplicial complex is a finite collection K of geometric  ... 
doi:10.4230/lipics.socg.2018.41 dblp:conf/compgeom/GoaocPPT018 fatcat:do6soxkr2fdp3c7tlruljp6m64

Sheafifiable homotopy model categories

TIBOR BEKE
2000 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
If a Quillen model category can be specified using a certain logical syntax (intuitively, "is algebraic/combinatorial enough"), so that it can be defined in any category of sheaves, then the satisfaction  ...  This suggests that one borrows the logical description of weak equivalences and one of the classes of cofibrations and fibrations.  ...  It is quite ironic to observe that while the simple (inductive) injective replacement arguments break down for unbounded complexes, the proof strategy of 3.13 can only apply to a category of complexes  ... 
doi:10.1017/s0305004100004722 fatcat:2x5gqilwpfbnzbptncdflwstt4

Computability of homotopy groups of nilpotent complexes

Kathryn Weld
1987 Journal of Pure and Applied Algebra  
In Section 6 we obtain the rewards of the application of logic. For a nilpotent complex, the Postnikov tower does not directly yield a presentation of x,X.  ...  A simplicial set X is called locally finite if the set of n-simplices X" is finite for all n 2 0. Let Y be a Kan complex and let Hom,(l, Y) be the simplicial path complex.  ...  Each complex Yn,r,q will be a subcomplex of Y,,i, and for q> n, a q-deformation retract of Yn,j.  ... 
doi:10.1016/0022-4049(87)90106-x fatcat:ej3rwx35bzbdnmtciiw2epbfim

Freeness conditions for quasi 3-crossed modules and complexes of using simplicial algebras with CW−bases

Ali Mutlu, Berrin Mutlu
2013 Mathematical Sciences  
Quasi 3-crossed complexes are introduced and similar freeness results are given for these are discussed.  ...  Using free simplicial algebras with given CW−basis, it is shown how to construct a free or totally free quasi 3-crossed module on suitable construction data.  ...  The following notation and terminology is due to [6] . We give an explicit description of the construction of a totally free 2-crossed module.  ... 
doi:10.1186/2251-7456-7-35 fatcat:cazwzgr76zbrlnyvvmwalqvpnu
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