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Multivariable, parameterized, and colored extensions of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
A sheaf is a set of multiple edges between two vertices. Duality properties yield results for sheafs analogous to those for chains. In particular, the sheaf polynomial is a “dual polynomial” for the chain polynomial. Following [1074], it may be defined as Sh(G;ω,w)=(−1)v(G)−1∑A⊆E∏e∈A(we−1)(1−ω)r(E)−r(A).
Operators in the Cowen-Douglas Class and Related Topics
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Proof The sheaf Iℳ(Ω) is generated by the set of functions {f : f ∈ ℳ}. Let Ijℳ(Ω) be the subsheaf generated by the set of functions J={f1,…,fl}⊆ℳ⊆O(Ω).
Category-theoretic approaches to semantic technologies
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
Spivak et al. (2016), draw on sheaf theory in their categorical approach to dynamic systems, which allows them to account for both continuous- and discrete-time through the use of functors in a 2-category framework. This approach also accounts for synchronized continuous time, in which each moment is assigned a phase θ ∈ [0, 1).
Constructing condensed memories in functorial time
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023
A sheaf is, in other words, a presheaf in which ‘nothing is missing’ – all of the information that can be consistently assigned to the points in has been assigned. In this regard, a sheaf on a topological space packs all the local data attached to open sets of . As such, sheaves are tools used to transfer between local and global data: here data are global in the sense that some data are assigned to every open set of , and local in the sense that data assigned on every open set can be restricted, in a compatible way, to data assigned on coverings of that open set, such that all of these data assignments are equivalent.
A set-theoretic proof of the representation of MV-algebras by sheaves
Published in Journal of Applied Non-Classical Logics, 2022
Alejandro Estrada, Yuri A. Poveda
MV-algebras serve as the algebraic semantics of Łukasiewicz infinite-valued propositional logic, analogous to the role that the class of Boolean algebras plays in characterising classical propositional logic (Chang, 1959). MV-algebras are categorically equivalent to unital lattice-ordered abelian groups (Dubuc & Poveda, 2015; Mundici, 1986). In their work ‘Representation theory of MV-algebras’ (Dubuc & Poveda, 2010), the authors proved that every MV-algebra is isomorphic to the MV-algebra of all global sections of a sheaf of MV-chains on a compact topological space. In this paper, which is essentially self-contained except for the use of the Chinese remainder theorem (Theorem 2.4), we provide a simple proof of this result using basic ideas from MV-algebras, avoiding knowledge of sheaf theory in our exposition.