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Groupoids and Conditional Symmetry
"... We introduce groupoids – generalisations of groups in which not all pairs of elements may be multiplied, or, equivalently, categories in which all morphisms are invertible – as the appropriate algebraic structures for dealing with conditional symmetries in Constraint Satisfaction Problems (CSPs). We ..."
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Cited by 2 (0 self)
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We introduce groupoids – generalisations of groups in which not all pairs of elements may be multiplied, or, equivalently, categories in which all morphisms are invertible – as the appropriate algebraic structures for dealing with conditional symmetries in Constraint Satisfaction Problems (CSPs
Symmetry groupoids and patterns of synchrony in coupled cell networks
 SIAM J. Appl. Dynam. Sys
, 2003
"... Abstract. A coupled cell system is a network of dynamical systems, or “cells, ” coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells t ..."
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Cited by 70 (20 self)
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the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings
Groupoids: unifying internal and external symmetry. A tour through some examples
 Notices Amer. Math. Soc
, 1996
"... Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program ..."
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Cited by 104 (4 self)
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Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program
Galois symmetries of fundamental groupoids and noncommutative geometry
 Duke Math. Journal
"... ..."
Groupoids, Hypergraphs, and Symmetries in Finite Models
"... We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. The universal algebraic and combinatorial properties of groupoids are abstracted form the composition behaviour of partial injections and support a ve ..."
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Cited by 3 (3 self)
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We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. The universal algebraic and combinatorial properties of groupoids are abstracted form the composition behaviour of partial injections and support a
Causal Groupoid Symmetries and Big Data
"... The big problem of Big Data is the lack of a machine learning process that scales and finds meaningful features. Humans fill in for the insufficient automation, but the complexity of the tasks outpaces the human mind’s capacity to comprehend the data. Heuristic partition methods may help but still ..."
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need humans to adjust the parameters. The same problems exist in many other disciplines and technologies that depend on Big Data or Machine Learning. Proposed here is a fractal groupoidtheoretical method that recursively partitions the problem and requires no heuristics or human intervention
On Groupoids and Hypergraphs
, 2012
"... We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (subgroupoid), and only counts transitions between colour classes (cosets). These groupoi ..."
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Cited by 3 (3 self)
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the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing
The Monodromy Groupoid Of A Lie Groupoid
, 1996
"... this paper can be summarised roughly as follows. Recall that the stars of a groupoid are the fibres of the source map. ..."
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Cited by 10 (6 self)
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this paper can be summarised roughly as follows. Recall that the stars of a groupoid are the fibres of the source map.
Nonlinear dynamics of networks: the groupoid formalism
 Bull. Amer. Math. Soc
, 2006
"... Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which ..."
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Cited by 76 (13 self)
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be more flexible. A recent generalization of the grouptheoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group
Results 1  10
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3,386