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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. Dyn. Syst.
, 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 th ..."
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Cited by 71 (20 self)
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idea is to replace 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
ON LOCAL INTEGRABILITY CONDITIONS OF JET GROUPOIDS
, 708
"... Abstract. A Jet groupoid Rq over a manifold X is a special Lie groupoid consisting of qjets of local diffeomorphisms X → X. As a subbundle of Jq(X× X), a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept of formal integrability. ..."
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. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, modelled as a section ω of a natural bundle F. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects. 1.
A GENERALIZATION OF COXETER GROUPS, ROOT SYSTEMS, AND MATSUMOTO’S THEOREM
, 2006
"... Abstract. The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context ..."
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Cited by 3 (2 self)
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Abstract. The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context
BochnerKähler metrics
"... Abstract. A Kähler metric is said to be BochnerKähler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain welldefined sense, the space of BochnerKähler metrics in ..."
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Cited by 25 (1 self)
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projective space carries a BochnerKähler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a BochnerKähler metric whose associated momentum mapping has the orbits of its symmetry pseudogroupoid as fibers.
On biological homochirality
, 2009
"... Generalizing Landau’s spontaneous symmetry breaking arguments using the standard groupoid approach to stereochemistry allows reconsideration of the origin of biological homochirality. On Earth, limited metabolic free energy density may have served as a low temperatureanalog to ‘freeze ’ the syst ..."
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Generalizing Landau’s spontaneous symmetry breaking arguments using the standard groupoid approach to stereochemistry allows reconsideration of the origin of biological homochirality. On Earth, limited metabolic free energy density may have served as a low temperatureanalog to ‘freeze
A formal approach to the molecular fuzzy lockandkey
"... Abstract The fuzzy lockandkey (FLK) powers a vast array of sophisticated logic gates at interand intracellular levels. We invoke representations of groupoid tiling wreath products analogous to the study of nonrigid molecules or of related fuzzy symmetry extensions to build a Morse Function th ..."
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Abstract The fuzzy lockandkey (FLK) powers a vast array of sophisticated logic gates at interand intracellular levels. We invoke representations of groupoid tiling wreath products analogous to the study of nonrigid molecules or of related fuzzy symmetry extensions to build a Morse Function