Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bound...Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .展开更多
The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left...The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.展开更多
Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid s...Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it describes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are confirmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as system autonomy, robotics, systems integration, and image and voice recognition.展开更多
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...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 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 groupoid-theoretical method that recursively partitions the problem and requires no heuristics or human intervention. It takes two steps. First, make explicit the fundamental causal nature of information in the physical world by encoding it as a causal set. Second, construct a functor F: C C′ on the category of causal sets that morphs causal set C into smaller causal set C′ by partitioning C into a set of invariant groupoid-theoretical blocks. Repeating the construction, there arises a sequence of progressively smaller causal sets C, C′, C″, … The sequence defines a fractal hierarchy of features, with the features being invariant and hence endowed with a physical meaning, and the hierarchy being scale-free and hence ensuring proper scaling at all granularities. Fractals exist in nature nearly everywhere and at all physical scales, and invariants have long been known to be meaningful to us. The theory is also of interest for NP-hard combinatorial problems that can be expressed as a causal set, such as the Traveling Salesman problem. The recursive groupoid partition promoted by functor F works against their combinatorial complexity and appears to allow a low-order polynomial solution. A true test of this property requires special hardware, not yet available. However, as a proof of concept, a suite of sequential, non-heuristic algorithms were developed and used to solve a real-world 120-city problem of TSP on a personal computer. The results are reported.展开更多
After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection ...After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.展开更多
文摘Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .
基金The National Natural Science Foundation of China(No.10871042)the Natural Science Foundation of Jiangsu Province(No.BK2009258)
文摘The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.
文摘Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it describes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are confirmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as system autonomy, robotics, systems integration, and image and voice recognition.
文摘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 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 groupoid-theoretical method that recursively partitions the problem and requires no heuristics or human intervention. It takes two steps. First, make explicit the fundamental causal nature of information in the physical world by encoding it as a causal set. Second, construct a functor F: C C′ on the category of causal sets that morphs causal set C into smaller causal set C′ by partitioning C into a set of invariant groupoid-theoretical blocks. Repeating the construction, there arises a sequence of progressively smaller causal sets C, C′, C″, … The sequence defines a fractal hierarchy of features, with the features being invariant and hence endowed with a physical meaning, and the hierarchy being scale-free and hence ensuring proper scaling at all granularities. Fractals exist in nature nearly everywhere and at all physical scales, and invariants have long been known to be meaningful to us. The theory is also of interest for NP-hard combinatorial problems that can be expressed as a causal set, such as the Traveling Salesman problem. The recursive groupoid partition promoted by functor F works against their combinatorial complexity and appears to allow a low-order polynomial solution. A true test of this property requires special hardware, not yet available. However, as a proof of concept, a suite of sequential, non-heuristic algorithms were developed and used to solve a real-world 120-city problem of TSP on a personal computer. The results are reported.
基金National Key Basic Research Project of China under Grant No.2004CB318000National Natural Science Foundation of China under Grant Nos.10375038 and 90403018
文摘After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid,we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection betweengroupoids variation and the methods of the first and second discrete variational principles.
