243 research outputs found

    Torsors and ternary Moufang loops arising in projective geometry

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    We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.Comment: 15 p., 5 figure

    The splitting process in free probability theory

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    Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work

    A discussion on the origin of quantum probabilities

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    We study the origin of quantum probabilities as arising from non-boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).Comment: Improved versio

    Remarks on the GNS Representation and the Geometry of Quantum States

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    It is shown how to introduce a geometric description of the algebraic approach to the non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but also Hermitian realization of a `quantum Poisson algebra'. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Hamiltonian systems.Comment: 20 page

    Commuting UU-operators in Jordan algebras

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    The Whitehead group of the Novikov ring

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    The Bass-Heller-Swan-Farrell-Hsiang-Siebenmann decomposition of the Whitehead group K1(Aρ[z,z1])K_1(A_{\rho}[z,z^{-1}]) of a twisted Laurent polynomial extension Aρ[z,z1]A_{\rho}[z,z^{-1}] of a ring AA is generalized to a decomposition of the Whitehead group K1(Aρ((z)))K_1(A_{\rho}((z))) of a twisted Novikov ring of power series Aρ((z))=Aρ[[z]][z1]A_{\rho}((z))=A_{\rho}[[z]][z^{-1}]. The decomposition involves a summand W1(A,ρ)W_1(A,\rho) which is an abelian quotient of the multiplicative group W(A,ρ)W(A,\rho) of Witt vectors 1+a1z+a2z2+...Aρ[[z]]1+a_1z+a_2z^2+... \in A_{\rho}[[z]]. An example is constructed to show that in general the natural surjection W(A,ρ)abW1(A,ρ)W(A,\rho)^{ab} \to W_1(A,\rho) is not an isomorphism.Comment: Latex file using Diagrams.tex, 36 pages. To appear in "K-theory

    Acta Cybernetica : Volume 16. Number 4.

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