ABSTRACT We prove that gauge fields satisfying the Yang-Mills equations are characterized by the ... more ABSTRACT We prove that gauge fields satisfying the Yang-Mills equations are characterized by the property of being harmonic functions for the Lévy Laplacian defined on the space of piecewise smooth paths. More precisely: a parallel transport can be associated with a connection satisfying the Yang-Mills equations if and only if the transport satisfies the Laplace equation for the Lévy Laplacian on the path space.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1998
The non-parametric version of Information Geometry has been developed in recent years. The first ... more The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳμ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳμ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.
ABSTRACT Si mostra che una connessione in Rn soddisfa le equazioni di Yang-Mills se e solo se il ... more ABSTRACT Si mostra che una connessione in Rn soddisfa le equazioni di Yang-Mills se e solo se il trasporto parallelo ad essa associato è una funzione armonica rispetto al laplaciano di Lévy sullo spazio dei cammini su Rn. It is shown that a connection in Rn satisfies the Yang-MilIs equation if and only if the associated parallel transport is a Lévy harmonic function on the space of paths on Rn.
Rapp. interno/Dip. di matematica, Politecn. di Torino, Jul 6, 1998
Information Geometry is a field where one can measure the deep impact of geometry and analysis in... more Information Geometry is a field where one can measure the deep impact of geometry and analysis in statistics, information theory and related applied fields. The present contribution has the goal of showing also the impact that statistics and information theory can have in geometry and analysis. Indeed it is clear that the development of the non-parametric and non-commutative versions of Information Geometry need a massive use of mathematical instruments of infinite-dimensional analysis, geometry and operator theory. On the other ...
ABSTRACT We prove that gauge fields satisfying the Yang-Mills equations are characterized by the ... more ABSTRACT We prove that gauge fields satisfying the Yang-Mills equations are characterized by the property of being harmonic functions for the Lévy Laplacian defined on the space of piecewise smooth paths. More precisely: a parallel transport can be associated with a connection satisfying the Yang-Mills equations if and only if the transport satisfies the Laplace equation for the Lévy Laplacian on the path space.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1998
The non-parametric version of Information Geometry has been developed in recent years. The first ... more The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳμ, the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳμ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.
ABSTRACT Si mostra che una connessione in Rn soddisfa le equazioni di Yang-Mills se e solo se il ... more ABSTRACT Si mostra che una connessione in Rn soddisfa le equazioni di Yang-Mills se e solo se il trasporto parallelo ad essa associato è una funzione armonica rispetto al laplaciano di Lévy sullo spazio dei cammini su Rn. It is shown that a connection in Rn satisfies the Yang-MilIs equation if and only if the associated parallel transport is a Lévy harmonic function on the space of paths on Rn.
Rapp. interno/Dip. di matematica, Politecn. di Torino, Jul 6, 1998
Information Geometry is a field where one can measure the deep impact of geometry and analysis in... more Information Geometry is a field where one can measure the deep impact of geometry and analysis in statistics, information theory and related applied fields. The present contribution has the goal of showing also the impact that statistics and information theory can have in geometry and analysis. Indeed it is clear that the development of the non-parametric and non-commutative versions of Information Geometry need a massive use of mathematical instruments of infinite-dimensional analysis, geometry and operator theory. On the other ...
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