Tomoaki Kawaguchi

Relativistic geometrical optics

We investigate the gravitational and electromagnetic fields on the generalized Lagrange space endowed with the metricgij(x, y) = γij(x) + {1 + 1/n2(x, y)}yiyj. The generalized Lagrange spacesMm do not reduce to Lagrange spaces. Consequently, they cannot be studied by methods of symplectic geometry. The restriction of the spacesMm to a sectionSν(M) leads to the Maxwell equations and Einstein equations for the electromagnetic and gravitational fields in dispersive media with the refractive indexn(x, V) endowed with the Synge metric. Whenn(x, V) = 1 we have the classical Einstein equations. If 1/n2=1−1/c2 (c being the light velocity), we get results given previously by the authors. The present paper is a detailed version of a work in preparation.

On the relation between the statistical divergence and geodesic distance

Abstract Various order α Kullback-Leiblers divergences are given based on the concept of order α relative entropy of Renyi and on that of Havrda-Charvat. The properties of their divergences are studied by using geodesic distance from the viewpoint of parameter space.

On a unification of divergences by means of information amounts

Abstract The notion of discriminating amount associated with the notion of a divergence is given. It is based on the concept of Renyis order α relative entropy and Burbea-Raos K α -divergence measure. Moreover, the K α -divergence measure and some order α Kullback-Leiblers divergence measures are unified by means of (α,β) discriminating amounts. Local properties of these amounts are studied on the statistical parameter space.

On a Riemannian approach to the order α relative entropy

To generalize Shannon’s entropy, A. Renyi [1] introduced order a entropy in 1960. Also in 1967, J. Havrda and F. Charvat [2] gave another generalization of Shannon’s entropy. Moreover A. Renyi [1] generalized the mutual information of random variables to order α relative entropy. Also N. Muraki and T. Kawaguchi [3] exhibited order α relative entropy based on Havrda-Charvat’s order α entropy in 1987 in the same way as an extension of Renyi’s order α relative entropy. These relative entropies are discriminating amount of difference between two distinct probability distributions, it is well known as the divergence in statistics. I. Csiszar [4] defined f-divergence by the generalization of Kullback-Leibler’s I-divergence [5] making use of an arbitrary convex function f defined on (0, ∞). On the other hands, J. Burbea and C.R. Rao made Kα-divergence by substituting Havrda-Charvat’s entropy in the Φ-entropy function which was defined on stochastic spaces by them [6]. Also we defined other divergences from a different standpoint [7]. These divergences do not satisfy the axiom of distance.

The electromagnetic field in the higher order relativistic geometrical optics

Continuing the paper[3] we are to study here the higher order relativistic geometrical optics, and describe the theory of the electromagnetic field. In the first two sections there are some old and new results concerning the canonical metrical N-connection CT(JV) of the generalized higher order Lagrange space GL(k)n endowed with the fundamental metric tensor (1.2). The coefficients of CT(N) are given in (2.4).

On the parameter space derived from the joint probability density functions and the property of its scalar curvature

Abstract Extended parameter spaces were introduced by Kawaguchi et al. (1992) to define the geometrical distance between two probability distributions having different function forms. A statistical interpretation of the extended parameter spaces was introduced. In this paper, the property of the scalar curvature of extended parameter spaces is given.

Generalized Lagrange metric derived from a Finsler function

Abstract One investigates the geometrical properties, Einstein equations and the laws of conservation of the energy-momentum tensors in the generalized Lagrange spaces endowed with the fundamental tensor field g ij ( x , y ) = γ ij ( x , y )+(1/ c 2 ) y i y j derived from a Finslerian function.