Structure of super-families

At present the study of nuclear interactions induced by cosmic rays is the unique source of information on the nuclear interactions in the energy region above 10 to the 15th power eV. The phenomena in this energy region are observed by air shower arrays or emulsion chambers installed at high mountain. An emulsion chamber is the pile of lead plates and photo-sensitive layers (nuclear emulsion plates and/or X-ray films) used to detect electron showers. High spatial resolution of photographic material used in the emulsion chamber enables the observation of the phenomena in detail, and recent experiments of emulsion chamber with large area are being carried out at high mountain altitudes by several groups in the world

Discrete Hamilton-Jacobi Theory

We develop a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some well-known results, such as the Bellman equation (discrete-time HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton-Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.Comment: 26 pages, 2 figure

Extremely high energy hadron and gamma-ray families(3). Core structure of the halo of superfamily

The study of the core structure seen in the halo of Mini-Andromeda 3(M.A.3), which was observed in the Chacaltaya emulsion chamber, is presented. On the assumption that lateral distribution of darkness of the core is exponential type, i.e., D=D0exp(-R/r0), subtraction of D from halo darkness is performed until the cores are gone. The same quantity on cores obtained by this way are summarized. The analysis is preliminary and is going to be developed

Symmetry Reduction of Optimal Control Systems and Principal Connections

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points.Comment: Significant revisions made; simpler estimates; very mild strengthening of the hypotheses on Theorem 1.2 to get much stronger conclusions than in ver.1. To appear in Math. An