Four panelists gave their reports. Three of them are complied here. Additional comments are included in the report by the moderator Sugimoto.

We show the effectiveness of the KS regularization and the method of variation of parameter for integrating of the perturbed two body problems.

We describe results of N-body simulations of star clusters moving on elliptic orbits. Our aim is to follow the evolution of these clusters and study how their dissolution times depend on particle number and eccentricity.

We find that the scaling of the lifetimes is nearly independent of the eccentricity, despite the fact that the amount of tidal shocking changes with eccentricity. In addition, no leveling off of the lifetimes at the high-N end, which would indicate the influence of tidal shocking, is seen.

In order to study the capabilities of SPH in self-gravitating hydrodynamical problems, we have performed a series of three-dimensional numerical simulations of the collapse of a rotating homogeneous cloud, varying the number of particles and the artificial viscosity prescription.

Halo initial mass functions (IMFs) heavily biased toward white dwarf (WD) precursors (∼ 1 - 8M⊙) have been suggested as a suitable mechanism for explaining MACHO events. However, by simple chemical evolution argument, Gibson & Mould (1997; GM97) pointed out that such WD-heavy IMFs (wdIMF) cause the overproduction of carbon and nitrogen. We re-examine this problem using numerical simulations.

We present a new statistical algorithm of Chemical Evolution implemented into the Tree-SPH code developed by Lia & Carraro (2000).

In this article we report the results of our numerical investigation of warped accretion discs in binary stellar systems. We perform complete 3-D hydrodynamic simulations of binary discs. The disc is rendered unstable to the warp mode under the action of the magnetic field of the companion star in the binary. The disc thus warped is noted to undergo retrograde precession with a precession period just slightly less than the binary period. This small difference in periods can explain the phenomenon of negative superhumps observed in a number of binaries. Besides the modal analysis based on Fourier transforms, warps were also studied by a simple and robust technique that we developed; this is based on an analysis of the azimuthal distributions of particles that lie above and below the mid-plane of the disc.

We present the formation and evolution of a structure in particles obtained from a local N-body simulation of a dense planetary ring like Saturn's ring. Our simulations show in a particle system the spontaneous formation of a spatial structure like wakes, clumps, and a structure which could be induced by the viscous overstability. Such a formation depends on parameters characterizing a ring system: the wake is likely to form in Saturn's ring and the existence of the wake is consistent with observations. The viscous overstability would be a good candidate for the explanation of subring structures in the ring.

A fast algorithm for studying one-dimensional systems that are Lagrangian integrable between collisions is presented. The algorithm is based on an event-driven scheme, and uses a heap ordered set of predicted future events. As an application, we discuss the case of an expanding one-dimensional Universe.

We investigate recent suggestions that substructure in cold dark matter (CDM) halos has potentially destructive effects on galactic disks. N-body simulations of disk/bulge models of the Milky Way, embedded in a dark matter halo with substructure similar to that found in cosmological simulations, show that tides from substructure halos play only a minor role in the dynamical heating of the stellar disk. This suggests that substructure might not preclude CDM halos from being acceptable hosts of thin stellar disks.

The first results of Monte Carlo simulations of the evolution of tidally limited, multi-mass million star clusters with stellar evolution and dynamically formed binaries are presented.

We are using SPH to model the evolution of triaxial molecular cores to investigate the fragmentation of these cores and the possible production of multiple systems. Molecular cores appear to be triaxial in shape and this may have a significant effect on their evolution as they collapse to form filament-like structures which can then fragment into multiple 'protostars'.

We study the differential energy distribution of dark matter halos, carrying out cosmological N-body simulation. From our simulation, we give an analytical formula of the differential energy distribution of dark matter in the halos. Density distribution from the analytical formula is consistent with the Navarro, Frenk, and White (NFW) profile. We find that a parameter in our analytical formula of differential energy distribution is related with the slope of inner cusp of dark halo. We discuss physical reason of form of the analytical formula.

We critically examine the reliability of clustering below the mean separation length of particles in cosmological high-resolution N-body simulations. The particle discreteness effect imposes two fundamental limitations on those scales; the lack of the initial fluctuation power and the finite mass resolution. We address this problem applying the dark matter halo approach which makes us possible to examine separately how those two limitations affect the simulation results at later epochs. We find that the reliability of the simulations on scale below the mean particle separation is primarily determined by the mass of particles. Since the small scale matter clustering is dominated by the matter in a virialized halo, it is naturally expected that the small scale clustering becomes unreliable when the characteristic nonlinear mass for gravitationally collapsed objects at that time becomes smaller than the particle mass. This is confirmed by a detailed comparison with three major cosmological simulations. The characteristic mass, MNL(z), is approximately given by the condition σ(MNL, z) = 1. This mass scale should be much larger than the simulation particle mass mpart so as to reproduce the proper amplitude of the small scale clustering. This requirement can be translated to the criterion for the critical redshift zcrit when simulations reasonably resolve the clustering on scales below the mean separation: MNL(zcrit) = nhalompart, where we introduce a fudge factor, nhalo, of around ten. We conclude that at z > zcrit high-resolution N-body simulations are not reliable on scales below the mean particle separation. Although it is not obvious if the opposite is true at z < zcrit, our detailed comparison suggests that it is indeed the case at least as far as the two-point correlation functions of the matter are concerned. The detail discussions are presented in Hamana, Yoshida & Suto (2001).

In most of previous theoretical and numerical works on the orbital evolution of satellite galaxies, the classical Chandrasekhar's formula has been adopted. However, a recent direct N-body result “(van den Bosh et al. 1999)” demonstrated that the orbital evolution is rather different from what is predicted from the Chandrasekhar's formula. The reason for this discrepancy, however, was not clear.

The evolution of substructure in dark matter halos is investigated in a series of simulations of N = 105 satellite halos on elliptical orbits in the gravitational potential of a much larger host system. The bound mass of the satellite decreases with each pericentric passage and most of the mass is lost from the outer region of the satellite halo. We parameterize the change in its density profile by modifying the initial profile by a factor proportional to (1 + r-3), which results in reasonable fits to the mass profiles of tidally stripped subhalos.

We are showing first results from simulations studying the effect of Brownian motion of the central mass on a cusp. N-body simulations of cuspy stellar systems around supermassive black holes represent a very challenging computational task. The wide range of dynamical timescales combined with large particle numbers required here leads us to the implementation of a systolic force calculation into a Hermite direct force block time-step scheme. With this new algorithm, we are able to handle particle numbers as large as 900000 on the parallel computers accessible to us. Due to their extraordinary parallelism (up to 98% and more) we suggest to use the described schemes on a computer cluster where each computing node is connected to a GRAPE board.

In N-body problem, it is necessary to regularize forces between close encountering particles to conserve their total energy. Without regularization, these particles has rapid change of their energy. We can, therefore, select close encountering particles looking at their energy and apply regularization to them. We discuss application of this scheme to spherically collapsing case.

It has long been known that some spiral galaxies have a large-scale lopsided structure (e.g., Baldwin, Lynden-Bell, & Sancisi 1980). The frequency of the lopsidedness in disk galaxies reaches to half of H i disks (e.g., Richter & Sancisi 1994) and one third of stellar disks (e.g., Zaritsky & Rix 1997). The fraction of lopsided disks is large, nevertheless their origin is not understood well. Although some theoretical models are proposed to explain the lopsidedness, dynamical friction between lopsided disks and spherical halos has not been taken into account. Then, in this contribution, we have examined the effect of dynamical friction on lopsided disks. To avoid discreteness noise due to small number of particles on the estimate of dynamical friction, we have used the matrix method.

Smoothed particle hydrodynamics (SPH) is one of the widely used methods to calculate the various astrophysical fluid dynamics. However, standard SPH cannot accurately describe the long-term evolution of shear flows: The large density error emerges within a dynamical timescale, and the amplitude of the error becomes larger than the value of density itself (Δρ ≳ ρ), when we take the mean separation of the particles as the smoothing length. The origin of error is due to the inaccurate description of the continuity equation in the standard SPH formalism. To ensure the local mass conservation property, we have reformulated SPH, in which we distinguish the particle velocity and the fluid velocity for the updation of the particle positions. We find that the present modification provides an accurate description of the density evolution in SPH.