We study the one-dimensional asymmetric simple exclusion process (ASEP) with open boundary conditions. Particles are injected and ejected at both boundaries. It is clarified that the steady state of the model is intimately related to the AskeyWilson polynomials. The partition function and the n-point functions are obtained in the integral form with four boundary parameters. The thermodynamic current is evaluated to confirm the conjectured phase diagram.

High-speed motion of a mechanical arm is necessary to speed up a job done by the arm. In high speed, however, the desired trajectory of motion of the arm cannot be obtained simply by applying the trajectory function to the servo system as the reference function because the time lag in the servo system is not negligible.A solution to this problem is to apply dynamically compensating computed torques to the servo system. By this method, however, for increasing the accuracy of the mathematical model of the arm necessary to compute the compensating torques, a very large effort would be required.To avoid this difficulty, an alternative method of correcting the reference function by trial will be useful. Repeating a proper process of trial and correction, the reference function which realizes the desired pattern of trajectory may be obtained.In this paper, correcting algorithm of a reference function for this method is investigated theoretically from the standpoint of stability or convergency of the process of trial and correction, and a stable correcting algorithm is obtained. Through the experiment using a mechanical arm of six degrees of freedom controlled by a digital computer, it is confirmed that the process of trial and correction by this algorithm is stable and the response of the servo system converges rapidly to the desired pattern of trajectory.

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We study dark soliton solutions of a multi-component Gross-Pitaevskii equation for hyperfine spin F = 1 spinor Bose-Einstein condensate. The interactions are supposed to be inter-atomic repulsive and anti-ferromagnetic ones of equal magnitude. The solutions are obtained from those of an integrable 2 × 2 matrix nonlinear Schrödinger equation with nonvanishing boundary conditions. We investigate the one-soliton and two-soliton solutions in detail. One-soliton is classified into two kinds. The ferromagnetic state has wavefunctions of domain-wall shape and its total spin is nonzero. The polar state provides a hole soliton and its total spin is zero. These two states are selected by choosing the type of the boundary conditions. In two-soliton collisions, we observe the spin-mixing or spin-transfer. It is found that, as "magnetic" carriers, solitons in the ferromagnetic state are operative for the spin-mixing while those in the polar are passive.