We have evaluated by numerical simulation the average size R(K) of random polygons of fixed knot topology K=,3(1),3(1) musical sharp 4(1), and we have confirmed the scaling law R(2)(K) approximately N(2nu(K)) for the number N of polygonal nodes in a wide range; N=100-2200. The best fit gives 2nu(K) approximately 1.11-1.16 with good fitting curves in the whole range of N. The estimate of 2nu(K) is consistent with the exponent of self-avoiding polygons. In a limited range of N (N greater, similar 600), however, we have another fit with 2nu(K) approximately 1.01-1.07, which is close to the exponent of random polygons.

This paper presents a taxonomy of errors in chat-oriented dialogue systems. Compared to human-human conversations and task-oriented dialogues, little is known about the errors made in chat-oriented dialogue systems. Through a data collection of chat dialogues and analyses of dialogue breakdowns, we classified errors and created a taxonomy. Although the proposed taxonomy may not be complete, this paper is the first to present a taxonomy of errors in chat-oriented dialogue systems. We also highlight the difficulty in pinpointing errors in such systems.

This paper aims to find errors that lead to dialogue breakdowns in chat-oriented dialogue systems. We collected chat dialogue data, annotated them with dialogue breakdown labels, and collected comments describing the error that led to the breakdown. By mining the comments, we first identified error types. Then, we calculated the correlation between an error type and the degree of dialogue breakdown it incurred, quantifying its impact on dialogue breakdown. This is the first study to quantitatively analyze error types and their effect in chat-oriented dialogue systems.

The knotting probability is defined by the probability with which an N -step self-avoiding polygon (SAP) with a fixed type of knot appears in the configuration space. We evaluate these probabilities for some knot types on a simple cubic lattice. For the trivial knot, we find that the knotting probability decays much slower for the SAP on the cubic lattice than for continuum 1 models of the SAP as a function of N . In particular the characteristic length of the trivial knot that corresponds to a 'half-life' of the knotting probability is estimated to be 2.5 × 10 5 on the cubic lattice.

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagnet. Transient behavior from chaotic to steady oscillation is also observed in another range of the feedback parameter. An analytical theory is also developed, which indicates the appearance of multiple attractors in a phase space due to the feedback current. An instantaneous imbalance between the spin-transfer torque and damping torque causes a transition between the attractors, and results in the complex dynamics.PACS numbers: