There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

Chaos, fractional kinetics, and anomalous transport

The computational brain: Patricia S. Churchland and Terrence J. Sejnowski (MIT Press, Cambridge, MA, 1992); xi, 544 pages, $39.95

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

Spatially periodic vegetation patterns are observed in many semiarid regions. They are classified according to their symmetry as vegetation stripes on hillsides or spotted patterns on flat ground. We propose a simple model based on the interactions among water, soil and vegetation, which gives a systematic description of the variety of vegetation patterns. First, it describes most characteristics of vegetation stripes investigated in field observations and agrees with the hypothesized origin of vegetation stripes deduced from field observations, the spatially heterogeneous infiltration rate of surface water into soil. Moreover, it indicates that spotted patterns emerge from the same dynamics. Second, the pattern selection derived from the model is consistent with that found in real phenomena, i.e. predominantly bare ground with spotted vegetation in the most severe environments, banded patterns in intermediate environments and nearly continuous vegetation with spotted bare ground in the most favorable environments.

/pdf/systematic-analysis-of-periodic-vegetation-patterns-4jv5hs91y2.pdf

Systematic Analysis of Periodic Vegetation Patterns

The stability of isolating integrals against the perturbation of potential function was studied by the method of surface of section in the classical dynamical system of two degrees of freedom with the hamiltonian, \begin{aligned} H{=}\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+x^{2}+y^{2}+2x^{2}y+\frac{2}{3}y^{3}-\frac{4}{3}\varepsilon y^{3}\right), \end{aligned} in the range -1< e <1. This system has isolating integrals in this range. Their structure is found to change discontinuously at e =-1/2 and e =0.

On the Stability of Isolating Integrals. I. Effect of the Perturbation in the Potential Function

/pdf/stagnant-motions-in-hamiltonian-systems-3wiemgdk13.pdf

Stagnant Motions in Hamiltonian Systems

Various computer experiments based on the numerical integration of dynamical equation of motion are reviewed and their importance is emphasized in connection with ergodic theory. To do this a short review of the present status of the ergodic theories is also given and the interrelations among various results of ergodic theories are clarified. The induction period discovered in the previous papers (H. Hirooka and N. SaitO, ]. Phys. Soc. Japan 26 (1969), 624; N. Ooyama, H. Hirooka and N. SaitO, J. Phys. Soc. Japan 27 (1969), 815) is further examined in one·dimensional anharmonic lattices including exponential lattices. The apparent randomness which the system exhibits after elapsing the induction period is, strictly speaking, quasi·stochastic in the sence that the process is described completely as deterministic. The "true" stocha­ sticity which seems required in the thermodynamical state will be realized in the thermodynamical limit. The ergodic theory is regarded as most fundamental in statistical me­ chanics. Since the age of Boltzmann various discussions were given to this problem. The development is mainly done by mathematicians, and especially important results were obtained by Birkhoff, Von Neumann, Hop£, Khinchine and others. The theories, however, were not necessarily satisfactory for phys­ icists, since one can say nothing about ergodicity in real physical systems. The fundamental postulates of thermodynamics and statistical mechanics, such as the zeroth law of thermodynamics or ergodicity etc., are introduced from passive experiences or from purely theoretical requirements. The theoretical investigation of elucidating these postulates from mechanical point of view is an important task of theoretical physics. But there was devised no active experi­ ment to clarify the essential aspects of these problems. This is quite peculiar compared with other branches of theoretical physics. The unsatisfactory stage of mathematical theory of ergodicity, and the lack of experiments which are seductive to theorists may be the reason why many physicists have been almost indifferent to ergodic theory. However in 1950's two epochmaking progresses which opened the new ways were made. One is the mathematical studies done by Kolmogorov and his school and the other is the numerical experiments on computer carried

/pdf/computer-experiments-on-ergodic-problems-in-anharmonic-5bsxks9bn0.pdf

Computer Experiments on Ergodic Problems in Anharmonic Lattice Vibrations

We study subexponential instability to characterize a dynamical instability of weak chaos. We show that a dynamical system with subexponential instability has an infinite invariant measure, and then we present the generalized Lyapunov exponent to characterize subexponential instability.

Yoji Aizawa

Papers

Systematic Analysis of Periodic Vegetation Patterns

On the Stability of Isolating Integrals. I. Effect of the Perturbation in the Potential Function

Stagnant Motions in Hamiltonian Systems

Computer Experiments on Ergodic Problems in Anharmonic Lattice Vibrations

Subexponential instability implies infinite invariant measure