For a system consisting of bistable units with global and asymmetric local interactions, there exist two different types of dynamical transition between stick-slip and steady motions: continuous and discontinuous in the amplitude of stickslip motion. These two types are consistent with the experimental findings in a dry friction system. The dynamical phase diagram and other types of transition are also discussed.

We propose a network percolation model, called cut-off model, which exhibits strongly discontinuous transition by an averaging effect of cluster sizes. In this model, a randomly selected bond is added if the size of the cluster formed by the bond is less than a times of the mean cluster size. It is shown that the model is strongly discontinuous when a is a finite constant.

Ultradiscrete equations are derived from a set of reaction diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.

We propose a simple stochastic model describing the position-dependent ball-passing network in football (soccer) games. In this network, a player in a certain area in a divided field is a node, and a pass between two nodes corresponds to an edge. Our stochastic process model is characterized by the consecutive choice of a node depending on its intrinsic fitness. We derive an explicit expression for the degree distribution and find that the derived distribution reproduces that for actual data reasonably well.

Statistical properties of position-dependent ball-passing networks in real football games are examined. We find that the networks have the small-world property, and their degree distributions are fitted well by a truncated gamma distribution function. In order to reproduce these properties of networks, a model based on a Markov chain is proposed. (C) 2014 Elsevier B.V. All rights reserved.

This paper proposes a numerical model of the network of two-Chinese-character compound words (two-character network, for short). In this network, a Chinese character is a node and a two-Chinese-character compound word links two nodes. The basic framework of the model is that an important character gets many edges. As the importance of a character, we use the frequency of each character appearing in publications. The direction of edge is given according to a random number assigned to nodes. The network generated by the model is small-world and scale-free, and reproduces statistical properties in the actual two-character network quantitatively. (C) 2014 Elsevier B.V. All rights reserved.

By means of ultradiscretization and probabilistic treatment, we derive a stochastic cellular automaton (CA) model from a dynamical model for bistable units with global and asymmetric local interactions. The obtained CA model is composed of the elementary CA rule 254 with a probabilistic rule that regulates the average of the total cell states.

There exists a threshold-sensitive dynamical transition between uniform and periodic growth modes in the domain growth of ascorbic acid crystals from its aqueous supersaturated solution film. The crystal growth induces solution flow. Humidity controls the fluidity of the solution. The solution flow varies the film thickness. The threshold exists in the thickness of the solution film. If the thickness becomes lower than the threshold, the solution flow and the crystal growth almost stop.

In this study, we examine the validity of the transition of the human height distribution from the log-normal distribution to the normal distribution during puberty, as suggested in an earlier study [Kuninaka et al.: J. Phys. Soc. Jpn. 78 (2009) 125001]. Our data analysis reveals that, in late puberty, the variation in height decreases as children grow. Thus, the classification of a height dataset by age at this stage leads us to analyze a mixture of distributions with larger means and smaller variations. This mixture distribution has a negative skewness and is consequently closer to the normal distribution than to the log-normal distribution. The opposite case occurs in early puberty and the mixture distribution is positively skewed, which resembles the log-normal distribution rather than the normal distribution. Thus, this scenario mimics the transition during puberty. Additionally, our scenario is realized through a numerical simulation based on a statistical model. The present study does not support the transition suggested by the earlier study.

In this paper, we focus on the "minimal payment" which minimizes the number of coins left after the payment. Two kinds of multifractal properties of the minimal payment are studied. The first one is a frequency distribution of change amounts, and the second one is a visiting probability on the delay plot of successive change amounts. When the face values of coins are power of two (1, 2, 4, 8, ... ), we find that these two distributions are related to well-known multifractal models, and derive analytical expressions of multifractal spectra.

Some statistical properties of origami (the art of paper folding) are investigated. In a crease pattern of an origami work, a paper sheet is divided into many "cells" surrounded by creases. We compute the cell-size distributions of four origami works and find that they are described by the superposition of lognormal distributions with different averages and variances. This lognormal behavior can be explained by a successive-folding process that works as a multiplicative effect. To confirm this idea, we introduce a numerical model by only incorporating a successive-folding process, and discuss its analytical properties. Moreover, we show that a group-separation structure is also essential in origami, which is different from pure crumpling. © 2013 The Physical Society of Japan.

The 'minimal' payment-a payment method which minimizes the number of coins in a purse-is presented. We focus on a time series of change given back to a shopper repeating the minimal payment. By using the delay plot, the set of successive change possesses a fine structure similar to the Sierpinski gasket. We also estimate effectivity of the minimal-payment method by means of the average number of coins in a purse, and conclude that the minimal-payment strategy is the best to reduce the number of coins in a purse. Moreover, we compare our results to the rule-60 cellular automaton and the Pascal-Sierpinski gaskets, which are known as generators of the discrete Sierpinski gasket. (C) 2012 Elsevier Ltd. All rights reserved.

We propose the rule ''254'' cellular automaton model with a probabilistic global rule. The global rule regulates the proportion of total cell states. The model reproduces spatiotemporal patterns obtained by an experiment of peeling an adhesive tape. The statistical properties of the spatiotemporal patterns are characterized from the fractal point of view.

The present paper proposes a stochastic model to be solved analytically, and a power-law-like distribution is derived. This model is formulated based on a cascade fracture with the additional effect that each fragment at each stage of a cascade ceases fracture with a certain probability. When the probability is constant, the exponent of the power-law cumulative distribution lies between -1 and 0, depending not only on the probability but the distribution of fracture points. Whereas, when the probability depends on the size of a fragment, the exponent is less than -1, irrespective of the distribution of fracture points. The applicability of our model is also discussed.

The statistics of isothermal lines and loops of the cosmic microwave background (CMB) radiation on the sky map is studied and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the fractal dimension D-e of the entire pattern of isothermal lines, the fractal dimension D-e of a single isothermal line, the exponent zeta in Korcak's law for the size distribution of isothermal loops, the two kind of Hurst exponents, H-e for the profile of the CMB radiation temperature, and H-c for a single isothermal line. We also perform fractal analysis of two artificial sky maps simulated by a standard model in physical cosmology, the WMAP best-fit Lambda cold dark matter (Lambda CDM) model, and by the Gaussian free model of rough surfaces. The temperature fluctuations of the real CMB radiation and in the simulation using the Lambda CDM model are non-Gaussian, in the sense that the displacement of isothermal lines and loops has an antipersistent property indicated by H-e similar or equal to 0.23 < 1/2.

Competition between global and asymmetric local interactions among bistable units brings about the characteristic collective behavior: the state of each unit is nonsteady, while their average property remains almost constant. The existence of noise is necessary for sustaining this nonsteady bistable dynamics. In the dynamics, the interface motion between the two different stable states is characterized. The difference from the case with global and diffusive (symmetric) local interactions is discussed.

The method of branch ordering evaluates the hierarchical structure of branching patterns, and the bifurcation ratio is an important variable in the topological analysis of branching patterns. In a simple mathematical model of a branching system called the random model, a central limit theorem for the inverse bifurcation ratio has been confirmed. In the present paper, we make a qualitative comparison between the random model and an ensemble of diffusion-limited aggregation (DLA) clusters from the viewpoint of the central limit theorem. From numerical data, we extrapolate asymptotic behaviors of the average and variance of the inverse bifurcation ratio of DLA clusters. These asymptotic properties are in good agreement with those of the random model qualitatively. Hence, the central limit theorem also holds for DLA clusters.

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented.

In order to formulate and examine the central limit theorem for a binary tree numerically, a method for generating random binary trees is presented. We first propose the correspondence between binary trees and a certain type of binary sequences (which we call Dyck sequences). Then, the method for generating random Dyck sequences is shown. Also, we propose the method of branch ordering of a binary tree by means of only the corresponding Dyck sequence. We confirm that the method is in good consistency with the topological analysis of binary trees known as the Horton-Strahler analysis. Two types of central limit theorem are numerically confirmed, and the obtained results are expressed in simple forms. Furthermore, the proposed method is available for a wide range of the topological analysis of binary trees.

Humidity-temperature conditions for the emergence of various domain patterns of ascorbic acid crystal are identified, where the crystal grows from its thinly spreading aqueous solution by solvent evaporation. In particular, under a low-humidity condition, the coexistent pattern is formed by two domains having different growth rates. In the faster growing domain, there exists a clear time delay between the crystal growth in the solution and the drying of the crystal surface. From the shape of the domain boundary in the coexistent pattern, the growth rate ratio between the two domains can be estimated.

Some statistical properties of a network of two-Chinese-character compound words in the Japanese language are reported. In this network, a node represents a Chinese character and an edge represents a two-Chinese-character compound word. It is found that this network has properties of being "small-world" and "scale-free". A network formed by only Chinese characters for common use (joyo-kanji in Japanese), which is regarded as a subclass of the original network, also has the small-world property. However, a degree distribution of the network exhibits no clear power law. In order to reproduce the disappearance of the power-law property, a model for a selecting process of the Chinese characters for Common Use is proposed. (C) 2009 Elsevier B.V. All rights reserved.

The bifurcation ratio for the equiprobable binary tree model is formulated. We obtain the exact expression of the kth moment of the second-order streams. We also obtain a recursive equation between rth and (r+1)th order streams. Horton's law is confirmed numerically by calculating this recursive equation and asymptotic properties of the bifurcation ratio are discussed.

A scaling hypothesis for the standard deviation sigma of the height of growing interfaces is proposed by extending the Family-Vicsek (FV) scaling hypothesis. A data-collapsing method is adopted for estimating values of three exponents alpha,beta, and z, which characterize, respectively, the roughness, growth, and dynamic properties of growing interfaces. The estimation is carried out through sigma, which is a function of both the time and the width of the interfaces. The advantages of the present extended scaling hypothesis are as follows: (A) The value of,B can be obtained even if the data for Q in terms of t are few so that its value is not determined precisely from the slope of the In a vs In t plot. (B) Different scaling relations can be obtained during the time evolution of interface growth. (C) By introducing a new exponent, which represents the time dependence of Q for a short width, a scaling argument is possible even for growing interfaces that do not satisfy the FV scaling relation. Successful applications are carried out to a few numerical models and a paper-wetting experiment.

A model for the collective motion of granular particles induced by an interaction with interface motion is constructed. By numerical simulation, it is found that the model can qualitatively reproduce labyrinthine patterns that are quite similar to those experimentally obtained in the drying process of a water-granule mixture. It is found that the characteristic scale of the patterns is proportional to the inverse of the strength of dynamic friction for the particles. Moreover, the stick-slip motion of the interfaces is confirmed in our model.

Dynamical properties of peeling an adhesive tape are investigated experimentally as an analogy of sliding friction. An adhesive tape is peeled by pulling an elastic spring connected to the tape. Controlling its spring constant k and pulling speed V, peel force is measured and spatiotemporal patterns formed on the peeled tape by deformed adhesive are observed. It is found that there exist two kinds of adhesive state in peeling front. The emergence of multiple states is caused by the stability of a characteristic structure (tunnel structure) formed by deformed adhesive. Tunnel structures are distributed spatiotemporally on adhesive tape after peeling. Based on the spatiotemporal distribution, a morphology-dynamical phase diagram is constructed on k-V space and is divided into the four regions: (A) uniform pattern with tunnel structure, (B) uniform pattern without tunnel structure, (C) striped pattern with oscillatory peeling, and (D) spatiotemporally coexistent pattern.

In the aggregation of granular particles caused by the motion of the sweeping front. percolation occurs in the universality class of two dimensions due to a change in the particle concentration corresponding to the occupation ratio. The collective motion of the granular particles is described as the process of self-avoiding invasion percolation. The cluster structure of the aggregated granular particles exhibits a scaling property in a wide range of the particle concentration.

We examined various spatiotemporal patterns evolving in deformed adhesives during peeling. The patterns are regarded as a spatiotemporal distribution of a tunnel structure formed by the deformed adhesive, and are classified into the following four types: (A) uniform pattern with tunnel structure, (B) uniform pattern without tunnel structure, (C) striped pattern alternating between A and B in time, and (D) spatiotemporal pattern made by the coexistence of two regions with and without the tunnel structure. We are able to reproduce these patterns and the mechanical and statistical behaviors by constructing a model with a state variable representing the stability of the tunnel structure. The essential factor of the pattern formation is the competition between a local asymmetric interaction and a global coupling of the state variables. (c) 2006 Elsevier B.V. All rights reserved.

Generation of earthquakes has not been still understood. However, some statistical properties of seismicity are established such as magnitude-frequency distribution indicating power law. In this study, we numerically investigate one-dimensional spring-block earthquake model, considering self-affine fault surfaces, in order to understand such statistical properties.

We have examined whether rough interfaces of bacterial colonies are multi-affine. We have used the bacterial species called Bacillus subtilis, which has been found to exhibit a variety of colony patterns when varying both the concentration of nutrient and solidity of agar medium. Consequently, we have found that the colony interface on a nutrient-rich, solid agar medium is multi-affine. On the other hand, the colony interface on a nutrient-rich, semi-solid agar medium is self-affine.

Our experimental results about the dynamical behavior of adhesive tape in peeling at a constant peel speed are reported. We focus our attention on spatio-temporal patterns formed by deformed adhesive in peeling on the tape. Because of the viscoelastic nature of adhesive, it is found that the patterns are classified into four types by changing both the peel speed and the stiffness of the system, which is controlled by a spring connected to the tape. In order to explain the experimental results, we propose a model for the pattern formation on adhesive tape in peeling by introducing the state variable representing the stability of a characteristic structure, which is called tunnel structure.

We have extended dynamic scaling hypothesis to describe the wide range of experiments on growing rough interfaces. An alternative approach of original Family-Vicsek scaling hypothesis was not successfully applied to the growing interfaces of paper wetting in that we obtained unphysical values of roughness exponents. Since very nice data collapse suggested the existence of some kind of dynamic scaling, we have tried to extend the dynamic scaling hypothesis. It was found that the extended dynamic scaling hypothesis can be nicely applied to the growing rough interfaces in paper wetting, yielding values of roughness and growth exponents alpha similar or equal to 0.73, beta similar or equal to 0.60, respectively.

The formation of concentric ring colonies by bacterial species Bacillus subtilis and Proteus mirabilis has been investigated experimentally, focusing our attention on the dependence of local cell density upon the bacterial motility. It has been confirmed that these concentric ring colonies reflect the periodic change of the bacterial motility between motile cell state and immotile cell state. We conclude that this periodic change is macroscopically determined neither by biological factors (i.e., biological clock) nor by chemical factors (chemotaxis as inhibitor). And our experimental results strongly suggest that the essential factor for the change of the bacterial motility during concentric ring formation is the local cell density. (c) 2005 Elsevier B.V. All rights reserved.

Pattern dynamics of directional crack propagation phenomena observed in drying process of starch-water mixture is investigated. To visualize the three-dimensional structure of the drying-fracture process two kinds of experiments are performed, i.e., resin solidification planing method and real-time measurement of water content distribution with MR instruments. A cross section with polygonal structure is visualized in both experiments. The depth dependency of cell size is measured. The phenomenological model for water transportation is also discussed.

Characteristic patterns of expansion were generated by bacterial cell populations of Serratia marcescens on media with different concentrations of nutrient and agar. These patterns were classified as Eden-like, dense-branching-morphology-like, flower-like, concentric ring-like and diffusion-limited aggregation-like. Although flower-like was specific to Serratia marcescens, the other patterns were exhibited by Bacillus subtilis. Through macro- and microscopic tracing of the processes generating these patterns, physico-chemical principles of bacterial growth, collaborative and independent properties of bacteria, structural organization for population expansion, and the division of labor among bacterial cells (i.e., wall composer, pressure generator, and logistic supporter) were brought to light. © 2005, Japanese Society of Microbial Ecology &amp
The Japanese Society of Soil Microbiology. All rights reserved.

We have numerically investigated the self-affinity of the bacterial colony model. In order to examine the growth exponent beta, we find a new approach to the dynamic scaling hypothesis for the growing rough surface. We have first investigated the self-affinity of the well-known Eden model to check the validity of the new approach. The obtained results of the exponents alpha congruent to 0.50 and beta congruent to 0.33 coincide with the wellknown values of the exponents of the Eden model. We have then applied it successfully to growing interfaces produced by the bacterial colony model. We have found that the self-affinity of the bacterial colony model is characterized as alpha congruent to 0.80 and beta congruent to 0.65.

We investigated (i) the relationship between the spatiotemporal pattern of a deformed adhesive and peel load during peeling, and (ii) the dependence of the pattern formation on peel speed in the hard spring limit case. In the hard spring limit case, it is found that elastic and viscous peeling states coexist. It is also found that the occupation ratio of tunnel structures in a separation front is determined by the peel speed and is a monotonically decreasing function of peel speed. We experimentally confirmed that the value of peel load has a linear dependence on the occupation ratio of tunnel structures. An explanation of dynamical behavior is also given on the basis of the creation-annihilation dynamics of tunnel structures.

The formation of concentric ring-like colonies by bacterial species Bacillus subtilis has been investigated, focusing our attention on the effect of local cell density upon the bacterial motility: (i) Neither any chemicals nor a pacemaker at the center of the ring takes part in the concentric ring formation. (ii) Phase entrainment between two colonies having different phase of concentric ring does not occur. (iii) From the measurement of lag-phase time when varying the initial cell density, the start of the first migration phase is found to depend on the cell density. (iv) When cutting the part of a colony which is behind a migration phase just after the start of migration, the migration phase becomes shorter. On the other hand, the following consolidation phase becomes longer. (v) By the replica-printing method, active bacteria move collectively from inside to outside of the outermost consolidation terrace. Our present experimental results are qualitatively consistent with the results of the other bacterial species P. mirabilis, although the individual cell motility is quite different from each other. The present results suggest that the essential factor of the change of the bacterial motility of B. subtilis during concetric ring formation is the local cell density.

Morphological diagram with respect to crystal growth of L-ascorbic acid (C6H8O6; so-called vitamin C) from methanol solution on a flat glass dish is presented. Varying humidity and initial concentration of L-ascorbic acid in methanol solution, the following three distinct kinds of growing patterns have been observed: homogeneous disk, concentric ring and dendrite. In addition, in higher concentration clearly faceted small single crystals grow in any humidity less than 90%. Crossovers from one pattern to another were observed, too.

We formulate a theory for the aging phenomena in AuCd-type shape-memory alloys. The kinetics of the martensitic transformation is represented in terms of the local strain as the order-parameter field. We introduce a secondary slow variable coupled with the order parameter. This theory is an extension of the previous paper by one of the authors, where the secondary variable was assumed to be a nonconserved quantity. In the present paper, we explore the consequence that the secondary variable is conserved. It is shown that the conserved slow variable is more favorable to account for martensite stabilization after aging observed experimentally. In order to study the rubberlike elasticity, the stress-strain relation is also derived both by the direct simulations of the set of kinetic equations and by the interface (twin boundary) dynamics. We propose experiments to elucidate whether the slow variable is conserved or not.

We show experimental results regarding how density waves of granular particles flowing through a vertical glass pipe emerge by controlling two kinds of parameters; air flow out of a flask attached to the bottom-end of the pipe and inflow of granular materials from a hopper attached to the top-end. It was found, by controlling the former parameter, that the most important factor for the onset of density waves is the strong interaction between granular particles and the air. It was also found, by changing the latter parameter, that there is a sharp transition from the uniform flow to the density waves at the threshold value of the inflow rate of granules. Power spectra of density waves show a clear power-law form S(f) similar to f(-alpha), where alpha similar or equal to 4/3. The value of alpha is robust in that, once density waves occurred, it was independent of the two parameters described above.

We present experimental results regarding the dynamical property of granular flow through a vertical glass pipe under the control of inflow rate. It was found that there exists a critical inflow rate for the formation of density waves. The power spectrum of the density waves exhibits clear power-law behavior, S(f) similar to f(-alpha), with alpha similar or equal to 4/3. This value of alpha is robust under variation of inflow rate.

We experimentally investigated the dynamical behavior of adhesive tape in peeling with emphasis on the emergence of slow and fast peeling motions. The dynamical-morphological phase diagram for peeled adhesive tape as a function of peel speed and spring constant was obtained. The spatiotemporal patterns of peeled adhesive were classified into four types: low-speed pattern, high-speed pattern, oscillatory pattern, and spatiotemporal intermittent pattern.

Spatio-temporal patterns emerged inside a colony of bacterial species Proteus mirabilis on the surface of nutrient-rich semisolid agar medium have been investigated. We observed various patterns composed of the following basic types: propagating stripe, propagating stripe with fixed dislocation, expanding and shrinking target, and rotating spiral. The remarkable point is that the pattern changes immediately when we alter the position for observation, but it returns to the original if we restore the observing position within a few minutes. We further investigated mesoscopic and microscopic properties of the spatiotemporal patterns. It turned out that whenever the spatio-temporal patterns are observed in a colony, the areas are composed of two superimposed monolayers of elongated bacterial cells. In each area they are aligned almost parallel with each other like a two-dimensional nematic liquid crystal, and move collectively and independently of another layer. It has been found that the observed spatio-temporal patterns are explained as the moire effect.

We introduce a quite simple model for a phase-separating binary mixture where one of the components is cross-linked. It is shown by computer simulations that target patterns are generated. We use a Cahn-Hilliard equation with a bulk free energy density which varies as a molecular weight changes with time. Target patterns are obtained in the following way. We first quench the system, having an off-critical composition, into a spinodal region, and find that disconnected droplets emerge in the system. With cross-linking one of the components, the phase diagram of the system shifts to a location where a bicontinuous structure is formed. As a result, target patterns grow from the droplets without any additional nuclei.

Computer simulations based on the time-dependent Ginzbug-Landau approach have been performed for the formation of domain structure in the cubic-tetragonal transformation under an external field. The fcc-L1(0) ordering has been studied as a model case of the transformation, and the Landau free-energy function has been determined so as to reproduce the free energy calculated by the Monte Carlo simulation. The simulations have demonstrated the features observed in our experiments on FePd alloy under an external stress, and have clarified the formation mechanism of a single variant structure. The internal stress field developed by the preferential formation of a variant favored by the external stress further accelerates the trend cooperatively, and eventually leads to a single variant structure.

The aggregation process of granules driven by the motion of water-air fronts in the drying process of two-dimensional wet granular systems is experimentally studied. The fronts and aggregated granules interact with each other during this drying, and a labyrinthine pattern is formed as a result. A qualitative framework of the pattern formation is given, and topological and metrical properties of the pattern are also discussed.

We performed numerical simulations for pattern formations in cubic-tetragonal structural transitions. Our model can reproduce a series of patterns which are quite similar to morphologies observed in experiments. And the evolution of patterns is consistent with the results of the interface dynamics. The mechanism of pattern formations is discussed from an elastic point of view.

Interface equations of motion for a single tetragonal nucleus growth and for twin formation under elastic fields in cubic-tetragonal structural transitions are derived. We give an expression to the elastic energy in the form of non-local, long-range interactions between nonconserved order parameters. We find that a single tetragonal nucleus grows anisotropically into oval or square and that twin structure is a stationary state in the coarsening process. These results are consistent with the numerical simulation.

Elastic effects by cubic-tetragonal symmetry breaking on the ordering kinetics are investigated. We propose a coarse-grained model characterized by a Ginzburg-Landau free energy with non-conserved order parameters coupled to an elastic field. By numerical study we obtain an oval shaped domain and a square domain in two dimensions and a plate-like domain and a rod-like domain in three dimensions. And in the coarsening process, a striped pattern is obtained. These patterns are remarkably similar to morphologies observed in real cubic-tetragonal order-disorder alloys. Moreover, we study the dynamics during the coarsening process by analyzing the change of the total domain boundaries LDB(t). LDB(t) decreases as a power of time: t-α(α &gt
0). The exponent a becomes small with increasing the magnitude of the elastic effects.

We have made a numerical scheme of solving two fluid hydrodynamics near the λ point, which includes the mass density, momentum density, entropy density, and complex order parameter. In our model the normal fluid velocity vanishes at the boundaries due to finite viscosity. As preliminary examples we examine coexistence of a superfluid region and a normal fluid region in two space dimensions firstly in heat flow and secondly in gravity and heat flow. We observe formation of a vortex from a wall in the second case. © 1996 Plenum Publishing Corporation.