Table of contents

Population explosions in systems with birth-death processes are studied. The author shows that the fluctuations in local birth and death rates are strongly enhanced by the exploding system and thus the intermittent spatial distribution of the reproducing particles is formed. It is characterized by presence of strong bursts separated by large areas with low population density. The author estimates the probability of such rare bursts and find their typical forms. The analysis reveals a close mathematical relationship between the bursts and the solitons of the nonlinear Schrodinger equation.

Introduces a new numerical method to determine eigenvectors of a Hamiltonian matrix. The method is particularly useful for matrices of large dimension. The essence of the method is to determine each phase in the expansion of an eigenvector by computing the projection weight to the eigenvector on a trial function with an arbitrary phase. The trial function depends on the unknown phase in the expansion of the eigenvector, and it is shown that the weight takes its maximum value if the phase in the trial function takes the correct value of the phase in the expansion of the eigenvector.

Two evolution equations are considered of the form u_t=H_i(x, t, u, u_x, u_xx, . . .), i=1, 2, which are related by the general point transformation x*=P(x, t, u), t*=Q(x, t, u), u*=R(x, t, u). It is shown that the time transformation must necessarily be of the form t*=Q(t) and that if H_i are polynomials in the spatial derivatives of u then x*=P(x, t).

The time evolution of operators for q-oscillators is derived for the first time by exploiting the connection between q-deformation algebras and Lie-admissible algebras.

The adsorption of a directed polymer chain is studied in two dimensions using a restricted solid-on-solid model and allowing for a short range attraction between monomers. All the thermodynamic properties of the chain are evaluated exactly. The system does not exhibit a collapse or chain folding transition.

The Lie symmetry vector fields are derived for a nonlinear equation in plasma physics.

Considers the pseudo-Euclidean Hurwitz pairs, as formulated in a recent study by Randriamihamison (1990). From these pairs, the author constructs solutions of sigma models defined on real hyperquadrics and solutions of pure spinor models. This provides a non-trivial extension of previous works of Fujii (1988).

A new integral with respect to the index of Bessel functions of the first kind is evaluated.

In this paper, a hierarchy of bilinear Caudrey-Dodd-Gibbon-Kotera-Sawada equations with a unified structure is proposed. A nonlinear superposition formula for the CDGKS equation is proved under certain conditions. A Backlund transformation for a higher-order CDGKS equation is presented.

Similarity solutions are derived for the nonlinear diffusion equation c_t= Del (C Del C) in one dimension and with cylindrical symmetry. Some applications are indicated.

Dynamical arguments are presented which suggest that there are non-integrable systems without clustering of singularities, without infinite singularities, or singularities with an infinite number of branches in the complex t-plane. Several examples with only algebraic singularities are studied, for which strong numerical evidence is presented for non-integrability and infinitely sheeted solutions. 'Weak-Painleve' potentials are also analysed from this point of view, and all integrable cases are found to possess only finitely sheeted solutions.

In the framework of the nonlinear theory of elasticity the new perturbation method based on the gauge approach is developed to study static dislocations continuously distributed in materials. The self-consistent system of field equations for defect dynamics is presented up to the second-order approximation. The solution for the straight screw dislocations is obtained. The problem of an electron localization in crystals with screw dislocations is considered. The localized electron states are found to appear only in the second-order approximation for M=0 where M is the angular quantum number.

A special quaternion representation is constructed for a pair of relativistic vectors and skew-symmetric tensors on the basis of the group theory of Lorentz transformations. The construction has considerable advantages over the conventional vector-tensor description. It is pointed out that pairs of Minkowski vectors as well as certain scalars and skew-symmetric tensors can also be interpreted as simple components of more complex physical quantities, each of them expressed by a single quaternion. As an example a concise relativistic quaternion formulation of Maxwell's electrodynamics is presented. The relativistic covariance can be maintained even for the existence of magnetic monopoles.

The adiabatic quantum evolution of a two-state system without energy-level crossings is an example of the Stokes phenomenon. In the latter, a small (subdominant) exponential is an asymptotic expansion appears when a Stokes line is crossed; truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth, compact and universal manner across the Stokes line. In quantum evolution this corresponds to a smooth transition, universal in form, between 'superadiabatic' basis states (high-order WKB approximate solutions of the time-dependent Schrodinger equation). The authors give a numerical demonstration of this previously predicted universality by constructing, for two Hamiltonians, the superadiabatic quantum bases asymptotic to the actual evolving state. Universality when a Stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state, and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term.

The properties of q-deformed boson operators with non-generic q (q is a root of unity) are analysed by using the representation theory method and their finite-dimensional representations are thereby obtained. Based on this discussion, reducibilities and decompositions of q-deformed boson-realized representations of quantum universal enveloping algebra U_qSL(l) are studied for non-generic cases. The explicit matrix elements of some indecomposable representations are obtained on the q-deformed Fock spaces. Necessary details are provided for U_qSL(2) and U_qSL(3). In particular, the Lusztig operator extension of U_qSL(2) is discussed in an explicit form.

The high-temperature expansion of the grand thermodynamic potential of a nonconformally invariant spin-0 gas in an arbitrary ultrastatic spacetime with boundary is given in terms of the Minakshisundaram-Pleijel coefficients of the heat-kernel and the zeta function of the spatial section. The general formula is then used to find the expansion in the case of a massive bosonic field subject to Dirichlet boundary conditions on hypercuboids in a flat n-dimensional spacetime. A detailed analysis of inhomogeneous multidimensional Epstein zeta functions is necessary and some new properties of them are derived. Finally the thermodynamics of the system is considered.

The author shows the space of n-magnon states in integrable isotropic models of magnetic chains, the structure of the space of states in the RSOS model. The S-matrix in these models is described in terms of weights in integrable RSOS models.

Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.

Presents a computationally efficient, cell dynamical system which mimics the two-component Ginzburg-Landau equation. This model is used to numerically study the time-dependent behaviour of the vorticity number for a two-dimensional lattice. The author finds that the vorticity number initially decays exponentially in time and subsequently shows a power-law decay in time ( approximately 1/t, where t is time).

The majority of lattice gases have, apart from the physical conserved quantities of particle number, momentum and energy, spurious ones, usually staggered in space and time. At the level of linear excitations these staggered modes may be purely diffusive or damped propagating waves. In the eight- and nine-bit model on the square lattice one finds a large number of new spurious modes and the authors derive Green-Kubo relations for the diffusivities and damping constants and calculate them in the Boltzmann approximation.

A new kind of random walk named bounded Levy flights (BLFs), where the step length is a bounded random variable, is proposed and their properties are studied with the aid of mean field and Monte Carlo techniques. BLFs are characterized by the Levy exponent ( sigma ) and the length of the longest possible flight (R_M). It is found that in one dimension (1D), the mean number of distinct sites visited by the walker (S_N) and the average square displacement (R_N²) behave like S_N varies as R^Q_MN^ds(Q=f^sigma ,ds=1/2) and R²_N varies as R_M^{f( sigma )}N^nu (v=1), where f( sigma ) is a continuously tunable function of sigma with f( sigma )0.9 ( sigma 0.1) and f( sigma )0 ( sigma 2). In addition, the long-time behaviour of annihilation reactions between BLFs, which react via exchange in 1D is found to be anomalous because the density of walkers ( rho _A) behaves like d rho _A/dt-R^{f( sigma )}_M rho _A^X with X=1+(1/ds)=3(t) while, shortly after the beginning of the reaction, the classical behaviour X=2(t0) holds.

Levy flight is a random walk in which the step length r is a random variable with probability distribution proportional to 1/r^1+u, where u is the Levy parameter and 0<u< infinity . The authors study the critical behaviour of fully directed Levy flight on Sierpinski carpets using the Monte Carlo method. The obtained critical exponent v/sub /// is independent of the parameter u, but v_{perpendicular to} is found to be dependent on u. This seems to be interesting compared with the directed Levy flight on ordinary Euclidean lattices previously discussed by Hu and one of the authors, for which v/sub /// and v_{perpendicular to} are independent of u. These results indicate that directed Levy flights on different fractals belong to different universality classes.

The authors discuss generalizations of Feigenbaum's constants alpha and delta to complex polynomial maps of degree higher than 2, and present some numerical estimates. Universality classes are found to depend on the nature of the critical points of the polynomial.

The authors develop a general class of approximations of mean-spherical (MSA) type as a method for studying lattice percolation problems. They review the MSA and test certain extensions of it on lattice spin models. The relations between thermal spin models and percolation models are then reviewed in order to identify natural extensions of the MSA to percolation models. These extensions are used to treat both site and bond percolation models. In one 'low-density' formulation of MSA, the threshold for bond percolation on a lattice is found to equal the value at the origin of the corresponding lattice Green's function. This formula gives accurate results for all lattices studied, and in all space dimensions d>or=3. An accurate treatment is also given of the general site-bond problem. The entire percolation locus for this problem agrees closely with the results of simulation. They also introduce a new method for studying percolation transitions which is a hybrid of the Kikuchi cluster approximation scheme and the MSA. The method is shown to give good values for percolation thresholds while preserving the valuable features of the standard MSA. In particular, it provides a convenient means of computing the pair connectedness function. They also explore extensions of their approximations to treat directed site and bond percolation.

The author investigates the critical properties of the three-matrix chain model corresponding to the Blume-Emery-Griffiths model in a homogeneous magnetic field on a fluctuating lattice. The third-order critical line and the tricritical point disappear after the magnetic field is switched on. It is, however, found that another critical line exists as a result of the naive perturbative expansion and the string susceptibility exponent is -1/3 on the line. It is also shown that the critical exponent of the magnetic moment beta is 1/6 which coincides with the KPZ solution.

The long-time limit of the alignment function (or remanent energy E=-( sigma h)/2) of the fully asymmetric SK model is investigated analytically for parallel (E_p), sequential (E_s) and random-sequential (E_rs) update. As expected one finds E_p<E_s<E_rs. The results are supported by numerical simulations.

The authors calculate the universal part F of the free energy of rectangular domains at critical points by use of conformal field theory. F includes a term logarithmic in the size (or area), due to the corners. In addition, there is a term F₀ depending on the aspect ratio, which they determine by integrating the stress tensor (T) over the rectangle. This term involves complete elliptic integrals, but may be more simply expressed in terms of the Dedekind eta -function. For central charge c>0, they find that F₀ is maximal for squares, providing a thermodynamic driving force for the elongation of small domains, and argue that this should be a general tendency.

Presents new results on a probabilistic approach to parallel dynamics of the Little-Hopfield model. The authors propose a truncated auxiliary dynamics method to control a feedback noise in this symmetrical neural network with full connection. It allows them to propose an ansatz for derivation of the explicit recurrence relations for the main and residual (noisy) overlap evolution for arbitrary discrete moment t.

The authors describe null spinning strings with U(1) gauge symmetry in terms of a phase-space Langrangian and then quantize it in the light-cone gauge. With normal ordering they come to the conclusion that the critical dimension is D=2.

This comment purports to respond to some remarks made on the author's paper (Shivamoggi, ibid., vol.23, p.1689, 1990) in a recent comment (Paladin and Vulpiani ibid., vol.23, p.4717, 1990) in an effort to clarify a somewhat controversial situation in the enstrophy cascade of two-dimensional turbulence.

The formalism of the generalized fermionic Bogolyubov transformation, previously by Fan and VanderLinde (1990), is extended to the bosonic case. The corresponding bosonic quasi-particle and quasi-vacuum state are derived; the bosonic generalized Bogolyubov operator is decomposed as a normal product form by the technique of integration within an ordered product (IWOP).