Table of contents

It is pointed out that the Eckhaus nonlinear PDE i Psi _t+ Psi _xx+(2( mod Psi mod ²)_x+ mod Psi mod ⁴-V(x)) Psi =0 remains C-integrable (i.e., integrable by a change of dependent variable) even in the presence of any given external potential V(x).

A new conservation theorem is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only, without using either Lagrangian or Hamiltonian structures (which may even fail to exist for the equations at hand). One example and implications of the theorem on the structure of point symmetry transformations are presented.

The authors use the semiclassical periodic orbit theory to identify the recently discovered one-parameter family of stable periodic orbits in the x²y² potential occupying an area of 0.005% on the surface of section. They also indicate the presence of another stable family of periodic orbits of higher length. The sensitivity of the method provides hope for ruling out ergodicity in other systems.

Both quantum and classical analyses are performed to study the barrier crossing dynamics in a driven quartic oscillator. The regions of phase space with regular classical motion are found to be smaller than the size of a quantum state. However, coherent tunnelling is still possible due to the existence of Floquet states localized on two small regular islands. The quantum evolution of localized wavepackets and the classical evolution of the corresponding distributions show similar coherence properties although the degree of coherence is quantally enhanced.

In Jahn-Teller (JT) systems with N equivalent sites on which electrons (holes) may localize, one encounters the interaction of many electronic states through JT vibrations. The total multi-dimensional Hamiltonian is reduced to N equivalent Hamiltonians of low dimension and distorted symmetry. Explicit examples are presented and general implications discussed.

It is shown that an appropriate triple-well version of the binding potential V(x)=ax²+bx⁴+cx⁶ (a, c)0, b(0, b²)3ac) provides a solvable model whereby the discontinuous behaviour of the energy eigenvalues of any finite set of lowest-lying levels can be deduced analytically. The reason for the occurrence of such discontinuities is discussed.

The authors present a (2+1)-dimensional Skyrme-like model with a symmetry-breaking potential, which in R₃ has charge-n instanton solutions, and in the static limit in R₂ a sphaleron-like solution.

The authors find out the Bose realization of a generalized Heisenberg algebra, in which the bracket of the annihilation and creation operators is proportional to a polynomial function of the number operator. The eigenvalues of the corresponding oscillator are derived in a special case. They stress also the connection between non-canonical commutation relations and q-algebras.

The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q<or=1. In the range - infinity (q(0, these coincide with the q-boson operators and for 0(q<or=1 they coincide with q-fermions. The ordinary bosons and fermions may be identified with the limits q=-1 and +1 respectively. Generalized q-fermion coherent states are constructed by utilizing a nonlinear shift automorphism of the algebra of q-fermion operators. These are compared with the coherent states defined as eigenstates of annihilation operator. Matrix elements of the shift operator in the Fock space basis are evaluated.

The quantum algebras associated with the R-matrix obtained from the two parameter free fermion model are examined. It is shown that in addition to U_q,s(gl(1, 1)) there exists another algebra which is isomorphic to U_q,s(gl(1, 1)) as an algebra but has a different Hopf structure. It does not have a classical limit and is related to U_p(sl(2, c)) at a root of unity (p=i, i²=-1).

The explicit symmetry breaking phenomenon in supersymmetric quantum mechanics (SUSY QM) is analysed in terms of anomalies. The anomalous behaviour can be assigned to the supersymmetric charges which define the two Hamiltonians of the model. The relation between the presence of anomalies and the factorization of the Schrodinger equations is also discussed.

Geometrical and statistical properties of non-equilibrium crumpled surfaces (CS) and crumpled wires (CW) are investigated and compared. The relationship between the geodesic distance x and the Pythagorean distance r in CS and CW and their dependence on the linear (uncrumpled) size L is studied. Among other results the authors show that the moments of the probability distribution P(x, r) for CS requires an infinite hierarchy of critical exponents.

Peres and Weinberg have debated the definition of entropy in nonlinear quantum mechanics. This issue can be resolved, it is contended, if one adopts an entropy measure consistent with the approach of Guiasu (following work of Wiener and Siegel) in which a probability distribution over the space of wavefunctions is estimated. This is tantamount to rejecting the von Neumann measure (S_N), -Tr rho ln rho , of the entropy of a density matrix ( rho ), which is at the basis of Peres' argument. As authority for such a rejection of S_N, a detailed critique of this measure, given by Band and Park in the mid-1970s, can be cited.

An inequality, similar to Kraus's proposal proved by Maassen and Uffink, concerning the lower limit for the sum of entropies for unbiased quantum measurements is given. The author shows that Sigma ^N+1_i=1S(W⁽ⁱ⁾)>or=(N+1) ln((N+1)/2) where W⁽ⁱ⁾ are results of N+1 mutually unbiased measurements performed on the same initial state.

The authors have adapted the dimerization algorithm for enumerating self-avoiding walks to exploit the parallelism of the Connection Machine System. By this approach they have extended the series for the number of self-avoiding walks on the 2D square lattice by five terms. These data may permit more accurate estimates of critical properties.

Ultradiffusion is defined on a family of fractal branching Koch curves. By an exact renormalization decimation transformation the anomalous long-time behaviour of the autocorrelation function is obtained. For the particular hierarchy the authors find a non-universal crossover C₁=(m-1)/2M and a universal one C₂=1/2 as the effective temperature C is increased, where M is the maximum ramification number of the fractal. They also find that for large M, the dynamic behaviour of ultradiffusion on the fractal is independent of the effective temperature C.

A new systematic method is proposed for the analysis of the storage capacity of analogue neural networks with general input-output relations. It is based on the self-consistent signal-to-noise analysis in which renormalization of the signal part in the local field is properly performed. A remarkable feature of the present recipe, which in the case of symmetric analogue networks yields the same result as obtained by replica calculations, is the capability of dealing with asymmetric networks of analogue neurons. The theory is applied for the asymmetric network in which each neuron is loaded with biased patterns while some neurons are assigned only to extend inhibitory synaptic couplings free of learning.

A theory is proposed for the nonlinear excited states in the hydrogen bonded chain of peptide groups, where the interaction between intramolecular and intermolecular vibrations is taken into account. It is an interpolation between the behaviour in two limiting cases, i.e. the vibron soliton in the subsonic region, and the acoustic soliton in the supersonic region. Using this framework, the mechanism of energy migration in hydrogen bonded systems of peptide groups is analysed.

A one-dimensional (1D) system associated with a gl(m mod n) vertex model is investigated. The author adds a chemical potential term so that the model reduces to the supersymmetric t-J model for (m, n)=(2, 1). It is conjectured that the ground state of the model possesses SU(m) symmetry in the rational limit. The observation by Kawakami and Yang on the t-J model is extended to a more general theorem of the 1D field theory: the t-J model is just an example of the present model, and such systems can be described by a multi-component Gaussian model compactified on tori with different radii.

The authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries. The Laplacian potential around a linearly self-similar square Koch tree is studied in detail. The multiplicative nature of this potential, and the consequent multifractality of the harmonic measure are discussed. On prefractal stages, the density d mu of the harmonic measure and the corresponding Holder alpha =-ln d mu are well defined along the boundary, except in the folds where the tangent is undefined. A regularization scheme is introduced to eliminate these local effects. They then consider the probability distributions P( alpha ) d alpha of successive stages, and discuss their collapse into an f ( alpha ) curve. Both the left- and right-hand sides of this curve show good convergence. Other studies indicate that, for DLA, the right-hand tail does not converge. A brief comparison is made between the multifractality of these two cases.

The eigenvalues of the Fateev-Zamolodchikov Z_N invariant model transfer matrix are found for N odd. Their zeros in the complex plane of the rapidity variable are shown to satisfy a set of Bethe-ansatz type equations similar to those obtained for the integrable XXZ chains. The eigenvalue for a filled sea of (N-1)-strings gives the free energy found by the matrix inversion method.

Recent work has drawn attention to the utility of algebraic approximants for series expansion analysis. The special case of quadratic approximants are applied to lattice trees (weakly) embedded on the simple cubic lattice. The resulting estimates of x_c, the reciprocal of the growth constant, are compared with those obtained from other methods of analysis.

Both high-temperature and low-temperature series are used to locate and characterize the first-order transition in the 3-state Potts model in (2+1) dimensions on both square and triangular lattices. Estimates are presented for the vacuum energy, latent heat, magnetization, susceptibility and mass gap at the transition. The spontaneous magnetization and latent heat appear to display an 'approximate universality' at this weak first-order transition.

The generating function for the number N_c,n of almost-convex polygons on the square lattice with concavity index c=1 and perimeter n is derived rigorously. The asymptotic behaviour of N_c,n for large n is determined and this result confirms a conjecture by Enting et al. (1992).

The space groups of the orthorhombic approximant lattices to the primitive icosahedral quasilattice are classified. There exist three Bravais classes: Pmmm, Cmmm and Immm. The basis vectors of the Bravais lattice of an approximant are parallel to two-, three- and/or fivefold axes of the quasilattice. It is found that there exist many nonsymmorphic space groups with a common Bravais lattice in addition to symmorphic ones. This is because glides commonly appear.

The authors develop a method for solving 1D kinetic boundary layer problems for linear equations that uses separate Hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall. This so-called two-stream method is tested for two especially simple kinetic equations, the linear BGK equation and the Klein-Kramers equation, the kinetic equation for Brownian particles. For both of these equations, extensive exact information is available for two simple boundary layer problems, the Milne and the albedo problem. A comparison of these exact results with those obtained with this version of the two-stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude. In particular, the method allows them to obtain accurate results not only for the moments of the distribution function, but also for the distribution function itself.

An exact relation between the bulk effective Seebeck coefficient alpha _e of a composite conductor and the bulk effective electrical and thermal conductivities sigma _e and gamma _e is used to study the scaling behaviour of alpha _e near percolation. The behaviour turns out to be quite rich, as a result of its dependence on three dimensionless small parameters, namely, the electrical and thermal conductivity ratios of the two components sigma _I/ sigma _M, gamma _I/ gamma _M, and the distance away from the percolation threshold Delta p identical to p_M-p_c. The behaviour of the thermoelectric figure of merit Z_e in the different parts of the critical region is also discussed.

The disordered magnetic lattice gas (DMLG) as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely-long-ranged model by incorporating short-range order (SRO). The authors have shown rigorously that the local thermodynamic properties of the DMLG on a Cayley tree of finite coordination number z are identical to the thermodynamic properties of the DMLG in a pair approximation obtained by using the method of the distribution function. Further, a modified pair approximation for the DMLG is presented which is exactly solvable. It is formulated for general random bond distribution functions, and is then examined for the special case of Gaussian distributions.

In his seminal paper Cover (1965) used geometrical arguments to compute the probability of separating two sets of patterns with a perceptron. The authors extend these ideas to feedforward networks with hidden layers. There are intrinsic limitations to the number of patterns that a net of this kind can separate and they find quantitative bounds valid for any net with d input and h hidden neurons.

The authors have developed a new diagnostic tool for the analysis of the order-to-chaos transition: the partial Lyapunov exponents, defined through the dynamics in the tangent space. They allow the dynamics of single variables to be analysed, and are suitable for systems with several degrees of freedom. The authors have numerically simulated the dynamics of a model of five nonlinearly coupled oscillators; the partial Lyapunov exponents have been used to compute a characteristic coherence time for each degree of freedom. These quantities give information which is complementary to the usual statistical correlation times, and show that the high-frequency degrees of freedom, while losing their correlation during the order-to-chaos transition, may keep their coherence over long times.

Builds a one-to-one correspondence between some polyominoes, called parallelogram polyominoes, and some heaps of segments. Then the author derives explicit expressions for the generating function of convex polyominoes, according to their height, width and area. The author also enumerates two subsets of convex polyominoes, namely the directed and convex polyominoes and the parallelogram polyominoes, according to these three parameters. Generating functions of polyominoes having a fixed width are also studied.

Uses the so-called DSV methodology, that links some enumeration problems to the theory of algebraic languages, to get a system of q-difference equations involving the generating functions of convex polyominoes, of convex and directed polyominoes, and of parallelogram polyominoes, according to their height, width and area. Then, the author shows various applications of this system.

q-integration is defined for the complex quantum plane. The existence of classical and Grassmanian limits of q-integration is proved. Quantum versions of Cauchy's and Stokes' theorems are formulated.

The quantum algebraic structures associated with a family of R-matrices one extracts from the Perk-Schultz model are studied. Starting with the quantum spaces one can define from the R-matrices, the author shows how different duality conditions at the level of quantum spaces (Manin's construction) translate in different quantum groups and quantized universal enveloping algebras.

Discusses various methods for estimating the bias from a repeated set of measurements that exhibit a spread larger than expected on the basis of their statistical errors. Monte Carlo simulations are used to test how well the methods perform.

A new method of solving analytically coupled Riccati equations by means of embedding them into a matrix Riccati equation is proposed. This process imposes certain conditions onto the coefficients of the original equations. Whenever it is impossible to match these requirements, the calculation of less demanding invariants is discussed. The algebraic structure of these is simple enough to be practically used for lowering the system's dimension. In general, the newly soluble equations do not possess the Painleve property.

For a kth-order singular Lagrangian the presymplectic equation plus the different 'mth-order differential equation' conditions that can be considered in the Lagrangian space yield different dynamics which are studied. This leads to a new classification of the constraints. This study is also performed in the 'intermediate formalisms' which can be defined between the Lagrangian and the Hamiltonian ones; the corresponding classification schemes are then related.

Periodic stationary-wave solutions of the intermediate long-wave (ILW) equation are derived using the bilinear transformation method, and a new expression for the dispersion relation is obtained. The class of physically important real-valued solutions is identified. These solutions may be represented as an infinite superposition of solitary-wave profiles, a property shared by the related Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equation. This nonlinear superposition principle, which has been the subject of various interpretations in the literature, is discussed. The ILW periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. For intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave. In the shallow-water (KdV) limit the ILW periodic solution leads to the familiar cnoidal wave, whereas the deep-water (BO) limit yields Benjamin's periodic wave. A previously unknown expression for the cnoidal-wave dispersion relation in terms of theta functions is obtained.

The exact propagator of this dynamic system is derived for a rational wedge. For an irrational wedge, the proposed propagator can be confirmed by expanding it in terms of eigenfunctions and eigenvalues, which agree with those obtained from the corresponding Schrodinger equation. The results are also valid for an inverse square potential interacting with a wedge. Finally the authors investigate the classical path's contributions to the propagator for the free particle and the rational wedge case.

A recently proposed generalized von Neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator. Utilizing Lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value (x) is extracted, including both amplitude and phaseshift relative to the driving force. A differential equation for (x) is derived and the damping constant is identified.

The level shift of Dirac particles under the influence of point interaction potentials has been determined. A Green function method is used to obtain closed expressions for the level shift and perturbed wavefunctions, provided that the corresponding eigenvalue problem without point interaction potentials can be solved exactly. Several applications are discussed in detail.

A generalization of Berry's connection is proposed for quantum mechanical models whose wavefunctions are sections of a vector bundle over a bundle space with fibre, the configuration space of the model and base space a space of parameters. The transformation properties of the generalized Berry's connection under diffeomorphisms of the parameter space are also studied.