Table of contents

The authors find the differential equation for the generating function or a multiplicative stochastic process and they apply to it the group analysis. They give the general form of the Lie generators and find the conditions for the existence of similarity solutions. Two classes of similarity solutions are presented and the analytic expression of the generating function is given.

The functional connection between Blankenbecler-Goldberger (1962) amplitude and Glauber amplitude have been found, so that a fast way to reduce the integral expression of BG amplitude in the scattering problem can be acquired. Two equivalent expressions are discussed.

The actions for two q-Heisenberg algebras are constructed by extending a contact metric structure of a Heisenberg group manifold to the corresponding line bundle. The author obtains the q-Heisenberg algebras under a canonical quantization and regularization. Also the q parameter is shown to be the regularizing parameter.

The possibility of obtaining the cylindrically symmetric solution in the case of low energy string theory is investigated.

The author shows that the state determination of a pure spin state can be obtained from the results of three Stern-Gerlach type measurements. If the initial state is m=0 state, the selection of measurements cannot be made in advance.

The integrable nonlinear discrete system of Ablowitz-Ladik (1975) is investigated from the viewpoint of the quantum inverse scattering transform with non-trivial boundary conditions at the two ends. A new type of quantum R-matrix is obtained, which has been used to set up the algebraic Bethe ansatz equations. The asymptotic R-matrices R_+or- (as lambda , or mu +or- infinity ) are similar to the non-standard solution of the Yang-Baxter equation.

A class of simply solvable long-range ferromagnetic models is studied in terms of the eigenvalues and eigenvectors of the interaction matrix. The validity of this approach was first systematically studied by Canning (1992). The generalized ferromagnetic models studied in this paper are a ferromagnetic equivalent of Hopfield neural networks and site-disorder spin glass models, although the interactions of the examples studied are chosen in a deterministic way. These ferromagnetic models, in the same way as the separable disordered models, are described by Curie-Weiss mean-field equations of the form (S_i)=tanh beta ( Sigma _jJ_ij(S_j)), and have a free energy surface with many minima (but finite in number) separated by infinite energy barriers. They have stable states (in the sense that they have an infinite lifetime in the thermodynamic limit) which are non-ferromagnetic, although the ferromagnetic stable states always have the lowest free energy.

The dynamics of a system quenched below the critical point with explicitly broken symmetry (off-critical quench) is considered in the framework of an O(N) vector model in the large-N limit with non-conserved order parameter. Considering the behaviour of fluctuations in the transverse directions, the author finds ordering and scaling of the transverse structure factor in the intermediate time regime between the usual early and late stages.

The authors consider the Q-state two-dimensional Potts model for Q<g, i.e. in the first order phase transition regime. Following a scheme given by P. Martin (1991), the authors prove an identity between the spectra of the transfer matrices of the Potts model and the transfer matrices connecting the diagonals of a 6-vertex model. By using a Bethe ansatz for the latter, they obtain an exact expression for the correlation length of the Potts model at the transition point.

Two-dimensional invasion percolation simulations into regular fractals are performed to study the dependence of the invading cluster fractal dimension on the geometry of the medium. The fractal dimension of the invading cluster D_i neither depends on the average coordination number of the network, nor on the links of the backbone. The authors find that D_i only depends on the fractal dimension of the backbone of the medium D_bb and varies linearly with it.

The multifractal structure of the flow (channel discharge) distribution is investigated in river network models which are extended versions of Scheidegger's river network model (1967). Two models are proposed: in the first model the constant injection rate in Scheidegger's model is replaced by an uncorrelated random variable and in the second model the injection rate is given by that of a power law L^beta where L is the length in the downstream direction. The effect of the injection rate on the multifractality of the flow distribution is studied by Monte Carlo simulation. It is shown that the partition function Z(q) identical to Sigma _iI_i^q scales as Z(q) approximately=L^{zeta (q)} where I_i is the flow (channel discharge) of water passing over the bond i within the river network and the summation ranges over all bonds. In the first model, for a large L, the multifractal structure of the flow distribution agrees with that of Scheidegger's model. In the second model, it is found that the power law injection rate has an important on the multifractality.

A new approach is proposed to derive a stochastic differential equation from a master equation. Starting from the Liouvillian in a Fock space formulation and using a functional integral representation in terms of coherent states the authors get the corresponding Langevin equation. As examples aggregation and segregation processes with different probability rates are analyzed. In the case of a state dependent growth rate it results in a generalized Kardar-Zhang equation including the kind of noise which is a purely Gaussian one in this model.

Previously, directed animals on square and triangular lattices have been enumerated by area, and have been found to have simple generating functions, whilst the hexagonal lattice generating function has not been obtained. Directed animals on several new lattices are enumerated, one class of which is solved exactly. Directed animals by bonds (with and without loops) are also enumerated. In each case an asymptotic growth like n^-1/2 mu ⁿ is observed and precise estimates for mu are given.

The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T_c, and becomes essentially independent of temperature below T_c. Both in the vicinity of T_c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.

Bethe ansatz equations (BAE) are investigated for the isotropic spin-¹/₂ Heisenberg chain as well as the XXZ chain with real solutions characterized by identical integers in the logarithmic form of the BAE. Such states do not belong to the usual classification scheme (string hypothesis). They always exist in critical sectors with respect to the total spin of the chain and the anisotropy. The conception of holes has to be changed in this case, because there occur more holes than expected. The finite-size corrections for the XXX model are calculated including complex roots and the new type of real solutions with repeating integers. The 'tower structure' predicted by the hypothesis of conformal invariance is shown to remain unchanged.

Lattice nonlinear sigma models, where the Stiefel manifold O(n)/O(n-p) is attached to each lattice site are introduced to analyse phase transitions in frustrated antiferromagnets, with non-collinear spin orderings. A Monte Carlo study of these models in three dimensions indicates either first-order transitions or second-order ones corresponding to new universality classes. There is evidence for a defect mediated transition in some two-dimensional models. None of the results can be reproduced by RG techniques (2+ in and 4- in expansions).

The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.

A cluster hopping mechanism for transport in two-component random networks below p_c is proposed and shown to account for observed conductivity exponents. Key structures forming paths of least resistance are identified. A novel, computationally efficient method for determining exponents results.

Optimal capacities of perceptrons with graded input-output relations are computed within the Gardner approach (1988). The influence of desired output precision, stability with respect to input errors, and output-pattern statistics are analysed and discussed.

Neural networks storing Hadamard patterns have been completely classified with respect to permutation symmetry. The symmetry group of the Hadamard patterns is found to be isomorphic to GL(n,F₂), and the symmetry groups of the networks are explicitly constructed for the most important classes. The volumes of different equivalence classes have been calculated.

The effect of correlations in neural networks is investigated by considering biased input and output patterns. Statistical mechanics is applied to study training times and internal potentials of the MINOVER and ADALINE learning algorithms. For the latter, a direct extension to generalization ability is obtained. Comparison with computer simulations shows good agreement with theoretical predictions. With biased patterns, the authors find a decrease in training times and internal potentials for the MINOVER algorithm, which, however, does not lead to faster storage of a given information measure. In ADALINE training, characteristic times undergo a transition from order 1 to order N at any finite bias, for the learning of patterns as well as for the decay of the generalization error. This leads to a rescaling of the gain parameters.

The authors present the first study of the quantum phase-space behaviour (Wigner functions) for non-hydrogenic atoms in magnetic fields as well as a comprehensive study of spectral properties. They consider primarily an energy regime (scaled energy -0.5) where hydrogen is near-integrable and hydrogenic wavefunctions would be localized on tori. They find that the quantum energy level statistics for non-hydrogenic atoms are at the 'chaotic' (Wigner) limit. However, the quantum phase space distributions, contrary to what one would expect if the underlying classical motion were chaotic, remain dominated by torus-like structures. But the wavefunctions do explore a larger fraction of phase-space than in the hydrogenic case where, in the integrable regime, Wigner wavefunctions are generally localized on a single torus. Due to the non-semiclassical nature of the core they are localized on more than one torus; additional structures other than the tori are also present. Possible interpretations of the results in terms of models of the underlying classical dynamics are discussed.

By using the method proposed by Gomez and Sierra (1991), the author gives the explicit expression of the intertwiner for semiperiodic representations of SU_q(2) where q^N=1. The quantum Clebsch-Gordan coefficient for this case is also discussed.

Some algebraic structures in elliptic solutions of the Yang-Baxter equations are investigated. The author proves the crossing symmetry in Belavin's model (1981) as well as in the A_n-1⁽¹⁾ face model and constructs a new family of L-operators for Belavin's R-matrix as an application.

A computationally effective method for decomposing r-fold tensor products of irreducible representations of U(N) in a basis-independent fashion is given. The multiplicity arising from the tensor decomposition is resolved with the eigenvalues of invariant operators chosen from the universal enveloping algebra generated by the infinitesimal operators of the dual (or complementary) representation. Shift operators which commute with the U(N) invariant operators, but not the dual invariant operators, are introduced to compute the eigenvectors and eigenvalues of the dual invariant operators algebraically. A three-fold tensor product of irreducible representations of SU(4) is decomposed to illustrate the power and generality of the method.

The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.

In activated stochastic processes-such as diffusion in a condensed medium, reaction kinetics, the evolution of the genetic constitution of a population, or the motion of the index cases in a pandemic-there are barriers separating initial and final states. This work investigates the nature of the optimal path between states, such as the path followed by the quickest diffusers, and the timescale associated with it; i.e. the time required to establish communication-and the manner in which that communication takes place. The optimal path is a property of a stochastic matrix defining transition rates in terms of the Pauli-Kolmogorov master equation. The path can therefore be computed, as shown in a numerical case study.

The success of simulated annealing depends strongly upon the choice of a suitable annealing schedule. For a class of small sample systems the optimal annealing schedules are determined. They show distinct scaling behaviour as a function of the number of Metropolis steps carried out at each temperature of the schedule. This behaviour can be traced back to the influence of dominating barriers during cooling. Knowing the optimal schedule for a few different total annealing steps allows one to predict the optimal annealing schedule for intermediate times.

The conditions are found under which the static and nonstatic charge and current densities generate the electromagnetic field confined to the finite region of space. The prescriptions are given for the construction of static magnetic solenoids of an arbitrary geometrical form.

A procedure is given to obtain the quantum state of a free-electron laser with an axial-guide field from a coherent state. It is shown that the coherent state corresponds to the classical trajectories. The small-signal gain is obtained and the effects of the guide field on the gain are bounded in order to keep the orbital and electrostatic stabilities.

By emphasizing a common kinematic core at the expense of the specific dynamics appropriate to individual configurations a unified view is obtained of a large part of the field of interferometry whether using photons or material particles. The only restriction on the coordinate system is that the metric tensor is independent of time. Within this framework a comparison is made of the main features of various types of interferometer, and the effect of dispersion is calculated for the type based on the ring laser oscillator. For a ring interferometer using material particles the particle proper time for a complete transit around the ring is shown to be the same for both directions of travel and a paradoxical consequence is noticed.

The authors investigate an exactly solvable potential class which contains the Coulomb potential as a special case. These potentials have an angular-momentum-independent repulsive core, but retain several important characteristics of the Coulomb problem. Some possible fields of their application are also proposed.

The ideas of supersymmetric quantum mechanics are applied to the nonrelativistic hydrogen atoms to give a first principles derivation of its Lenz vector.

The Pancharatnam phase for displaced number states of the harmonic oscillator is discussed. In particular, it is examined how a single quantum oscillator, driven by a suitable transient external force, evolves form the initial eigenstate u_m(x) to the final, displaced number state modified with a suitable phase factor. The significance of this, usually neglected, phase factor for the solution of the relevant time-dependent Schrodinger equation and for the geometric phase accumulated in the wavefunction during the time evolution of the system is examined. The general expression for the geometric phase for a noncyclic evolution from an initial displaced number state at time t₁ to the final state at time t₂ is derived and two applications, that of delta and harmonic forcing, are worked out. The special case of cyclic evolution is subsequently discussed and, in particular, the conditions leading to such an evolution are derived. The relationship to the classical notion of cyclic evolution is also examined in some detail and it is demonstrated that the general expression for the Pancharatnam phase reduces to the corresponding Berry phase.

It is shown that any chiral superfield has an anomalous dimension equal to zero when all the couplings of the theory in which it is contained are at a renormalization group fixed point. It is argued that super-QED only has such a fixed point at zero coupling, and thence that the theory is trivial.

The connection between the Green function for an isotropic harmonic oscillator in two-dimensional complex space and that for a hydrogen-like atom in the Dirac monopole field plus the Aharonov-Bohm field is established by using path integrals. The connection found provides a simple method for constructing the Green function for atomic systems.

The universal field equations, constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent Lagrangians, are shown to be linearized by a Legendre transformation. This establishes the conjecture that these equations describe integrable systems. While this construction is implicit in general, there exists a large class of solutions for which an explicit form may be written.