Table of contents

We show that computing the coefficients of the Taylor expansion of the solution of the off-equilibrium dynamical equations characterizing models with quenched disorder is a very effective way to understand the long-time asymptotic behaviour. We study the p=3 spherical spin-glass model, and we compute the asymptotic energy (in the critical region and down to T=0) and the coefficients of the time decay of the energy.

The traffic-flow problem in two dimensions is formulated as a three-state model on a square lattice in terms of Pauli operators. Using a Fock-space representation of the master equation we get the Liouvillian for the problem with asymmetric exclusion. Three different realizations, symmetric, right-before-left model and an exchange model will be analysed within the mean-field approximation (MFA). The resulting kinetic equations for the average occupation number of cars in the upward and sideward directions are coupled. The average velocity can be calculated in MFA. It results in a jamming transition that depends on the total concentration of cars.

A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.

A generalized real-space renormalization scheme is developed for geometrical critical phenomena. The renormalization group is parametrized by the standard length-scaling factor and a new rectangular area-fraction factor. This rectangular renormalization scheme utilizes relatively small rectangular sublattices to effectively renormalize large square lattices. With the area-fraction factor, one can systematically study rectangular generalizations of the conventional square-cell renormalization theories. Application to self-avoiding random walks yields critical descriptors that are comparable to, and in most cases better than previous results obtained from more complex renormalization schemes.

An approach is developed for the calculation of fluctuations in an ensemble of systems obeying Maxwell-Boltzmann, Fermi-Dirac or Bose-Einstein statistics, and in which a specified set of physical quantities is conserved. The technique is applied to two situations of interest: in the first only energy is conserved while in the second both energy and number are conserved.

The performance of three recently presented training algorithms in neural networks is investigated. These algorithms are robust to infeasible problems, in which case an appropriate error function is minimized. In the infeasibility regime, simulations are performed and compared to recently published analytical work in one-step replica symmetry broken theory. A careful analysis explains insufficiencies in these analytic results. A new stability result in the infeasibility regime is derived and shown to match simulation data.

The Gelfand-Zetlin basis for representations of U_q(sl(N)) is improved to better fit the case when q is a root of unity. The usual q-deformed representations, as well as the nilpotent, periodic (cyclic), semi-periodic (semi-cyclic) and also some atypical representations are now described with the same formalism.

A new transformation is defined that connects a function and its Legendre transform by means of a continuous free parameter. The cyclic behaviour of consecutive Legendre transformations is reflected in the periodic dependence of the new transform on this parameter. This transformation opens new options wherever the conventional Legendre transformation is used (including mechanics, thermodynamics and optics) and is suggestively derived here by considering the geometrical-optics limit of a diffraction integral. The connection to a classical limit of the fractional Fourier transformation is also established and the mathematical and geometrical properties of the transformation are demonstrated.

The nonlinear dynamics of both the isotropic and single ion anisotropic Heisenberg ferromagnetic spin chains with Dzialoshinski-Moriya-type weak anisotropic interaction has been studied in the classical continuum limit. In both cases integrable weak ferromagnetic models exhibiting soliton-like elementary spin excitations have been identified. A class of spin wave solutions have also been reported in the anisotropic case.

We describe a form of piecewise deterministic dynamics, where solutions evolve deterministically throughout most of phase space, but, in the presence of noise, make nondeterministic jumps to other solutions when the trajectory passes near a singularity in the equations of motion. The type of singularity we consider in this paper is a single point where the Lipschitz conditions fail and many closed-loop trajectories share a common tangent point. It is shown that there is a finite uncertainty associated with the behaviour that is independent of the magnitude of the noise. The long-term behaviour, while similar in appearance to deterministic chaos, has rather different implications for prediction and control.

In the jet-bundle description of first-order classical field theories there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior choice of a connection. Bearing these facts in mind, we analyse the situation in the jet-bundle description of time-dependent classical mechanics. We prove that this connection-dependence also occurs in this case, although it is usually hidden by the use of the "natural" connection given by the trivial bundle structure of the phase spaces under consideration. However, we also prove that this dependence is dynamically irrelevant, except where the dynamical variation of the energy is concerned. In addition, the relationship between first integrals and connections is shown for a sufficiently large class of Lagrangians.

All systems of (n+1)-dimensional quasilinear second-order evolution equations invariant under chain of algebras AG(1.n) contained in/implied by AG₁(1.n) contained in/implied by AG₂(1.n) are described. The results obtained are illustrated by the examples of the nonlinear Schrodinger equations, Hamilton-Jacobi-type systems and reaction-diffusion equations.

We compute the eta function for Chern-Simons quantum field theory with complex gauge group. The calculation is performed using the Schwinger expansion technique. We discuss, in particular, the role of the metric on the field configuration space, and demonstrate that for a certain class of acceptable metrics the one-loop phase contribution to the effective action can be calculated explicitly. The result is found to be proportional to a gauge invariant part of the action.

The level-1 integrable highest weight modules of U_q(sl₂) admit a level-0 action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-1 module generated by components of vertex operators. Each level-1 module is a direct sum of finite-dimensional irreducible level-0 modules, whose highest weight vector is expressed in terms of Macdonald polynomials. This decomposition leads to the fermionic character formula for the level-1 modules.

Two representations in multifractal analysis, the so-called q and tau representations, are discussed theoretically and computed practically. Complementary to the standard q-representation, the so-called tau -representation is especially suited to resolving the most rarified subsets of the distributed measure. Moreover, these two representations are especially adapted, respectively, to the well known fixed-size and fixed-mass box-counting algorithms. Both strategies are first applied to iteratively constructed mathematical measures. Once tested in this way, we use them to analyse the mass distribution and the growth probability distribution of an experimental electrodeposited pattern.

Embedding logical operations in non-dissipative physical processes requires the use of reversible logic. Following Feynman's (1986) approach, the sixteen distinct truth tables of classical logic are shown to be contained in the 8! reversible logic operations covered by the symmetric group S₈, which permute the eight values of three logical variables. Small subgroups of S₈ are shown to cover, respectively, reversible logic, reversible switching and reversible arithmetic. A new universal primitive is found which generates a covering group of reversible logic. It is shown that the octahedral group in four dimensions covers both reversible logic and switching and, hence, that the orthogonal group O(4) provides a covering group for quantum gates.

Two of the simplest integrable Hamiltonians H(x,y,p_x,p_y,)=(p_x²+p_y²)/2+V(x,y) with a second integral quartic in the momenta are those with potentials V₃(x,y)=by(3x²+16y²)+d(x²+16y²)+ eta y and V₄(x,y)=a(x⁴+6x²y²+8y⁴)+c(x²+4y²)+vy^-2. We show how V₃ can be obtained from V₄. In the process we obtain a new potential of the class, V_N, that includes both V₃ and V₄ as particular cases. For this potential we give the second integral of motion, separating variables, a Lax representation and a bi-Hamiltonian structure, thus synthesizing the corresponding results for potentials V₃ and V₄. The integrable extension V_N+ mu x^-2 is also discussed.

A theoretical investigation is made of static and dynamic effects when a nematic liquid crystal is subjected to crossed electric and magnetic fields. In the static problem a twist-wall solution is discussed for a semi-infinite sample of nematic; a control parameter, q, describes the relationship between the fields and their crossed angle and is used to characterize the solution. For an infinite sample of nematic this parameter also turns out to characterize the types of solution (travelling waves) which are available for the nonlinear dynamic equation when certain approximations are made. The type of solution which occurs is shown to depend crucially on the boundary conditions, the relative magnitudes of the electric and magnetic fields and their crossed angle.

Quasi-energy spectral series ( epsilon _nu (h(cross)), Psi ( epsilon _nu )) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross)^-1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma _{epsilon ( nu} ) to the quasi-energy spectrum is calculated. It is shown that the gamma _{epsilon ( nu} ) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.

In this paper, the KdV-type equations with variable coefficients appearing a lot in the literature are discussed generally. The explicit transformations which transform various KdV, mKdV and KP-type nonlinear evolution equations into their "canonical" forms with constant coefficients are given. The results presented here tie up many of the investigations scattered in the literature.

A new interpretation of the numerical data for self-avoiding walks in critical dimensions is suggested on the basis of a different renormalization scheme for the random walk with a long-term correlation.

For original paper see ibid., vol. 28, p. 5685 (1995). We show that the logical basis of using a different renormalization scheme is flawed compared to standard renormalization group predictions.

For original paper see ibid., vol. 24, p. 4721 (1991). In this note, we consider further the unified formulation of the spectra of temperature fluctuations in isotropic turbulence given by us previously. We indicate restrictions imposed on the model parameters by general physical considerations.