Table of contents

An infinite range spin-glass-like model for granular systems is introduced and studied through the replica mean-field formalism. Equilibrium, density-dependent properties under vibration and gravity are obtained that qualitatively resemble the results from real and numerical experiments.

The Cayley - Hamilton - Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras defined by a pair of compatible solutions of the Yang - Baxter equation. This class includes the RTT algebras as well as the reflection equation algebras.

A new formula describing the large-order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the Hamiltonian is suggested. A new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested. The branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic, sextic, octic and decadic oscillators.

The existence and uniqueness of generating partitions in non-hyperbolic dynamical systems is usually studied in a simple, exemplary system, namely the Hénon map at standard parameter values a = 1.4 and b = 0.3. We compare its standard partition with three other binary partitions, which are quite different from the standard partition but also appear to be generating. One of these partitions passes twice through the same orbit of homoclinic tangencies, providing a counterexample to a recent conjecture by Jaeger and Kantz ( J. Phys. A: Math. Gen. 30 L567). Introducing some simple rules to manipulate symbolic sequences, we show how to translate symbolic sequences produced by one partition into sequences produced by the other partitions. This proves that all these partitions are as good approximations to generating partitions as the standard partition. We also construct an infinite number of binary partitions, which are all quite similar to the standard partition, derive their translation rules, and prove the same equivalence. It is not known for sure whether any of these partitions is indeed generating. But if one of them is generating, then they all are.

New integrable boundary conditions for integrable quantum systems can be constructed by tuning of scattering phases due to reflection at a boundary and an adjacent impurity and subsequent projection onto subspaces. We illustrate this mechanism by considering a -impurity attached to an open gl(n)-invariant quantum chain and a Kondo spin S coupled to the supersymmetric t-J model.

A particle motion in a 2D periodic potential with the symmetry such that a particular motion can be restricted on the x-axis, subject to the external periodic force in the x-direction, is studied. It is found that by changing the amplitude of the external force, the 1D diffusive motion in the x-direction undergoes the instability at an amplitude, above which the diffusive motion in the y-direction, showing on-off intermittency, is observed. We call it on-off diffusion. By introducing a simple mapping model, the diffusion coefficient in the y-direction is found to take the scaling form slightly above the instability point where and . and L, respectively, are the transverse Lyapunov exponent evaluating the magnitude of instability and its fluctuation in the y-direction. The scaling function h(z) takes the asymptotic form, for and for .

Lattice animals with fugacities conjugate to the number of indepedent cycles, or to the number of nearest neighbour contacts, go through a collapse transition at a -point at a critical value of the fugacity. We examine the phase diagram of a model which includes both a cycle and a contact fugacity with Monte Carlo methods. Using an underlying cut-and-paste Metropolis algorithm for lattice animals, we implement in the first instance a multiple Markov chain simulation of collapsing animals to estimate the location of the collapse transitions and the values of the crossover exponents associated with these. Secondly, we use umbrella sampling to sample animals over a rectangle in the phase diagram to examine the structure of the phase diagram of these animals.

The reversible reactions and are investigated. From the exact Langevin equations describing our model, we set up a systematic approximation scheme to compute the approach of the density of C particles to its equilibrium value. We show that for a sufficiently long time t, this approach takes the form of a power law , for any dimension d. The amplitude A is also computed exactly, but is expected to be model dependent. For uncorrelated initial conditions, the C density turns out to be a monotonic time function. The cases of correlated initial conditions and unequal diffusion constants are investigated as well. In the former, correlations may break the monotonicity of the density or in some special cases they may change the long time behaviour. For the latter, the power law remains valid, only the amplitude changes, even in the extreme case of immobile C particles. We also consider the case of segregated initial condition for which a reaction front is observed, and confirm that its width is governed by mean-field exponent in any dimension.

We formulate a learning algorithm for online learning in neural networks using the extended Kalman filter approach, providing a principled and practicable approximation to the full Bayesian treatment. The latter, which constitutes optimal learning, does not require artificial setting of training parameters and allows for the estimation of a wide range of quantities of interest. We analyse the performance of the algorithm using tools of statistical physics in several scenarios: we look at drifting rules represented by linear and nonlinear perceptrons and investigate how different prior settings affect the generalization performance as well as learnability itself. We investigate the learning behaviour of stationary two-layer network, where the algorithm seems to avoid the, otherwise common, problem of long symmetric plateaus.

We consider the critical dynamics of a system with an n-component nonconserved order parameter coupled to a conserved field with long-range diffusion. An exponent characterizes the long-range transport, being the known locally conserved case. With renormalization group calculations done up to one loop order, several regions are found with different values of the dynamic exponent z in the -n plane. For n<4, there are three regimes, I: nonuniversal, dependent z, II: universal with z depending on n and III: conservation law irrelevant, z being equal to that in the nonconserved case. The known locally conserved case belongs to regions I and II.

We define the directed Abelian sandpile models by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, where c = 1 is for the isotropic case. We calculate some quantities characterizing the self-organized critical states on the one- and two-dimensional lattices. In particular, we obtain the expected number of topplings per added particle, (T), which shows the dependence on the lattice size L as L^x for large L. We show that the critical exponent x does not depend on the dimensionality d, at least for d = 1 and 2, and that when any anisotropy is included in the system x = 1, while x = 2 in the isotropic system. This result gives a rigorous proof of the conjecture by Kadanoff et al (1989 A 39 6524-37) that the anisotropy will distinguish different universality classes. We introduce a new critical exponent, θ, defined by X≡lim_L→∞(T)/L with as for . Both in d = 1 and 2, we obtain θ=1.

Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.

A new two-dimensional map is proposed to investigate the nonlinear and quantum effects on the tunnelling of carriers (electrons or holes) in a novel one-dimensional semimagnetic semiconductor superlattice. The magnetic-polaron effect resulting from the exchange interaction between the carrier in a mesoscopic dot and localized magnetic-ion spins leads to a nonlinear nature of the effective Schrödinger equation for the carrier. We find gaps and different multistability in the tunnelling properties of carriers that depend critically on the wavevector of the injected carriers. In particular, when the nonlinear coefficient is increased new nontunnelling regions appear adjacent to the regular instability regions. The properties can be useful for the transmission of information in microelectronic devices.

A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.

`Antisymmetric' correlation functions of the model of dense lattice polymers are proved to be given by the generalized Kirchhoff theorem. In the continuum limit they coincide with the correlation functions of the free complex Grassmann field that corresponds to the non-unitary conformal field theory (CFT) with c=-2. Explicit expressions for the correlation functions are found. These do not obey standard Wick rules due to the presence of zero mode. Nevertheless, the complex Grassmann field can be considered as primary with conformal weight . It is shown that the natural space of states wherein the operators of the theory act is the Krein space with indefinite metric.

For a finite Lie algebra of rank N, the Weyl orbits of strictly dominant weights contain number of weights, where is the dimension of its Weyl group . For any , there is a very peculiar subset for which we always have

For any dominant weight , the elements of are called permutation weights.

It is shown that there is a one-to-one correspondence between the elements of and where is the Weyl vector of . The concept of the signature factor which enters the Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant , calculation of the character for the irreducible representation will then be provided by multiplicity rules governing the generalized Schur functions. The main idea is again to express everything in terms of the so-called fundamental weights with which we obtain a quite relevant specialization in applications of the Weyl character formula. To provide simplifications in practical calculations, a reduction formula governing the classical Schur functions is also given. As the most suitable example, , which requires a sum over Weyl group elements, is studied explicitly. This will be instructive also for an explicit application of multiplicity rules.

As a result, it will be seen that the Weyl or Weyl-Kac character formulae find explicit applications no matter how large the rank of the underlying algebra.

We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible shear. Our treatment was motivated by the problem of the simulation of catheter navigation in a network of blood vessels, where this description is directly useful. In this context, we derive the Euler differential equations that characterize equilibrium configurations of stretch-free thin rods. We apply perturbation methods, used in time-dependent quantum theory, to the thin rod equations to describe incremental deformations of partially constrained rods. Further, our formalism leads naturally to a new and efficient finite element method valid for arbitrary deformations of thin rods with negligible stretch. Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.

We study a class of eigenfunctions of an analytic difference operator generalizing the special Lamé operator , paying particular attention to quantum-mechanical aspects. We show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume. We establish various orthogonality and non-orthogonality results by direct calculations, generalize the `one-gap picture' associated with the above Lamé operator, and obtain duality properties for the hyperbolic, trigonometric and rational specializations.

A new method for calculating multipole moments with eigenfunctions related to Sturm-Liouville problems is proposed. The method is based on an auxiliary third-order equation and its Laplace transform. The multipole moments for Coulomb and oscillator problems in quantum mechanics are calculated as an application of this method. The relation between the multipole moments for these problems and equations of Heun type is observed.

The classical solution is obtained for the system of two coupled harmonic oscillators with exponentially decaying mass. Using the Feynman path-integral method of quantization an exact propagator for the corresponding quantum system is derived.