Table of contents

The long-range properties of the random flux model (lattice fermions hopping under the influence of maximally random link disorder) are shown to be described by a supersymmetric field theory of non-linear -model type, where the group GL(n|n) is the global invariant manifold. An extension to non-Abelian generalizations of this model identifies connections to lattice QCD, Dirac fermions in a random gauge potential, and stochastic non-Hermitian operators.

We present an example of an exactly soluble bosonic coherent state path integral with non-polynomial action.

We have reconsidered the integrability of the Dodd-Bullough equation in the presence of the boundary condition. It is shown that the boundary condition itself can be deduced from the Lax equation. The forms of the infinite number of nonlocal conservation laws are shown to change due to the presence of the boundary condition.

We study a simple learning model based on the Hebb rule to cope with `delayed', unspecific reinforcement. In spite of the unspecific nature of the information-feedback, convergence to asymptotically perfect generalization is observed, with a rate depending, however, in a non-universal way on learning parameters. Asymptotic convergence can be as fast as that of Hebbian learning, but may be slower. Morever, for a certain range of parameter settings, it depends on initial conditions whether the system can reach the regime of asymptotically perfect generalization, or rather approaches a stationary state of poor generalization.

Diffusion of a non-biased walker through a composite (multi-phase) system is shown to be anomalous for length scales less than the correlation length (i.e., when the path length is measured with a ruler of length less than ) and Gaussian for length scales greater than . The values of and the fractal dimension d_w of the walker path in the anomalous regime reflect the phase properties and phase domain morphology of the composite. They are related to the diffusion coefficient D_w for walker diffusion in the Gaussian regime by D_w ^2-d_w, and to the macroscopic transport coefficient through the relation D_w. The correlation length thus gives the size above which the composite is effectively homogeneous with respect to the transport property of interest. Walker behaviour is compared for disordered (random), particulate-matrix, and labyrinthine two-phase microstructures.

We consider the heat kernel (and the zeta function) associated with Laplace-type operators acting on a general irreducible rank-1 locally symmetric space X. The set of Minakshisundaram-Pleijel coefficients {A_k(X)}_{k = 0} in the short-time asymptotic expansion of the kernel is calculated explicitly.

In the window approach to quasicrystals, the atomic position space E_|| is embedded into a space Eⁿ = E_|| + E. Windows are attached to points of a lattice Eⁿ. For standard fivefold and icosahedral tiling models, the windows are perpendicular projections of dual Voronoi and Delone cells from . Their cuts by the position space E_|| mark tiles and atomic positions. In the alternative covering approach, the position space is covered by overlapping copies of a quasi-unit cell which carries a fixed atomic configuration. The covering and window approach to quasicrystals are shown to be dual projects: D- and V-clusters are defined as projections to position space E_|| of Delone or Voronoi cells. Decagonal V-clusters in the Penrose tiling, related to the decagon covering, and two types of pentagonal D-clusters in the triangle tiling of fivefold point symmetry with their windows are analysed. They are linked, cover position space and have definite windows. For functions compatible with the tilings they form domains of definition. For icosahedral tilings the V-clusters are Kepler triacontahedra, the D-clusters are two icosahedra and one dodecahedron.

An analytical expression of the mean velocity for forced thermal ratchets is obtained under small amplitude of the periodic forcing. It gives quite accurate approximation of the mean velocity, in particular for fast periodic forcing, and reproduces the current reversal. Diffusion ratchets and forced ratchets with state-dependent noise are also considered.

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like - the `electron' is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions.

Firstly, that if the system is also stationary (time independent in some gauge) and isolated (in the sense that the fields belong to a suitable weighted Sobolev space), then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be asymptotically stationary and static.

Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged - all Coulomb external sources are electrically charged magnetic monopoles.

We use coherent states as a time-dependent variational ansatz for a semiclassical treatment of the dynamics of anharmonic quantum oscillators. In this approach the square variance of the Hamiltonian within coherent states is of particular interest. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the semiclassical equations of motion. Moreover, our approach allows for an estimate of the decoherence time of a classical object due to quantum fluctuations. We illustrate our findings with the example of the Toda chain.

The linearized Vlasov equation for a plasma system in a constant external magnetic field and the corresponding linear Vlasov operator are studied. The solution of the Vlasov equation is found by the resolvent method. The spectrum and eigenfunctions of the Vlasov operator are also found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are uncountably infinitely degenerate, which causes difficulty in solving this initial value problem by using the conventional eigenfunction expansion method. It also breaks the natural relation between the eigenfunctions and the resolvent solution in which the eigenfunctions can normally be considered as the coefficients of e^-it in the Laplace (or resolvent) solution.