Table of contents

A connection is made between the random-turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE (random Hermitian matrices). The scaling of the distribution gives the maximum mean displacement µ after t time steps as µ = (2t)^1/2 with standard deviation proportional to µ^1/3. The exponent 1/3 is typical of a large class of two-dimensional growth problems.

A bicomplex structure is associated with the Leznov-Saveliev equation of integrable models. The linear problem associated with the zero-curvature condition is derived in terms of the bicomplex linear equation. The explicit example of a non-Abelian conformal affine Toda model is discussed in detail and its conservation laws are derived from the zero-curvature representation of its equation of motion.

We classify the Fibonacci chains (F-chains) by their index sequences and construct an approximately finite-dimensional (AF) C*-algebra on the space of F-chains as Connes did on the space of Penrose tiling. The K-theory on this AF algebra suggests a connection between the noncommutative torus and the space of F-chains. A noncommutative torus, which can be regarded as the C*-algebra of a foliation on the torus, is explicitly embedded into the AF algebra on the space of F-chains. As a counterpart of that, we obtain a relation between the space of F-chains and the leaf space of Kronecker foliation on the torus using the cut-procedure of constructing F-chains. Our embedding of the C*-algebra of the foliation is consistent with the recent result of Landi, Lizzi, and Szabo that the C*-algebra of noncommutative torus can be embedded into an AF algebra.

We show that the spectrum of the Schrödinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.

The symmetric two-layer Ising model (TLIM) is studied by the corner transfer matrix renormalization group method. The critical points and critical exponents are calculated. It is found that the TLIM belongs to the same universality class as the Ising model. The shift exponent is calculated to be 1.773, which is consistent with the theoretical prediction of 1.75 with 1.3% deviation.

Using a description of defects in solids in terms of three-dimensional gravity, we will consider a charged quantum particle with spin-½ in the field of a finite magnetic flux in a space with a disclination. It will be demonstrated that the change in topology caused by the defect produces modifications in the energy spectrum of a charged particle which in this case depends on the magnetic flux and on the global aspects.

We show that the usual expression found in the literature, which tries to generalize the `Fermi golden rule' beyond second order in perturbation, is (surprisingly) incorrect. After identifying the weak steps of the two usual derivations, we derive a new expression of this generalized golden rule, intrinsically very different from the previous one, even though its form may look similar. We show that this new result already affects the next nonzero term of the transition probability expansion. From a direct comparison of this new result with its exact value - as given by the exact evolution operator - we also show that the usual approaches are inherently quite questionable beyond the second order in perturbation.

The super Moyal-Lax representation and the super Moyal momentum algebra are introduced and the properties of simple and extended supersymmetric integrable models are systematically investigated. It is shown that, much like in the bosonic cases, the super Moyal-Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. Similarly, we show that the parameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system. The super Moyal-Lax description allows us to go to the dispersionless limit of these models in a singular limit and we discuss some of the properties of such systems.

In this work we study the influence of a cosmic string on the classical and quantum dynamics of an electric dipole. We find the Lagrangian and the classical scattering. The Schrödinger equation for this problem is also solved and the quantum scattering is determined.

We introduce the dual vector | z,f (n)⟩_* of a single-mode nonlinear coherent state (NCS) by using the contour integral form of δ-function, and the explicit form of | z,f (n)⟩_* is obtained in enlarged Hilbert space. The nonlinear Fock states and their contour representation are deduced. A new completeness relation composed of | z,f (n)⟩_* and NCS in contour integral form is derived and its application in constructing a complex P-representation of density operators is shown.

We use symmetry vectors of nonlinear field equations to build alternative Hamiltonian structures. We construct such structures even for equations which are usually believed to be non-Hamiltonian such as heat, Burger and potential Burger equations. We improve on a previous version of the approach using recursion operators to increase the rank of the Poisson bracket matrices. Cole-Hopf and Miura-type transformations allow the mapping of these structures from one equation to another.

It is shown that time-independent circular currents and uniformly rotating charge distributions create heretofore unreported constant electric and magnetic fields associated with radial acceleration of the charges forming the circular currents and with radial acceleration of the charges comprising the rotating charge distributions. These fields are computed for several types of rotating charge distributions and for several types of circular currents. One of the consequences of the existence of these fields is that the Aharonov-Bohm effect can now be explained on the basis of classical electrodynamics.

We determine the iterative solution, on the complex plane, of the µ-holomorphy equation. Then we obtain the µ-holomorphic projective connection as a Neumann series in powers of the Beltrami differential µ. Since it has been shown that the Polyakov action of a conformal model with a central charge k is expressed in terms of the µ-holomorphic projective connection, we then prove the Polyakov conjecture.

The Leibniz rule for the fractional Riemann-Liouville derivative is studied in the algebra of functions defined by Laplace convolution. This algebra and the derived Leibniz rule is used in construction of an explicit form of stationary-conserved currents for linear fractional differential equations. The examples of fractional diffusion in 1+1 and fractional diffusion in d + 1 dimensions are discussed in detail. The results are generalized to the mixed fractional-differential and mixed sequential fractional-differential systems for which the stationarity-conservation laws are obtained. The derived currents are used in construction of stationary nonlocal charges.

We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with photo-detectors and passive interferometry (beamsplitters, polarization rotations, phase-shifters, etc). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of single-photon polarization states can be modelled using this description.

We present the exact solution for a set of nonlinear algebraic equations 1/z_l = πd + (2d/n)∑_m≠l1/(z_l-z_m). These were encountered by us in a recent study of the low-energy spectrum of the Heisenberg ferromagnetic chain. These equations are low-d (density) `degenerations' of a more complicated transcendental equation of Bethe's ansatz for a ferromagnet, but are interesting in themselves. They generalize, through a single parameter, the equations of Stieltjes, x<sub>l</sub> = ∑<sub>m≠l</sub>1/(x<sub>l</sub>-x<sub>m</sub>), familiar from random matrix theory. It is shown that the solutions of these set of equations are given by the zeros of generalized associated Laguerre polynomials. These zeros are interesting, since they provide one of the few known cases where the location is along a nontrivial curve in the complex plane that is determined in this work. Using a `Green function' and a saddle point technique we determine the asymptotic distribution of zeros.

A factor ρ(1−ρ) was missing in formula (77) for the off-diagonal long-range order parameter ω₀. This factor is the saturation value of the order parameter in the case of hard-core interactions. Equation (77) correctly reads