Table of contents

The possibility of detecting mutations in DNA from force measurements (as a first step towards sequence analysis) is discussed theoretically based on exact calculations. The force signal is associated with the domain wall separating the zipped from the unzipped regions. We propose a comparison method ('differential force microscope') to detect mutations. Two lattice models are treated as specific examples.

Modified Rayleigh conjecture (MRC) in scattering theory is proposed and justified. MRC allows one to develop numerical algorithms for solving direct scattering problems related to acoustic wave scattering by soft and hard obstacles of arbitrary shapes. It gives an error estimate for solving the direct scattering problem. It suggests a numerical method for finding the shape of a star-shaped obstacle from the scattering data.

We find transformation matrices allowing us to express a noncommutative three-dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following the 'philosophy of simplicity' approach. Noncommutative parameters have a physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.

An orthonormal set of functions is defined on a von Neumann lattice in phase space. There is one function assigned to each unit cell of area h. The functions are of the Wilson type in that in the x-direction they are obtained from one another by uniform translations, while in the p-direction they are double-peaked, and cannot be obtained by translations. A similar construction is also carried out with the x- and p-axes interchanged. The results apply to any dimension. An explicit example is worked out for the ground state of a harmonic oscillator, and a relation to coherent states is pointed out.

A criterion for the uniqueness of limiting Gibbs states in classical models with unique ground states is formulated. Various applications of this criterion formulated in the terminology of percolation theory are discussed.

The critical behaviour of two-dimensional stochastic lattice gas models with C_3v symmetry is analysed. We study the cumulants of the order parameter for the three-state (equilibrium) Potts model and for two irreversible models whose dynamic rules are invariant under the symmetry operations of the point group C_3v. By means of extensive numerical analysis of the phase transition we show that irreversibility does not affect the critical behaviour of the systems. In particular, we find that the Binder reduced fourth-order cumulant takes a universal value U* which is the same for the three-state Potts model and for the irreversible models. The same universal behaviour is observed for the reduced third-order cumulant.

We present a systematic analytical approach to the trapping of a random walk by a finite density ρ of diffusing traps in arbitrary dimension d. We confirm the phenomenologically predicted e^−c_dρtd/2 time decay of the survival probability, and compute the dimension-dependent constant c_d to leading order within an ε = 2 − d expansion.

A solution for the inhomogeneous telegraph equation for a point source moving with the velocity of light is constructed. We find relations describing both the transient and steady-state wave processes. The solutions obtained are used to define electromagnetic waves in a conductive medium. The case of a source moving faster than light is also given.

We discuss the Schrödinger equation in the presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed.

We study the pole structure of the ζ-function associated with the Hamiltonian H of a quantum mechanical particle living in the half-line ⁺, subject to the singular potential gx⁻² + x². We show that H admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter g. The ζ-functions of these operators present poles which depend on g and, in general, do not coincide with half an integer (they can even be irrational). The corresponding residues depend on the SAE considered.

The quasi-mutual entropy in the Jaynes–Cummings model is rigorously derived without using the diagonal approximation. The variation of the correlation in this model for the time development and the statistical mixture parameter is discussed.

In recent years, it has been shown that Lotka–Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as physics, population dynamics, chemistry and economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case. It will be demonstrated that Lotka–Volterra mappings play a central role in the quasipolynomial formalism for the discrete-time case. Moreover, the extension of the formalism into the discrete-time domain allows a significant generalization of Lotka–Volterra mappings as well as a whole transfer of algebraic methods into the discrete-time context. The result is a novel and more general conceptual framework for the understanding of Lotka–Volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of Lotka–Volterra mappings and their generalizations, but also for the development of new applications.

The semi-classical quantization of the two lowest energy static solutions of the boundary sine-Gordon model is considered. A relation between the Lagrangian and bootstrap parameters is established by comparing their quantum corrected energy difference and the exact one. This relation is also confirmed by studying the semi-classical limit of soliton reflections on the boundary.

Self-dual Yang–Mills equations on noncommutative spaces associated with pseudo-Euclidean space of signature (2, 2) are shown to be related via dimensional reductions to noncommutative formulations of Toda equations, of generalized nonlinear Schrödinger (NS) equations, of the super-Korteweg–de Vries (super-KdV) as well as of the matrix KdV equations. The noncommutative extensions of their linear systems and bicomplexes associated with conserved quantities are discussed as well. A q-plane version of the KdV equation with linear system is also shown.

Recently Ercolani and McLaughlin proved that the zeros of the biorthogonal polynomials with the weight function w(x, y) = exp(−V(x) − W(y) − 2τxy) are all real and distinct, and Mehta has extended their argument to the weight function w(x, y) = e^−x−y/(x + y) and to the more general case of the convolution (w₁ * w₂ * ... * w_m)(x, y), where w_i are functions of the same form as above. Using the concept of total positive and sign-regular functions, we further extend the argument to a large class of weight functions. Many examples are presented, including several whose pair of biorthogonal polynomials turn out to come from different families of classical orthogonal polynomials.

A class of quantum field theories invariant with respect to the action of an odd vector field Q on a source supermanifold Σ is considered. We suppose that Q satisfies the conditions of one of the localization theorems (Szabo R 1996 Equivariant localization of path integrals Preprint hep-th/9608068). The Q-invariant sector of a field theory from the above class is then shown to be equivalent to the quantum field theory defined on the zero locus of the vector field Q.

It is argued that the method recently proposed by Los (Los V F 2001 J. Phys. A: Math. Gen.34 6389) turning inhomogeneous time-convolution generalized master equations (GMEs) into homogeneous ones is problematic in the quantum case. It is also shown here that the proposed method is inapplicable to time-convolutionless GMEs.