Table of contents

We present a method for Baxterizing solutions of the constant Yang–Baxter equation associated with -graded Hopf algebras. To demonstrate the approach, we provide examples for the Taft algebras and the quantum group U_q[sl(2)].

The phenomenon of Fermi acceleration is addressed for a dissipative bouncing ball model with external stochastic perturbation. It is shown that the introduction of energy dissipation (inelastic collisions of the particle with the moving wall) is a sufficient condition to break down the process of Fermi acceleration. The phase transition from bounded to unbounded energy growth in the limit of vanishing dissipation is characterized.

Interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations. Forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area, especially in elasticity and acoustics. The theory has indeed been developed primarily in a quantum context; it is increasingly becoming clear, however, that important applications lie in the field of mechanical vibrations and acoustics. We provide an overview over basic concepts in this emerging field of wave chaos. This ranges from ray approximations of the Green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics—both experimentally and from a theoretical/numerical point of view.

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the eight-vertex model with parameter η given by 2Lη = 2m₁K + im₂K' from m₂ = 0 to the more general case of complex η. We find that there are several different cases depending on the parity of m₁ and m₂.

The third and fourth degree collisional moments for d-dimensional inelastic Maxwell models are exactly evaluated in terms of the velocity moments, with explicit expressions for the associated eigenvalues and cross coefficients as functions of the coefficient of normal restitution. The results are applied to the analysis of the time evolution of the moments (scaled with the thermal speed) in the free cooling problem. It is observed that the characteristic relaxation time toward the homogeneous cooling state decreases as the anisotropy of the corresponding moment increases. In particular, in contrast to what happens in the one-dimensional case, all the anisotropic moments of degree equal to or less than 4 vanish in the homogeneous cooling state for d ⩾ 2.

A trail on the square lattice with a fixed number, k, of vertices of degree 4 is called a k-trail. We model polymer collapse using k-trails by incorporating an interaction energy which is proportional to the number of nearest-neighbour contact edges of the trail. It is known that the number of square lattice n-edge closed (open) k-trails can be bounded above and below (to O(n^k)) by the number of n-step self-avoiding circuits (walks). This along with pattern theorems for self-interacting self-avoiding circuits and walks are used herein to establish upper and lower bounds (to O(n^k)) for the collapsing free energy of k-trails in terms of self-avoiding circuits or walks, as appropriate. We also use pattern theorems to obtain bounds on the limiting nearest-neighbour contact density for collapsing k-trails. Finally, we investigate k-trails with a fixed density of nearest-neighbour contacts and show that their limiting entropy per monomer is independent of k.

We have investigated the properties of a model of 1D anyons interacting through a δ-function repulsive potential. The structure of the quasi-periodic boundary conditions for the anyonic field operators and the many-anyon wavefunctions is clarified. The spectrum of the low-lying excitations including the particle–hole excitations is calculated for periodic and twisted boundary conditions. Using the ideas of the conformal field theory we obtain the large-distance asymptotics of the density and field correlation function at the critical temperature T = 0 and at small finite temperatures. Our expression for the field correlation function extends the results in the literature obtained for harmonic quantum anyonic fluids.

Representations of the quantum superalgebra U_q[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch–Gordan decomposition for the superalgebra U_q[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch–Gordan coefficients are derived, and is observed that they may be expressed in terms of the Q-Hahn polynomials. We next investigate representations of the quantum supergroup OSp_q(1/2) which are not well defined in the classical limit. Employing the universal -matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = −q is satisfied in all cases. Using the Clebsch–Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even-dimensional representations of the quantum supergroup OSp_q(1/2).

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation Y' = A(t)Y. The first one provides a bound on the convergence domain based on the norm of the operator A(t). The second condition links the convergence of the expansion with the structure of the spectrum of Y(t), thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.

Gauss in 1812, in his celebrated memoir on the hypergeometric series, presented a remarkable formula for the psi (or digamma) function, ψ(z), at rational arguments z, which can be expressed in terms of elementary functions. Davis in 1935 extended Gauss's result to the polygamma functions by using a known series representation of ψ⁽ⁿ⁾(z) in an elementary yet technical way. Kölbig in 1996, in his CERN technical report, also gave two extensions to ψ⁽ⁿ⁾(z) by using the series definition of polylogarithm function and the above-known series representation. Here we aim at deriving general formulae expressing ψ⁽ⁿ⁾(z) as rational arguments in terms of other functions, which will be obtained in two ways. In addition, several special cases are also considered and, as a by-product of our main results, we derive, in a simple and unified manner, all formulae given by Gauss, Davis and Kölbig. Finally, it should be noted that all our results, in view of the relationship between ψ⁽ⁿ⁾(z) and the Hurwitz zeta function, ζ(s, a), could be rewritten in the representation of ζ(s, a).

It is shown that under certain conditions the limit speed of electric charges moving in a space of type of dimension one or two, under isotropic friction, is preserved under some perturbations. These results hold when relativistic equations of motion are considered.

The solution corresponding to each initial condition for the dispersionless KP hierarchy can be found from the integration of a Hamilton–Jacobi equation by means of a transformation of coordinates. The solution is explicitly determined in a parametric form as well as a power series in a deformation parameter. In addition, a twistor formulation of the solution is given.

The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant γ involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy.

The algorithm rotating the complex spherical harmonics is presented. The convenient and ready to use formulae for ℓ = 0, 1, 2, 3 are listed. Any rotation in space is determined by the rotation axis and the rotation angle. The complex spherical harmonics defined in the fixed coordinate system is expanded as a linear combination of the spherical harmonics defined in the rotated coordinate system having 2ℓ + 1 terms, which are given explicitly. The derived formulae could be applied in quantum molecular calculations. The algorithm is based on the Cartesian representation of the spherical harmonics. The possible application of the algorithm to the evaluation of molecular integrals between slater type orbitals (STO) is described.

We set up a general formalism for models of spontaneous wavefunction collapse with dynamics represented by a stochastic differential equation driven by general Gaussian noises, not necessarily white in time. In particular, we show that the non-Schrödinger terms of the equation induce the collapse of the wavefunction to one of the common eigenstates of the collapsing operators, and that the collapse occurs with the correct quantum probabilities. We also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse process; such an approximation allows one to compute the leading-order terms for the deviations of the predictions of collapse models with respect to those of standard quantum mechanics. This analysis shows that to leading order, the 'imaginary noise' trick can be used for non-white Gaussian noise.

Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of (N + 1) mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of on finite phase space implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime is also a finite field.

The Lie point symmetries and corresponding invariant solutions are obtained for a Gaussian, irrotational, compressible fluid flow. A supersymmetric extension of this model is then formulated through the use of a superspace and superfield formalism. The Lie superalgebra of this extended model is determined and a classification of its subalgebras is performed. The method of symmetry reduction is systematically applied in order to derive special classes of invariant solutions of the supersymmetric model. Several new types of algebraic, hyperbolic, multi-solitonic and doubly periodic solutions are obtained in explicit form.

From the quantum mechanical viewpoint we derive the dielectric function of an electron plasma system in the presence of a radiation field. By using time-dependent wavefunctions for plasma electrons under the external ac field, we calculate the charge density fluctuation of the electronic system under a weak probing potential and the spectrum of the collective excitation is calculated and found to be strongly dependent upon the amplitude and frequency of the radiation field. We show, in the classical limit, that the reduction of the collective excitation frequency under the radiation field can be associated with the suppression of the plasma high-frequency reactive electrical conductivity. The result is consistent with the recent experimental observation of increased high-frequency mobility in two-dimensional electron gases under a radiation field.

In Gnutzmann et al (2005 J. Phys. A: Math. Gen.38 8921–33) the authors studied the 4-parameter family of isospectral flat 4-tori T^±(a, b, c, d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a, b, c, d). In this note we confirm the expectation expressed in Gnutzmann et al (2005 J. Phys. A: Math. Gen.38 8921–33) by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.

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