Table of contents

An interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments on that and previous analysis are provided here, together with considerations on the results of the purely mathematical zeta-function method, in an attempt at clarifying the issue. Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures.

When dealing with nonlinear time series of car traffic on highways, one of the outstanding problems to be solved is completion and continuation of data in space and time. To this end, the underlying process is decomposed into stochastic and deterministic components. The former is approximated by Gaussian white noise, while the latter refers, apart from always existing trends, to the space- and time-dependent jam propagation process. Jams are modelled in terms of dynamical Bayesian networks with radial basis functions involved. The models developed are used to tackle travel time estimation and prediction. Results are obtained for one of the most crowded traffic areas of Europe, namely the ring-like highway around Cologne.

Based on the equation of state of an ideal Bose gas, the heat capacities at constant volume and constant pressure of the Bose system are derived and used to analyse the low-temperature behaviour of the Bose system. It is expounded that some important thermodynamic processes such as a constant pressure and an adiabatic process cannot be carried out from the region of T > T_c to that of T < T_c, where T_c is the critical temperature of Bose–Einstein condensation of the Bose system. Consequently, some typical thermodynamic cycles such as the Carnot cycle, Brayton cycle, Otto cycle, Ericsson cycle, Diesel cycle and Atkinson cycle cannot be operated across the critical temperature T_c of Bose–Einstein condensation of an ideal Bose gas.

A Bethe Ansatz solution of the open spin- XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.

The high computational cost involved in modelling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve a new large set of linear equations every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algorithm based on the Sherman–Morrison–Woodbury formula for the simulation of progressive fracture in disordered quasi-brittle materials using discrete lattice networks. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to the same order as that of a simple backsolve (forward elimination and backward substitution) using the already LU factored matrix. That is, the computational cost is O(nnz(L)), where nnz(L) denotes the number of non-zeros of the Cholesky factorization L of the stiffness matrix A. This algorithm using the direct sparse solver is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) iterative solvers, and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the critical points. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.

In the tight-binding approximation we consider multi-channel transmission through a billiard coupled to leads. Following Dittes we derive the coupling matrix, the scattering matrix and the effective Hamiltonian, but take into account the energy restriction of the conductance band. The complex eigenvalues of the effective Hamiltonian define the poles of the scattering matrix. For some simple cases, we present exact values for the poles. We derive also the condition for the appearance of double poles.

We present a method to analytically compute means of functions on regular n-gons and to study cyclic quantities of the complex variable. To achieve this, we construct representations of complex functions and compact expressions of their mean based on the use of a scalar product. Applied in the field of celestial mechanics, this method leads to results concerning gravitational potential and relative equilibrium composed by nested polygons.

In early work, March and Murray gave a perturbation theory of the Dirac density matrix γ(r, r') generated by a localized potential V(r) embedded in an initially uniform Fermi gas to all orders in V(r). For potentials sufficiently slowly varying in space, they summed the resulting series for r' = r to regain the Thomas–Fermi density ρ(r) ∝ [μ − V(r)]^3/2, with μ the chemical potential of the Fermi gas. For an admittedly simplistic repulsive central potential V(r) = |A|exp(−cr), it is first shown here that what amounts to the sum of the March–Murray series for the s-wave (only) contribution to the density, namely ρ_s(r, μ), can be obtained in closed form. Furthermore, for specific numerical values of A and c in this exponential potential, the long-range behaviour of ρ_s(r, μ) is related to the zero-potential form of March and Murray, which merely suffers a μ-dependent phase shift. This result is interpreted in relation to the recent high density screening theorem of Zaremba, Nagy and Echenique. A brief discussion of excess electrical resistivity caused by nonlinear scattering in a Fermi gas is added; this now involves an off-diagonal local density of states. Finally, for periodic lattices, contact is made with the quantum-mechanical defect centre models of Koster and Slater (1954 Phys. Rev.96 1208) and of Beeby (1967 Proc. R. Soc. A 302 113), and also with the semiclassical approximation of Friedel (1954 Adv. Phys.3 446). In appendices, solvable low-dimensional models are briefly summarized.

We study a similarity reduction of the modified Yajima–Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A⁽¹⁾₂. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom 2, and admits a group of Bäcklund transformations isomorphic to the affine Weyl group of type A⁽¹⁾₂. We show that the system is equivalent to a two-parameter family of the fifth Painlevé equation.

We critically analyse the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and vice versa.

We exhibit a two-parameter class of states ρ_(α,γ) in a 2 ⊗ n quantum system for n ⩾ 3, which can be obtained from an arbitrary state by means of local quantum operations and classical communication, and which are invariant under all bilateral operations on a 2 ⊗ n quantum system. We calculate the negativity of ρ_(α,γ), and a lower bound and a tight upper bound on its entanglement of formation. It follows from this calculation that the entanglement of formation of ρ_(α,γ) cannot exceed its negativity.

The non-Hermitian Hamiltonian p² + x⁴ + iax, which spontaneously breaks PT-symmetry, and the subject of a recent study by Bender et al (2001 J. Phys. A: Math. Gen.34 L31), is amenable to a positivity representation, facilitating the generation of converging bounds to the complex-eigenenergies of the PT-breaking states. This system is much easier (i.e. fewer variational parameters) than the previously studied case of the Hamiltonian p² + ix³ + iax (2001 Handy J. Phys. A: Math. Gen.34 5065, Handy et al 2001 J. Phys. A: Math. Gen.34 5593), as first proposed by Delabaere and Trinh (2000 J. Phys. A: Math. Gen.33 8771), enabling the generation of low order algebraic spectral bounds (i.e. ), in addition to high order, numerically generated, converging bounds to the discrete states. We examine both approaches here.

We prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds. From a knowledge of the heat kernel in an infinite cylinder, some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension.

Due to an oversight, reference [13] was omitted from the above article:

[13] Smailagic E and Spallucci E 2003 J. Phys. A: Math. Gen.36 L467-L472