Table of contents

In a number of recent papers, the spectral properties of the Laplacian on randomly connected graphs have been studied within the replica formalism. In this letter we show how the replica formalism can be unravelled and we find an approximation for calculating the density of states which substantially simplifies the numerical resolution of the equations obtained, whilst giving results in excellent accord with the exact solution.

A simple and adaptable closure algorithm for the edge nodes of a lattice Boltzmann fluid simulation space is presented. Its rules are designed to be correct at every instant, to maintain local mass and to produce a specified fluid velocity and, crucially, the correct strain rate tensor at the resulting fluid boundary. Further, our algorithm models the fluid on boundary nodes to the same accuracy as on the bulk nodes and in a demonstrably equivalent manner, requiring only a specified boundary velocity, the fluid boundary pressure emerging. Illustrative results for steady and time-dependent flows, together with outline generalizations, are presented.

In previous work 0305-4470/29/19/031 (Skála L and Cízek J 1996 J. Phys. A: Math. Gen. 29 L129, 1996 J. Phys. A: Math. Gen. 29 6467), a new method of calculating perturbation energies for one-dimensional problems based on the linear dependence of the perturbation wavefunctions on the perturbation energies has been suggested. It is shown in this letter that this method can be extended to multi-dimensional problems and the linearity can be used not only at a boundary point but also at an arbitrary point inside the integration region. Degenerate eigenvalues are also discussed. The resulting perturbation theory is very simple and can be used at large orders.

We assign to any cofactor system a whole hierarchy of such systems. A sufficient condition for their complete integrability is given. The hierarchies admit construction of non-trivial integrable systems from trivial ones.

We present a six-loop calculation for Wilson's β-function and for the crossover exponent φ of the tricritical O(n) symmetric ϕ⁶ theory in d = 3 − dimensions. The counterterms of all but one of the 29 diagrams can be calculated analytically, while for one diagram high-precision numerical calculations together with the PSLQ integer relation search algorithm was used to come up with an analytical result. The divergences of the dimensionally regularized diagrams are removed by minimal subtraction.

Monte Carlo simulations have been used to explore the effects of the Eley-Rideal mechanism (reaction of CO molecule with already chemisorbed oxygen atom to produce CO₂) on a simple Langmuir-Hinshelwood model for the NO-CO catalytic reaction on a square surface. The diffusion of the CO and N adatoms on the surface and desorption of CO from the surface are also introduced into the model. Without diffusion and desorption, the model generates a very small reactive window of the order of 0.033. The moment CO partial pressure (y_CO) departs from zero, continuous production of CO₂ and N₂ starts. A first-order transition terminates the catalytic activity at y_CO = 0.033 and the surface is poisoned with a combination of CO and N. However, the diffusion of the N atom and CO molecule shifts the transition point from 0.033 to higher values of y_CO. The introduction of desorption of CO shows some interesting results. A very small desorption probability of CO (=0.01) increases the width of the reactive window to 0.12. However, this reactive window is separated by two transition points y₁(≈0.2) and y₂(≈0.32). For y_CO<y₁ (y_CO>y₂) the surface is poisoned by a combination of O and N (CO and N). With further increase in desorption probability the width increases significantly.

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n = 17. The crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent ϕ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that ϕ is equal to the Fisher exponent σ governing the size distribution of large directed percolation clusters.

We analyse the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the 'quantum coin toss' in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.

We study by Monte Carlo simulation the short-time exponent θ in an antiferromagnetic Ising system for which the magnetization is conserved but the sublattice magnetization (which is the order parameter in this case) is not. This system belongs to the dynamic class of model C. We use nearest-neighbour Kawasaki dynamics so that the magnetization is conserved locally. We find that in three dimensions θ is independent of the conserved magnetization. This is in agreement with the available theoretical studies, but in disagreement with previous simulation studies with a global conservation algorithm. However, we agree with both these studies regarding the result θ_C≠θ_A. We also find that in two dimensions, θ_C = θ_A.

We study self-programming in recurrent neural networks where both neurons (the 'processors') and synaptic interactions ('the programme') evolve in time simultaneously, according to specific coupled stochastic equations. The interactions are divided into a hierarchy of L groups with adiabatically separated and monotonically increasing time-scales, representing sub-routines of the system programme of decreasing volatility. We solve this model in equilibrium, assuming ergodicity at every level, and find as our replica-symmetric solution a formalism with a structure similar but not identical to Parisi's L-step replica symmetry breaking scheme. Apart from differences in details of the equations (due to the fact that here interactions, rather than spins, are grouped into clusters with different time-scales), in the present model the block sizes m_i of the emerging ultrametric solution are not restricted to the interval [0, 1], but are independent control parameters, defined in terms of the noise strengths of the various levels in the hierarchy, which can take any value in [0, ∞⟩. This is shown to lead to extremely rich phase diagrams, with an abundance of first-order transitions especially when the level of stochasticity in the interaction dynamics is chosen to be low.

A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton-Jacobi method and Jacobi's theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.

We study a family of chaotic maps with limit cases—the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius–Perron operator in different function spaces including spaces of analytical functions and study numerically the eigenvalues and eigenfunctions.

Making use of a dynamical invariant operator, we obtain a quantum mechanical solution of a damped harmonic oscillator having time-dependent frequency with and without inverse quadratic potential. We confirm that the uncertainty relation is always larger than ℏ/2 in both the number and thermal state. We obtain a density operator satisfying the Liouville-von Neumann equation, and use its diagonal elements to calculate various expectation values in the thermal state.

In a recent paper (Del Sol Mesa A and Quesne C 2000 J. Phys. A: Math. Gen.33 4059), we started a systematic study of the connections among different factorization types, suggested by Infeld and Hull, and their consequences for the construction of algebras. We devised a general procedure for constructing satellite algebras for all the Hamiltonians admitting a type E factorization by using the relationship between type A and E factorizations. Here we complete our analysis by showing that for Hamiltonians admitting a type F factorization, a similar method, starting from either type B or type C factorization, leads to other types of algebras. We therefore conclude that the existence of satellite algebras is a characteristic property of type E factorizable Hamiltonians. Our results are illustrated with the detailed discussion of the Coulomb problem.

We estimate an unknown qubit from the long sequence of n random polarization measurements of precision Δ. Using the standard Ito stochastic equations of the a posteriori state in the continuous measurement limit, we calculate the advancement of fidelity. We show that the standard optimum value 2/3 is achieved asymptotically for n ≫ Δ²/96 ≫ 1. We append a brief derivation of novel Ito equations for the estimate state.

Expansions in series of Coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations (GSWEs) are analysed and written together in pairs. Each pair consists of a solution in series of hypergeometric functions and another in series of Coulomb wavefunctions and has the same recurrence relations for the series coefficients, but the solutions inside it present different radii of convergence. Expansions without a phase parameter are derived by truncating the series with a phase parameter. For the Whittaker-Hill equation, solutions are found by treating that equation as a particular case of GSWE while, for the confluent GSWE, solutions, with and without a phase parameter, are given as pairs of series of Coulomb wavefunctions. Amongst the applications there are equations for the time dependence of Dirac test fields in some nonflat Friedmann-Robertson-Walker spacetimes, the radial Schrödinger equation for an electron in the field of two Coulombian centres and the Schrödinger equation for the Razavy-type quasi-exactly solvable potentials. For these problems it is possible to find wavefunctions in terms of infinite series, regular and convergent over the entire range of the independent variable, by matching expansions belonging to one or more of the above pairs. The infinite-series solutions for the Razavy-type potentials, in addition to the polynomial ones, suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi-classical methods of approximation.

We develop a new conception for the quantum mechanical arrival time distribution from the perspective of Bohmian mechanics. A detection probability for detectors sensitive to quite arbitrary spacetime domains is formulated. Basic positivity and monotonicity properties are established. We show that our detection probability improves and generalizes an earlier proposal by Leavens and McKinnon. The difference between the two notions is illustrated through application to a free wavepacket.

We construct Poisson structures for Ermakov systems using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them is degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.

In this paper, we set up a bimodule of the algebra on a fuzzy sphere. On the basis of differential operators in a moving frame, we generalize the ABS construction to the fuzzy-sphere case. By using the ABS construction, we obtain new soliton solutions for several physical systems.

We perform variable separation in the quasi-potential systems of equations of the form = −A⁻¹ ∇k = −Ã⁻¹∇, where A and à are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux–Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions.

This paper studies the application of the structure theory of infinite-dimensional pseudo-groups to computing symmetries of differential equations. The main tool is a combination of Cartan's method of equivalence and the moving coframe method introduced by Fels and Olver. Our approach does not require a preliminary computation of infinitesimal defining systems, their analysis and integration, and uses differentiation and linear algebra operations only. Examples of its main features are given.

For the quantum two-dimensional isotropic harmonic oscillator we show that the Infeld–Hull radial operators, as well as those of the supersymmetric approach for the radial equation, are contained in the constants of motion of the problem.

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field ϕ of dimension two. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of ϕ is shown to coincide with a simpler one, generated by a bilocal scalar field V(x₁,x₂) of dimension (1,1). The modes of V together with the unit operator span an infinite-dimensional Lie algebra _V whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of _V) is proven to be equivalent to c∊.

We have investigated analytically quantum tunnelling of large spin in the biaxial spin system with the magnetic field applied along the hard and medium anisotropy axes by using Schrödinger's interpretation of quantum mechanics. When the magnetic field parallels the hard axis, the tunnel splittings of all the energy level pairs become oscillatory as a function of the magnetic field. The quenching points are completely determined by the coexistence of solutions of Ince's equation. When the magnetic field points to the medium axis, the tunnel splitting oscillations disappear due to the absence of coexistence of solutions. These results coincide with the recent experimental observations in the nanomagnet Fe₈.

We shall try to show a relation between black hole (BH) entropy and topological entropy using the famous Baum-Connes conjecture for foliated manifolds which are particular examples of noncommutative spaces. Our argument is qualitative and it is based on the microscopic origin of the Beckenstein-Hawking area-entropy formula for BHs, provided by superstring theory, in the more general noncommutative geometric context of M-theory following the approach of Connes-Douglas-Schwarz.