Table of contents

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Neil Scriven Publisher Journal of Physics A: Mathematical and Generalneil.scriven@iop.org

We use an intertwining property of linear differential operators to construct the general solution of a type of coupled Ermakov-Pinney system with two distinct frequency terms.

Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and unbraided (usual) Yang-Baxter algebras is derived and also analysed.

A conserved lattice gas with random neighbour hopping of active particles is introduced which exhibits a continuous phase transition from an active state to an absorbing non-active state. Since the randomness of the particle hopping breaks long-range spatial correlations our model mimics the mean-field scaling behaviour of the recently introduced new universality class of absorbing phase transitions with a conserved field. The critical exponents of the order parameter are derived within a simple approximation. The results are compared with those of simulations and field theoretical approaches.

We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half-filling. Depending on the strength of the inter-layer coupling J, phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d_c = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d_c = 5 and a nonclassical anisotropy exponent. These two distinct critical lines meet at a nonequilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.

The Cauchy-Lorentz (CL) distribution has divergent lowest moments. This necessarily leads to the fact that the information theoretic approach is essential for the study of its statistical properties. Here, correlation measured by the mutual entropy is discussed for the multivariate CL distribution. It is found that correlation obeys a simple scaling law with respect to the dimensionality of the distribution. Then, regarding the CL distribution as a power-law quantum wavepacket, the information entropic uncertainty relation is also discussed both analytically and numerically. It is found that the sum of the position and momentum information entropies tends to the value of the lower bound for large dimensions.

We investigate the presence of irrelevant operators in the two-dimensional Ising model perturbed by a magnetic field, by studying the corrections induced by these operators in the spin-spin correlator of the model. To this end we perform a set of high-precision simulations for the correlator both along the axes and along the diagonal of the lattice. By comparing the numerical results with the predictions of a perturbative expansion around the critical point we find unambiguous evidence of the presence of such irrelevant operators. It turns out that among the irrelevant operators the one which gives the largest correction is the spin-4 operator T ² + ², which accounts for the breaking of the rotational invariance due to the lattice. This result agrees with what was already known for the correlator evaluated exactly at the critical point and also with recent results obtained in the case of the thermal perturbation of the model.

In this paper, a new method is developed to investigate the pore structure of finitely and even infinitely ramified Sierpinski carpets. The holes in every iteration stage of the carpet are described by a hole-counting polynomial. This polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters, from which dimensions governing the scaling of these quantities can be determined. Whereas the hole area is known to be two dimensional, the dimension of the hole perimeter may be related to the random walk dimension.

We apply the method of infrared bounds with the technique of Kennedy, Lieb and Shastry to the isotropic spin-1 Hamiltonian with bilinear (-J) and biquadratic (-J ') exchange interactions to examine the existence of antiferro-dipole long-range order (DLRO) and ferro-quadrupole long-range order (QLRO). We prove that DLRO exists at zero temperature in two and three dimensions for 0⩽J '<-0.188J and 0⩽J '<-1.954J, respectively. In three dimensions we also prove the existence of QLRO in the ground state for 0<2J⩽J '<2.664J.

We apply a newly proposed Monte Carlo method, the Wang-Landau algorithm, to the study of the three-dimensional antiferromagnetic q-state Potts models on a simple cubic lattice. We systematically study the phase transition of the models with q = 3, 4, 5 and 6. We obtain the finite-temperature phase transition for q = 3 and 4, whereas the transition temperature is down to zero for q = 5. For q = 6 there exists no order for any temperature. We also study the ground-state properties. The size dependence of the ground-state entropy is investigated. We find that the ground-state entropy is larger than the contribution from the typical configurations of the broken-sublattice-symmetry state for q = 3. The same situations are found for q = 4, 5 and 6.

Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of Lüders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the Lüders density matrix. Finally, we apply Girsanov's theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.

A new purification scheme is proposed which applies to arbitrary dimensional bipartite quantum systems. It is based on the repeated application of a special class of nonlinear quantum maps and a single, local unitary operation. This special class of nonlinear quantum maps is generated in a natural way by a Hermitian generalized XOR-gate. The proposed purification scheme offers two major advantages, namely it does not require local depolarization operations at each step of the purification procedure and it purifies more efficiently than other known purification schemes.

A method for geometric phase estimation from the scalar vector product of the initial- and final-state vectors of a system is proposed. The method is based on the time-frequency distribution, namely, the short-time Fourier transform (STFT) of a complex signal. By adjusting the width of the analysis window of the STFT, an estimated geometric phase can be obtained which closely matches a true geometric phase. The computational algorithm for geometric phase decomposition based on the Gabor expansion is presented. Numerical results are verified by several examples of geometric phase decompositions for SU(2) evolutions.

Free noncommutative fields constitute a natural and interesting example of constrained theories with higher derivatives. The quantization methods involving constraints in the higher derivative formalism can be nicely applied to these systems. We study real and complex free noncommutative scalar fields where momenta have an infinite number of terms. We show that these expressions can be summed in a closed way and lead to a set of Dirac brackets which matches the usual corresponding brackets of the commutative case.

We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.

As a parameter a is varied, the topology of nodal lines of complex scalar waves in space (i.e. their dislocations, phase singularities or vortices) can change according to a structurally stable reconnection process involving local hyperbolas whose branches switch. We exhibit families of exact solutions of the Helmholtz equation, representing knots and links that are destroyed by encounter with dislocation lines threading them when a is increased. In the analogous paraxial waves, the paraxial prohibition against dislocations with strength greater than unity introduces additional creation events. We carry out the analysis with polynomial waves, obtained by long-wavelength expansions of the wave equations. The paraxial events can alternatively be interpreted as knotting and linking of worldlines of dislocation points moving in the plane.

A new perspective for the anharmonic oscillator problem based on the SU(2) group method (SGM) is provided. One finds that the SGM is a possible unified approach to treat both the harmonic oscillator and the anharmonic oscillator, although for the latter only part of the energy spectrum can be obtained. Coordinate translation x → x + λ for the anharmonic potential is also discussed, as one expects that the energy spectrum of an anharmonic oscillator is not affected by the translation.

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the `de Sitter group' in 4+1 dimensions, SO(5,1). Using this approach, the non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

As an alternative to the covariant Ostrogradski method, we show that higher-derivative (HD) relativistic Lagrangian field theories can be reduced to second differential order by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Despite the intrinsic non-covariance of the Dirac procedure used to deal with the constraints, the explicit Lorentz invariance is recovered at the end. We develop this new setting on the basis of a simple scalar model and then its applications to generalized electrodynamics and HD gravity are worked out. For a wide class of field theories this method is better suited than Ostrogradski's for a generalization to 2n-derivative theories.

A novel mathematically simple Kontorovich–Lebedev representation of solutions to the Schrödinger equation for a three-particle problem where two of them interact via a zero-range potential is developed. The asymptotic limits and regularity properties are studied. The connection between the representations for E > 0 and E < 0 is also discussed.

New solutions to the Abelian U(1) Higgs model, corresponding to vortices of integer and half-integer winding number bound onto the edges of domain walls and possibly surrounded by annular current flows, are described. Independently of their stability issue, the existence of these states could have interesting consequences in different physical contexts.

In this paper we solve one-dimensional trapped SU(2) bosons with repulsive δ-function interaction by means of the Bethe-ansatz method. The features of ground state and low-lying excited states are studied by numerical and analytic methods. We show that the ground state is an isospin `ferromagnetic' state which differs from spin-½ fermion systems. There exist three quasi-particles in the excitation spectra, and both holon-antiholon and holon-isospinon excitations are gapless for large systems. The equilibrium thermodynamics of the system at finite temperature is studied using the thermodynamic Bethe ansatz. The thermodynamic quantities, such as specific heat, etc, are obtained for the case of the strong coupling limit.

We study the dynamics of supervised on-line learning of realizable tasks in feed-forward neural networks. We focus on the regime where the number of examples used for training is proportional to the number of input channels N. Using generating functional techniques from spin glass theory, we are able to average over the composition of the training set and transform the problem for N → ∞ to an effective single pattern system described completely by the student autocovariance, the student–teacher overlap and the student response function with exact closed equations. Our method applies to arbitrary learning rules, i.e., not necessarily of a gradient-descent type. The resulting exact macroscopic dynamical equations can be integrated without finite-size effects up to any degree of accuracy, but their main value is in providing an exact and simple starting point for analytical approximation schemes. Finally, we show how, in the region of absent anomalous response and using the hypothesis that (as in detailed balance systems) the short-time part of the various operators can be transformed away, one can describe the stationary state of the network succesfully by a set of coupled equations involving only four scalar order parameters.

Simple and accurate analytical approximations of integrals relating to overhead transmission lines are presented in this paper. These approximations are valid for all arguments of the integrals and are faster in comparison with numerical integration.

We discuss both the UV and IR origins of the one-loop triangle gauge anomalies for noncommutative non-Abelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only originate from planar triangle diagrams, the nonplanar triangle contributions giving rise to no breaking of the Ward identities. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free.

Using a new kinematical description of a free three-body problem in hyperspherical coordinates (Matveenko A V and Fukuda H 1998 J. Phys. A: Math. Gen.31 5371) we derive two infinite series of matrix identities interconnecting triangle angles, particle masses and internal hyperspherical angles. The corresponding relations for the matrix elements are practically all new.

Solutions to the random Fibonacci recurrence x_{n + 1} = x_n±βx_n-1 decrease (increase) exponentially, x_n~exp (λn), for sufficiently small (large) β. In the limits β→0 and β→∞, we expand the Lyapunov exponent λ(β) in powers of β and β^-1, respectively. For the classical case of β = 1 we obtain exact non-perturbative results. In particular, an invariant measure associated with Ricatti variable r_n = x_{n + 1}/x_n is shown to exhibit plateaux around all rational r.

Invariants of the coadjoint representation of two classes of Lie algebras are calculated. The first class consists of the nilpotent Lie algebras T(M), isomorphic to the algebras of upper triangular M×M matrices. The Lie algebra T(M) is shown to have [M / 2] functionally independent invariants. They can all be chosen to be polynomials and they are presented explicitly. The second class consists of the solvable Lie algebras L(M, f) with T(M) as their nilradical and f additional linearly nilindependent elements. Some general results on the invariants of L(M, f) are given and the cases M = 4 for all f and f = 1, or M-1 for all M are treated in detail.

We comment on a recent letter by de Albuquerque and Leite (2001 J. Phys. A: Math. Gen.34 L327), in which results to the second order in = 4 − d + m/2 were presented for the critical exponents ν_L2, η_L2 and γ_L2 of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.

In our paper [1] we failed to discuss an important earlier work by J Monroe [2], where the stability of recursion formulae for a Cayley tree has been studied. We regret this omission and withdraw the statement in section 3 that `In all previous publications this stability condition has been simply ignored'. Monroe made the appealing conjecture, that in the presence of more than one stable fixed point of the recursion formula the most stable (i.e. smallest derivative) corresponds to the physical solution. He showed that this criterion leads in the zero field case to the same solution as found by Peruggi et al [3] based on a free energy/point valid far from the border. Since most of the lattice points lie on the border or in the transition region, the latter method is hardly justified. Unfortunately Monroe's conjecture does not solve the border problems either, where one needs to prove that the basin of attraction of the unwanted solution shrinks in some sense to zero.

References

[1] Wagner F, Grensing D and Heide J 2000 J. Phys. A: Math. Gen.33 929

[2] Monroe J L 1994 Phys. Lett. A 188 80

[3] Peruggi F, di Liberto F and Monroy G 1983 J. Phys. A: Math. Gen.16 811