Table of contents

The existence of bi-Hamiltonian structures for the rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analysed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained.

We study the local time distribution of a Brownian particle diffusing along the links on a graph. In particular, we derive an analytic expression of its Laplace transform in terms of the Green function on the graph. We show that the asymptotic behaviour of this distribution has non-Gaussian tails characterized by a nontrivial large deviation function.

We present an exact solution for a catalytically activated annihilation A + A → 0 reaction taking place on a one-dimensional chain in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. An annihilation reaction takes place as soon as any two A particles land from the reservoir onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. We find that the disorder-average pressure P^(quen) per site of such a chain is given by P^(quen) = P^(Lan) + β⁻¹F, where P^(Lan) = β⁻¹ ln(1 + z) is the Langmuir adsorption pressure, (z being the activity and β⁻¹ the temperature), while β⁻¹F is the reaction-induced contribution, which can be expressed, under appropriate change of notation, as the Lyapunov exponent for the product of 2 × 2 random matrices, obtained exactly by Derrida and Hilhorst (1983 J. Phys. A: Math. Gen.16 2641). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.

The mutual learning process between two parity feed-forward networks with discrete and continuous weights is studied analytically, and we find that the number of steps required to achieve full synchronization between the two networks in the case of discrete weights is finite. The synchronization process is shown to be non-self-averaging and the analytical solution is based on random auxiliary variables. The learning time of an attacker that is trying to imitate one of the networks is examined analytically and is found to be much longer than the synchronization time. Analytical results are found to be in agreement with simulations.

A theorem of Hegerfeldt shows that if the spectrum of the Hamiltonian is bounded from below, then the propagation speed of certain probabilities does not have an upper bound. We prove a theorem analogous to Hegerfeldt's that appertains to asymmetric time evolutions given by a semigroup of operators. As an application, we consider a characterization of relativistic quasistable states by irreducible representations of the causal Poincaré semigroup and study the implications of the new theorem for this special case.

We apply the semiclassical spin coherent state method for the density of states by Pletyukhov et al (2002 Phys. Rev. Lett.89 116601) in the weak spin–orbit coupling limit and recover the modulation factor in the semiclassical trace formula found by Bolte and Keppeler (1998 Phys. Rev. Lett.81 1987; 1999 Ann. Phys., NY274 125).

We describe a mixing-length extension of the lattice Boltzmann approach to the simulation of an incompressible liquid in turbulent flow. The method uses a simple, adaptable, closure algorithm to bound the lattice Boltzmann fluid incorporating a law-of-the-wall. The test application, of an internal, pressure-driven and smooth duct flow, recovers correct velocity profiles for Reynolds number to 1.25 × 10⁵. In addition, the Reynolds number dependence of the friction factor in the smooth-wall branch of the Moody chart is correctly recovered. The method promises a straightforward extension to other curves of the Moody chart and to cylindrical pipe flow.

We calculate the exact density of states (DOS) for the three classical and two non-classical random matrix ensembles for finite matrix size N using supersymmetric integrals. The 1/N expansion yields, already in lowest order, good approximations to the exact result even for small values of N ∼ 5. We conjecture a connection between the N-dependence of the oscillating part of the DOS and the short-distance behaviour of the two-level correlation function.

The cellular automata (CA) approach to traffic modelling is extended to allow for spatially homogeneous steady state solutions that cover a two-dimensional region in the flow–density plane. Hence these models fulfil a basic postulate of a three-phase traffic theory proposed by Kerner. This is achieved by a synchronization distance, within which a vehicle always tries to adjust its speed to that of the vehicle in front. In the CA models presented, the modelling of the free and safe speeds, the slow-to-start rules as well as some contributions to noise are based on the ideas of the Nagel–Schreckenberg-type modelling. It is shown that the proposed CA models can be very transparent and still reproduce the two main types of congested patterns (the general pattern and the synchronized flow pattern) as well as their dependence on the flows near an on-ramp, in qualitative agreement with the recently developed continuum version of the three-phase traffic theory (Kerner B S and Klenov S L 2002 J. Phys. A: Math. Gen.35 L31 ). These features are qualitatively different from those in previously considered CA traffic models. The probability of the breakdown phenomenon (i.e. of the phase transition from free flow to synchronized flow) as function of the flow rate to the on-ramp and of the flow rate on the road upstream of the on-ramp is investigated. The capacity drops at the on-ramp which occur due to the formation of different congested patterns are calculated.

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U_q over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T–Q relations in which three of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.

If the physical agent (the 'pointer', or 'cursor', or 'clocking mechanism') that sequentially scans the T lines of a long computer program is a microscopic system, two quantum phenomena become relevant: spreading of the probability distribution of the pointer along the program lines, and scattering of the probability amplitude at the two endpoints of the physical space allowed for its motion.

We show that the first effect determines an upper bound O(T^−2/3) on the probability of finding the pointer exactly at the END line.

By adding an adequate number δ of further empty lines ('telomers'), one can store the result of the computation up to the moment in which the pointer is scattered back into the active region. This leads to a less severe upper bound O(√δ/T) on the probability of finding the pointer either at the END line or within the additional empty lines.

Our analysis is performed in the context of Feynman's model of quantum computation, the only model, to our knowledge, that explicitly includes a physically plausible quantum clocking mechanism in its considerations.

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. They are a class of left-symmetric algebras with commutative right multiplication operators, which can be viewed as bosonic. Fermionic Novikov algebras are a class of left-symmetric algebras with anti-commutative right multiplication operators. They correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we commence a study on fermionic Novikov algebras from the algebraic point of view. We will show that any fermionic Novikov algebra in dimension ≤3 must be bosonic. Moreover, we give the classification of real fermionic Novikov algebras on four-dimensional nilpotent Lie algebras and some examples in higher dimensions. As a corollary, we obtain kinds of four-dimensional real fermionic Novikov algebras which are not bosonic. All of these examples will serve as a guide for further development including the application in physics.

We propose an extension of quantum key distribution based on encoding the key into quNits, i.e. quantum states in an N-dimensional Hilbert space. We estimate both the mutual information between the legitimate parties and the eavesdropper, and the error rate, as a function of the dimension of the Hilbert space. We derive the information gained by an eavesdropper using optimal incoherent attacks and an upper bound on the legitimate party error rate that ensures unconditional security when the eavesdropper uses finite coherent eavesdropping attacks. We also consider realistic systems where we assume that the detector dark count probability is not negligible.

We test the consistency of the use of a noncommutative theory description for charged particles in a strong magnetic field, by deriving the induced Chern–Simons (CS) term for an external Abelian gauge field in 2 + 1 dimensions. In this description, the system is modelled by a noncommutative matter field coupled to a U(1) noncommutative gauge field, related to the original, commutative one, by a Seiberg–Witten transformation. We show that an Abelian CS term for the commutative gauge field is indeed induced, and moreover that it matches the result of previous commutative field theory calculations.

We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev–Petviashvili equation, the nonlinear Schrödinger equation, the λϕ⁴ model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions of nonlinear differential equations is successful as a consequence of some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions.

Quasiperiodic functions (QPFs) are characterized by their full vortex structure in one unit cell. This characterization is much finer and more sensitive than the topological one given by the total vorticity per unit cell (the 'Chern index'). It is shown that QPFs with an arbitrarily prescribed vortex structure exist by constructing explicitly such a 'standard' QPF. Two QPFs with the same vortex structure are equivalent, in the sense that their ratio is a function which is strictly periodic, nonvanishing and at least continuous. A general QPF can then be approximately reconstructed from its vortex structure on the basis of the standard QPF and the equivalence concept. As another application of this concept, a simple method is proposed for calculating the quasiperiodic eigenvectors of periodic matrices. Possible applications to the quantum-chaos problem on a phase-space torus are briefly discussed.

We show that quantum mechanics can be given a Lorentz-invariant realistic interpretation by applying our recently proposed relativistic extension of the de Broglie–Bohm theory to deduce non-locally correlated, Lorentz-invariant individual particle motions for the Einstein–Podolsky–Rosen experiment and the double-interferometer experiment proposed by Horne, Shimony and Zeilinger.

We show how higher-genus su(N) fusion multiplicities may be computed as the discretized volumes of certain polytopes. The method is illustrated by explicit analyses of some su(3) and su(4) fusions, but applies to all higher-point and higher-genus su(N) fusions. It is based on an extension of the realm of Berenstein–Zelevinsky triangles by including so-called gluing and loop-gluing diagrams. The identification of the loop-gluing diagrams is our main new result, since they enable us to characterize higher-genus fusions in terms of polytopes. Also, the genus-2 0-point su(3) fusion multiplicity is found to be a simple binomial coefficient in the affine level.

The three-dimensional long-wave approximation for electromagnetic waves in saturated ferromagnetic media is considered, taking into account damping and inhomogeneous exchange. The wave evolution is governed by a (3 + 1)-dimensional generalization of the Korteweg–de Vries, Burgers, Kadomtsev–Petviashvili and Zabolotskaya–Khokhlov equations. Neglecting the damping, we give plane-soliton and line-lump solutions, and show that they are unstable.

An analytical microscopic theory for the resonant multiple scattering of light by cold atoms with arbitrary internal degeneracy is presented. It permits us to calculate the average amplitude and the average intensity for one-photon states of the full transverse electromagnetic field in a dilute medium of unpolarized atoms. Special emphasis is laid upon an analysis in terms of irreducible representations of the rotation group. It allows us to sum explicitly the ladder and maximally crossed diagrams, giving the average intensity in the Boltzmann approximation and the interference corrections responsible for weak localization and coherent backscattering. The exact decomposition into field modes shows that the atomic internal degeneracy contributes to the depolarization of the average intensity and suppresses the interference corrections. Static as well as dynamic quantities such as the transport velocity, diffusion constants and relaxation times for all field modes and all atomic transitions are derived.

We introduce a self-dual, noncommutative and noncocommutative Hopf algebra _GT which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role to the Grothendieck–Teichmüller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for higher weak n-categories (n ≥ 3) by showing that these structures would not allow for any nontrivial deformations. Finally, we give an explicit description of the elements of _GT.

In this paper it is shown that the G^q,q_n−q,n−q-II system gives the Gauss–Codazzi–Ricci equations of a class of flat time-like n-submanifolds with index q in S²ⁿ⁻¹_2q(1), where G^q,q_n−q,n−q = O(2n − 2q, 2q)/O(n − q, q) × O(n − q, q) and 0 < q < n. Moreover, we construct a dressing action on the G^q,q_n−q,n−q-system space of solutions of the G^q,q_n−q,n−q-system which gives rise to Bäcklund transformations for flat time-like n-submanifolds with index q in S²ⁿ⁻¹_2q(1).

James H Simons Conference on Quantum and Reversible Computation

State University of New York at Stony Brook, 28-31 May 2003

Quantum information and quantum computation has been a rapidly developing field attracting interest from mathematicians, physicists and computer scientists.

The goal of this conference is to generate interaction between mathematicians interested in efficient quantum algorithms and quantum error-correcting codes, and physicists trying to build quantum computers. We will stimulate this interaction with lectures on the most important recent developments by world experts.

The conference will also cover reversible information processing, which is of interest in classical as well as quantum computing. The reversible nature of the quantum evolution of isolated systems makes logical reversibility on the classical level a prerequisite for any realization of a quantum computer. Reversible computing does not generate the heat associated with information erasure, and is therefore of interest in reducing the heat produced by classical computers.

This conference has been made possible through a generous donation by James H Simons.

Confirmed speakers:

• D Averin [SUNY at Stony Brook]

• I Cirac [Max-Planck-Institute, Munich]

• M Devoret [Yale University]

• D DiVincenzo [IBM]

• E H Farhi [MIT]

• R Fazio [Scuola Normale Superiore, Pisa]

• S Han [University of Kansas]

• M Hillery [Hunter College, CUNY]

• H K Lo [MagiQ Technologies]

• L Levitin [Boston University ]

• J Lukens [SUNY at Stony Brook]

• C Monroe [University of Michigan, Ann Arbor]

• L A Orozco [SUNY at Stony Brook]

• V Semenov [SUNY at Stony Brook]

• P Shor [AT and T]

• H Yuen [North West University]

• P Zoller [Institute for Theoretical Physics, University of Innsbruck]

More information and registration can be found on the web at:

http://insti.physics.sunysb.edu/itp/conf/simons-qcomputation.html

With effect from 1 January 2003 (volume 36), the following classification scheme will be introduced for research papers published in Journal of Physics A: Mathematical and General. We believe that this new scheme will help to clarify the journal's scope and enable authors and readers to more easily locate the appropriate section for their work.

We also list below some more detailed subject areas which should help define each section heading. These lists are by no means exhaustive and are intended only as a guide to the type of papers we envisage appearing in each section. We acknowledge that no classification scheme can be perfect and that there are some papers which might be placed in more than one section. Whenever possible, we will respect the author's wishes on placement and undertake to inform authors when we feel that a different classification is more appropriate.

We are happy to provide further advice on paper classification to authors upon request (please email jphysa@iop.org).

1. Statistical physics May include papers on

• statistical mechanics, lattice theory and thermodynamics

• quantum statistical mechanics and Bose-Einstein condensation

• phase transitions and critical phenomena

• numerical and computational methods

• theories of interacting particles (many-body theories)

• theoretical condensed matter and mesoscopic systems

• disordered systems, spin glasses and neural networks

• nonequilibrium processes

2. Chaotic and complex systems May include papers on

• nonlinear dynamics and classical chaos

• quantum chaos

• cellular automata

• biophysics

3. Mathematical physics May include papers on

• integrable systems

• random matrix theory

• special functions

• Lie algebras and quantum groups

• classical mechanics

• inverse problems

4. Quantum mechanics and quantum information theory May include papers on

• foundations of quantum mechanics

• quantum information, computation and cryptography

• theoretical quantum optics

• open quantum systems

5. Classical and quantum field theory May include papers on

• gauge and conformal field theory

• quantum electrodynamics and quantum chromodynamics

• string theory and its developments

• classical electromagnetism

6. Theory of continuous media May include papers on

• fluid dynamics and turbulence

• plasma physics