Table of contents

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.

We review the statistical mechanics approach to the study of the emerging collective behaviour of systems of heterogeneous interacting agents. The general framework is presented through examples in such contexts as ecosystem dynamics and traffic modelling. We then focus on the analysis of the optimal properties of large random resource-allocation problems and on Minority Games and related models of speculative trading in financial markets, discussing a number of extensions including multi-asset models, majority games and models with asymmetric information. Finally, we summarize the main conclusions and outline the major open problems and limitations of the approach.

We consider the quadrupolar glass model with infinite-range random interaction. Introducing a simple one-step replica symmetry breaking ansatz we investigate the para-glass continuous (discontinuous) transition which occurs below (above) a critical value of the quadrupole dimension m*. By using a mean-field approximation we study the stability of the one-step replica symmetry breaking solution and show that for m > m* there are two transitions. The thermodynamic transition at a temperature T_D is discontinuous but there is no latent heat. At a higher temperature we find the dynamical or glass transition temperature T_G and the corresponding discontinuous jump q_G of the order parameter.

Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has led to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.

In this paper, we establish the existence of non-trivial solutions for a semi-linear elliptic partial differential equation with a non-local term. This result allows us to prove the existence of standing wave (ground state) solutions for a generalized Davey–Stewartson system. A sharp upper bound is also obtained on the size of the initial values for which solutions exist globally.

We consider the differential equations y'' = λ₀(x)y' + s₀(x)y, where λ₀(x), s₀(x) are C^∞-functions. We prove (i) if the differential equation has a polynomial solution of degree n > 0, then δ_n = λ_ns_n−1 − λ_n−1s_n = 0, where λ_n = λ'_n−1 + s_n−1 + λ₀λ_n−1ands_n = s'_n−1 + s₀λ_k−1, n = 1, 2, .... Conversely (ii) if λ_nλ_n−1 ≠ 0 and δ_n = 0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kinds), Gegenbauer and the Hypergeometric type, etc obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.

The exact solution of the Dirac equation for a deformed form of the Woods–Saxon potential is obtained for the s-wave relativistic energy spectrum. The energy eigenvalues and two-component spinor wavefunctions are derived analytically by using a systematical method which is called Nikiforov–Uvarov. It is seen that the energy eigenvalues and the wavefunctions strongly depend on the parameters of the potential. In addition, it is also shown that the non-relativistic limit can be reached easily and directly for a special case of the standard Woods–Saxon potential.

We construct a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator. This representation is very different from the usual path integral forms suggested by Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), which involve the normal or the antinormal ordering of the Hamiltonian. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. We show that the semiclassical limit of the coherent state propagator in the Weyl representation involves classical trajectories that are independent of the width of coherent states. This propagator is also free from the phase corrections found in Baranger et al (2001 J. Phys. A: Math. Gen.34 7227) for the two Klauder forms and provides an explicit connection between the Wigner and the Husimi representations of the evolution operator.

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a 'canonical' unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on dmod9, and I support this claim with computer computations in all dimensions <48.

We explore the Hamiltonian operator , where is the Dirac delta function and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a spectral singularity at . For Re(z) < 0, H has an eigenvalue at E = −z²/4. For the case that Re(z) > 0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.

We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, 'as close as possible' to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows 'painless' quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.

Constructive algorithms are presented for controlling quantum systems evolving on the SU(1, 1) Lie group. These procedures are performed via structured decomposition of SU(1, 1), which achieve precise controls without any approximations or iterative computations, under the sufficient condition that examines the existence of such decomposition. The technique is applied to controlling transitions between SU(1, 1) coherent states. These results open up new perspectives on the control design of infinite-dimensional quantum systems involving discrete or continuous spectra.

The paper contains an application of the path-integration method to the quantization of the electromagnetic field in a linear material medium modelled as the Hopfield dielectric. Except for the frequency dispersion of the medium, non-dipole effects are included leading to the wave vector dependence of the dielectric function. Starting from a local microscopic Lagrangian, the elimination of the matter degrees of freedom leads to the effective action describing dynamics of the classical field in the medium, from which the classical constitutive relation, non-local, both in time and space variables, could be determined. Full quantization of the model is achieved by integration over all fields with source terms included into the Lagrangian, and taking into account the constraint and gauge fixing term, characteristic for a quantized gauge theory. This gives the generating functional from which quantum propagators could be constructed. Operators of effective quantum electromagnetic and polarization fields were further retrieved from the propagators. The quantum constitutive equation containing the absorption noise term was also determined. The canonical equal time commutation rules are examined using both the BJL-limiting procedure and using explicit expressions for the field operators—the results in both cases are the same.

Based on a gauge-invariant form of the electron propagation function, we propose a formalism for QED which preserves its gauge-invariant character when both photon and electron propagators are regularized with a sharp momentum-cutoff procedure. Perturbation calculations of the regularized fermion effective action functional of an external electromagnetic field are given. We study radiative corrections induced by a momentum-cutoff vacuum and derive the corresponding Ward–Takahashi identity. Several problems encountered in an attempt of constructing a momentum-cutoff QED model are discussed.

Equation (2.26) contains an error. The corrected version is shown in the PDF file.