Table of contents

The basic frameworks and techniques of the Bayesian approach to image restoration are reviewed from the statistical-mechanical point of view. First, a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem. Second, the basic formulation of the statistical estimation from the observed degraded image by using the Bayes formula is demonstrated. The relationship between Bayesian statistics and statistical mechanics is also listed. Particularly, it is explained that some correlation inequalities on the Nishimori line of the random spin model also play an important role in Bayesian image restoration. Third, the framework of Bayesian image restoration for binary images by means of the Ising model is reviewed. Some practical algorithms for binary image restoration are given by employing the mean-field and the Bethe approximations. Finally, Bayesian image restoration for a grey-level image using the Gaussian model is reviewed, and the Gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges. The line state is also extended to quantized values.

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n − 1)-body problem that incorporates the constant linear momentum and centre-of-mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.

We describe the twisted Yangians Y(g, h) which arise as boundary remnants of Yangians Y(g) in 1 + 1D integrable field theories. We describe and extend our recent construction of the intertwiners of their representations (the rational boundary S- or 'K'-matrices) and perform a case-by-case analysis for all pairs (g, h), giving the h-decomposition of Y(g, h)-representations where possible.

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley–Klein–Dickson family of spaces of 'complex Hermitian'-type and rank-1. The complex Hermitian elliptic P^N ≡ SU(N + 1)/(U(1) ⊗ SU(N)) and hyperbolic H^N ≡ SU(N, 1)/(U(1) ⊗ SU(N)) spaces, their analogues with indefinite Hermitian metric SU(p + 1, q)/(U(1) ⊗ SU(p, q)) and the non-compact symmetric space SL(N + 1, )/(SO(1, 1) ⊗ SL(N, )) are the generic members in this family; the remaining spaces are some contractions of the former.

The method encapsulates trigonometry for this whole family of spaces into a single basic trigonometric group equation, and has 'universality' and 'self-duality' as its distinctive traits. All previously known results on the trigonometry of P^N and H^N follow as particular cases of our general equations.

The following topics are covered rather explicitly: (0) description of the complete Cayley–Klein–Dickson family of rank-1 spaces of 'complex type', (1) derivation of the single basic group trigonometric equation, (2) translation to the basic 'complex Hermitian' cosine, sine and dual cosine laws, (3) comprehensive exploration of the bestiarium of 'complex Hermitian' trigonometric equations, (4) uncovering of a 'Cartan' sector of Hermitian trigonometry, related to triangle symplectic area and co-area, (5) existence conditions for a triangle in these spaces as inequalities and (6) restriction to the two special cases of 'complex' collinear and purely real triangles.

The physical quantum space of states of any quantum system belongs, as the complex Hermitian space member, to this parametrized family; hence its trigonometry appears as a rather particular case of the equations we obtain.

We discuss exact functional integral expressions for many-body matrix elements of the type ⟨ψ, A|e^−βĤ|ψ, A⟩, between particle number projected Hartree–Fock–Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen–Cooper–Schrieffer wavefunctions.

Starting with any R-matrix with spectral parameters, and obeying the Yang–Baxter equation and a unitarity condition, we construct the corresponding infinite-dimensional quantum group _R in terms of a deformed oscillator algebra _R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated with _R, we show that _R provides its integrals of motion. The construction can be applied to any infinite-dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N), the Yangian based on gl(N), using this construction we recover the nonlinear Schrödinger equation and its Y(N) symmetry.

We analyse the discrete form of the Chazy III equation proposed by Labrunie and Conte, with the help of two different integrability criteria: singularity confinement and algebraic entropy. We show that for all values of the free parameter this third-order mapping fails both criteria and thus cannot be integrable.

We study pairwise quantum entanglement in systems of fermions itinerant in a lattice from a second-quantized perspective. Entanglement in the grand-canonical ensemble is studied, both for energy eigenstates and for the thermal state. Relations between entanglement and superconducting correlations are discussed in a BCS-like model and for η-pair superconductivity.

Horton et al (2000 J. Phys. A: Math. Gen.33 7337) have proposed Bohm-type particle trajectories accompanying a Klein–Gordon wavefunction ψ on Minkowski space. From two vector fields on space–time, W⁺ and W⁻, defined in terms of ψ, they intend to construct a time-like vector field W, the integral curves of which are the possible trajectories, by the following rule: at every space–time point, take either W = W⁺ or W = W⁻ depending on which is time-like. This procedure, however, is ill-defined as soon as both are time-like, or both space-like. Indeed, they cannot both be time-like, but they can well both be space-like, contrary to the central claim of Horton et al. We point out the gap in their proof, we provide a counter example, and we argue that, even for a rather arbitrary wavefunction, the points where both W⁺ and W⁻ are space-like can form a set of positive measure.

We demonstrate that a straightforward limiting process can be used to continue trajectories in spacetime through exceptional regions in the velocity field (identified by Tumulka (2000 J. Phys. A: Math. Gen.35 7691)) using the formalism we previously proposed (Horton G, Dewdney C and Nesteruk A 2000 J. Phys. A: Math. Gen.33 7337).