Table of contents

I discuss Haldane's concept of generalized exclusion statistics (1991 Phys. Rev. Lett.67 937) and show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert subspace on which they act. These new definitions lead to properly defined particle distributions and thermodynamic properties. In another paper (Preprint 0710.0728), I show that the fractional exclusion statistics manifested in general systems with interaction have these, physically consistent, statistics parameters.

We study pseudo-optimal solutions to multi-objective optimization problems by introducing partial minima defined as follows. Point xk-dominates x' when at least k of the coordinates of x are smaller than the corresponding coordinates of x'. A point not k-dominated by any other point in the set is a k-minimum or a partial minimum, generalizing the global minimum. We study statistical properties of partial minima for a set of N points independently distributed inside the d-dimensional unit hypercube using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ∼ N^−(d−k)/k, when 1 ⩽ k < d. Interestingly, there are k − 1 distinct scaling laws characterizing the largest coordinates: the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, , with for 1 ⩽ j ⩽ k − 1. The average number of partial minima grows logarithmically, , when k = d. The full distribution of the number of minima is obtained in closed form in two dimensions.

Recently it was conjectured that nodal domains of random wavefunctions are adequately described by critical percolation theory. In this paper we strengthen this conjecture in two respects. First, we show that, though wavefunction correlations decay slowly, a careful use of Harris' criterion confirms that these correlations are unessential and nodal domains of random wavefunctions belong to the same universality class as non-correlated critical percolation. Second, we argue that level domains of random wavefunctions are described by the non-critical percolation model.

We study the statistics of quantum transmission through a one-dimensional disordered system modelled by a sequence of independent scattering units. Each unit is characterized by its length and by its action, which is proportional to the logarithm of the transmission probability through this unit. Unit actions and lengths are independent random variables, with a common distribution that is either narrow or broad. This investigation is motivated by results on disordered systems with non-stationary random potentials whose fluctuations grow with distance. In the statistical ensemble at fixed total sample length four phases can be distinguished, according to the values of the indices characterizing the distribution of the unit actions and lengths. The sample action, which is proportional to the logarithm of the conductance across the sample, is found to obey a fluctuating scaling law, and therefore to be non-self-averaging, in three of the four phases. According to the values of the two above-mentioned indices, the sample action may typically grow less rapidly than linearly with the sample length (underlocalization), more rapidly than linearly (superlocalization), or linearly but with non-trivial sample-to-sample fluctuations (fluctuating localization).

Models of directed paths have been used extensively in the scientific literature to model linear polymers. In this paper, we examine a directed path model of a linear polymer in a confining geometry (a wedge). We are particularly interested in c_n, the number of directed lattice paths of length n steps which take steps in the North–East and South–East directions and are confined in the wedge Y = ±X/p, where p is an integer. We examine the case p = 2 in detail and show that the generating function satisfies a functional equation with quartic kernel. We give a formal solution by using the kernel method, show that this model is not D-finite and determine asymptotics to leading order for c_n. In particular, the number of paths of length n in the wedge Y = ±X/2 is given by where the constant 0.67874... can be determined to arbitrary accuracy with little computational effort.

A statistical mechanical framework to analyze linear vector channel models in digital wireless communication is proposed for a large system. The framework is a generalization of that proposed for code-division multiple-access systems in Takeda et al (2006 Europhys. Lett.76 1193) and enables the analysis of the system in which the elements of the channel transfer matrix are statistically correlated with each other. The significance of the proposed scheme is demonstrated by assessing the performance of an existing model of multi-input multi-output communication systems.

The two-terminal reliability, known as the pair connectedness or connectivity function in percolation theory, may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account. When link and site probabilities are p and ρ, it obeys an asymptotic power-law behaviour, for which the scaling factor is the transfer matrix's eigenvalue of largest modulus. The location of the complex zeros of the two-terminal reliability polynomial exhibits structural transitions as 0 ⩽ ρ ⩽ 1.

We continue our investigation of the -Baxter–Bazhanov–Stroganov model using the method of separation of variables [1]. In this paper, we calculate the norms and matrix elements of a local -spin operator between eigenvectors of the auxiliary problem. For the norm the multiple sums over the intermediate states are performed explicitly. In the case N = 2, we solve the Baxter equation and obtain form-factors of the spin operator of the periodic Ising model on a finite lattice.

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrödinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we prove, for suitable values of the parameters, that the blow-up threshold (if the nonlinearity has the critical growth) is a universal constant.

In this paper, we study the competition of the linear and nonlinear lattices and its effects on the stability and dynamics of bright solitary waves. We consider both lattices in a perturbative framework, whereby the technique of Hamiltonian perturbation theory can be used to obtain information about the existence of solutions, and the same approach, as well as eigenvalue count considerations, can be used to obtain detailed conditions about their linear stability. We find that the analytical results are in very good agreement with our numerical findings and can also be used to predict features of the dynamical evolution of such solutions. A particularly interesting result of these considerations is the existence of a tunable cancellation effect between the linear and nonlinear lattices that allows for increased mobility of the solitary wave.

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a 'Laplacian') with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e. conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H = A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is also found that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.

We describe a new class of scattering matrices for quantum graphs in which back-scattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We also provide two methods to build the vertex scattering matrices needed for their construction.

General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state ρ₁₂ described by a 4 × 4 covariance matrix V, the Schur complement of a local covariance submatrix V₁ of it can be interpreted as a new covariance matrix representing a Gaussian operator of party 1 conditioned to local parity measurements on party 2. The connection with a partial parity measurement over a bipartite quantum state and the determination of the reduced Wigner function is given and an operational process of parity measurement is developed. Generalization of this procedure to an n-partite Gaussian state is given, and it is demonstrated that the n − 1 system state conditioned to a partial parity projection is given by a covariance matrix such that its 2 × 2 block elements are Schur complements of special local matrices.

Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4.

An iterative algorithm for determining a type of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This type includes the solution which underlies the large-N limit of the Hermitian matrix model in the one-cut case. It is also shown how the double scaling limit can be naturally formulated in this scheme

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.

As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N × Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.

Yu, Brown and Chuang investigated the entanglement attainable from unitary transformed thermal states in liquid-state nuclear magnetic resonance (NMR). Their research gave insight into the role of entanglement in a liquid-state NMR quantum computer. However, they assumed that the Zeeman energy of each nuclear spin which corresponds to a qubit takes a common value for all; there is no chemical shift. In this paper, we research a model with chemical shifts and analytically derive the physical parameter region where unitary transformed thermal states are entangled, by employing the positive partial transposition (PPT) criterion with respect to any bipartition. The analysis taking account of the chemical shift reveals how the difference between quantum gates reflects on the physical parameter region where unitary transformed thermal states are entangled. In addition, we examine the distillability of unitary transformed thermal states and the effect of the chemical shifts on the boundary between the separability and the nonseparability.

A confluence of numerical and theoretical results leads us to conjecture that the Hilbert–Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B_ν(a, b), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit–qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.

The general form of the Stefan–Boltzmann law for the energy density of black-body radiation is generalized to a spacetime with extra dimensions using standard kinetic and thermodynamic arguments. From statistical mechanics one obtains an exact formula. In a field-theoretic derivation, the Maxwell field must be quantized. The notion of electric and magnetic fields is different in spacetimes with more than four dimensions. While the energy–momentum tensor for the Maxwell field is traceless in four dimensions, it is not so when there are extra dimensions. But it is shown that its thermal average is traceless and in agreement with the thermodynamic results.

Surface tension effect on a two-dimensional channel flow against an inclined wall is considered. The flow is assumed to be steady, irrotational, inviscid and incompressible. The effect of surface tension is taken into account and the effect of gravity is neglected. Numerical solutions are obtained via series truncation procedure. The problem is solved numerically for various values of the Weber number α and for various values of the inclination angle β between the horizontal bottom and the inclined wall.

I show that the result of the paper by Neves et al (2006 J. Phys. A: Math. Gen.39 L293–6) is a special case of the well-known 'Gegenbauer finite integral'.