Table of contents

It is shown that the flow diagrams for the conductivities in the quantum Hall effect, arising from two ostensibly very different proposals based on modular symmetry, are in fact identical. The β-functions are different, the rates at which the flow lines are traversed are different, but the tangents to the flow lines are the same in both cases; hence, the flow diagrams are same in all aspects.

In this paper, several aspects of the dynamics of a toy model for long-range Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian mean field (HMF). For special cases, exact finite-N linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, N → ∞, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports the mean field modulus in the vicinity of its equilibrium value within some linear e-folding times.

We investigate a random integral which provides a natural example of an imaginary exponential functional of Brownian motion. This functional shows up in the study of the binary annihilation process, within the Doi–Peliti formalism for reaction-diffusion systems. The main emphasis is put on the complementarity between the usual Langevin approach and another approach based on the similarity with Kesten variables and other one-dimensional disordered systems. Even though neither of these routes leads to the full solution of the problem, we have obtained a collection of results describing various regimes of interest.

In this paper, we show that if each node of the Barabási–Albert (BA) network is characterized by the generalized degree q, i.e. the product of their degree k and the square root of their respective birth time, then the distribution function F(q, t) exhibits dynamic scaling F(q, t → ∞) ∼ t^−1/2ϕ(q/t^1/2) where ϕ(x) is the scaling function. We verified it by showing that a series of distinct F(q, t) versus q curves for different network sizes N collapse onto a single universal curve if we plot t^1/2F(q, t) versus q/t^1/2 instead. Finally, we show that the BA network falls into two universality classes depending on whether new nodes arrive with single edge (m = 1) or with multiple edges (m > 1).

Some chaotic properties of a family of stadium-like billiards with parabolic focusing components, which is described by a two-dimensional nonlinear area-preserving map, are studied. Critical values of billiard geometric parameters corresponding to a sudden change of the maximal Lyapunov exponent are found. It is shown that the maximal Lyapunov exponent obtained for chaotic orbits of this family is scaling invariant with respect to the control parameters describing the geometry of the billiard. We also show that this behavior is observed for a generic one-parameter family of mapping with the nonlinearity given by a tangent function.

This paper studies the problem of isochronal synchronization of time-delay chaotic systems featuring also coupling delay. Based on the Lyapunov–Krasovskii stability theory, sufficient conditions are derived for the stability of isochronal synchronization between a pair of identical chaotic systems. Such criteria permit the proper design of stable proportional linear feedback controller, more specifically, the design of adequate proportional feedback gain matrices. The proposed criteria are suited to systems with (i) intrinsic delay, (ii) coupling delay or (iii) both. Numerical simulations of the synchronization of delay-coupled systems are presented as examples of the application of the criteria.

Ultradiscretization with parity variables, which keeps the information of original variables' sign, is applied to the q-Painlevé II equation of type A₆ (q-PII). A special solution of the resulting ultradiscrete system, which corresponds to the special function solution of q-PII, is constructed. Ultradiscrete analogues of the q-Airy equation and its special solutions are also discussed in the process.

We consider a dynamical system moving in a Riemannian space and prove two theorems which relate the Lie point symmetries and Noether symmetries of the equation of motion, with the special projective group and the homothetic group of the space respectively. These theorems are used to classify the two-dimensional Newtonian dynamical systems, which admit Lie point/Noether symmetries. The results of the study, i.e. expressions of forces/potentials, Lie symmetries, Noether vectors and Noether integrals are presented in the form of tables for easy reference and convenience. Two cases are considered, Hamiltonian and non-Hamiltonian systems. The results are used to determine the Lie/Noether symmetries of two different systems. The Kepler–Ermakov system, which in general is non-conservative, and the conservative system with potential similar to the Hènon–Heiles potential. As an additional application, we consider the scalar field cosmologies in a FRW background with no matter, and look for the scalar field potentials for which the resulting cosmological models are integrable. It is found that the only integrable scalar field cosmologies are defined by the exponential and the unified dark matter potential. It is to be noted that in all aforementioned applications the Lie/Noether symmetry vectors are found by simply reading the appropriate entry in the relevant tables.

We consider the one-dimensional totally asymmetric simple exclusion process (TASEP model) with open boundary conditions and present the analytical computations leading to the exact formula for distance clearance distribution, i.e. probability density for a clear distance between subsequent particles of the model. The general relation is rapidly simplified for the middle part of the one-dimensional lattice. Both the analytical formulas and their approximations are compared with the numerical representation of the TASEP model. Such a comparison is presented for particles occurring in the internal part as well as in the boundary part of the lattice. Furthermore, we introduce the pertinent estimation for the so-called spectral rigidity of the model. The results obtained are sequentially discussed within the scope of vehicular traffic theory.

We present a complete analysis of the dynamics of a Bose–Einstein condensate trapped in a symmetric triple-well potential. Our classical analogue treatment, based on a time-dependent variational method using SU(3) coherent states, includes the parameter dependence analysis of the equilibrium points and their local stability, which is closely related to the condensate collective behaviour. We also consider the effects of off-site interactions, and how these 'cross-collisions' may become relevant for a large number of trapped bosons. Even in the presence of cross-collisional terms, the model still features an integrable sub-regime, known as the twin-condensate dynamics, which corresponds to invariant surfaces in the classical phase space. However, the quantum dynamics preserves the twin-condensate defining characteristics only partially, thus breaking the invariance of the associated quantum subspace. Moreover, the periodic geometry of the trapping potential allowed us to investigate the dynamics of finite angular momentum collective excitations, which can be suppressed by the emergence of chaos. Finally, using the generalized purity associated with the su(3) algebra, we were able to quantify the dynamical classicality of a quantum evolved system, as compared to the corresponding classical trajectory.

We analyze the concept of entanglement for a multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with an appropriate symmetry. For an arbitrary parastatistics, we define the S-rank generalizing the notion of the Schmidt rank. The S-rank, defined for all types of tensors, serves for distinguishing entanglement of pure states. In addition, for Bose and Fermi statistics, we construct an analog of the Jamiołkowski isomorphism.

The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.

We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.

Using the Voros star product, we investigate the status of the two-particle correlation function to study the possible extent to which the previously proposed violation of the Pauli principle may impact at low energies. The results show interesting features which are not present in the computations made using the Moyal star product.

In the framework of an ordinary-derivative approach, conformal gravity in spacetime of dimension 6 is studied. The field content, in addition to conformal graviton field, includes two auxiliary rank-2 symmetric tensor fields, two Stueckelberg vector fields and one Stueckelberg scalar field. The gauge invariant Lagrangian with conventional kinetic terms and corresponding gauge transformations are obtained. One of the rank-2 tensor fields and the scalar field have a canonical conformal dimension. With respect to these fields, the Lagrangian contains, in addition to other terms, a cubic potential. Gauging away the Stueckelberg fields and excluding the auxiliary fields via equations of motion, the higher derivative Lagrangian of 6D conformal gravity is obtained. The higher derivative Lagrangian involves quadratic and cubic curvature terms. This higher derivative Lagrangian coincides with the simplest Weyl invariant density discussed in the earlier literature. Generalization of de Donder gauge conditions to 6D conformal fields is also obtained.

Theories with axionic scalars admit three different Euclidean formulations, obtained by Wick rotation, Wick rotation combined with analytic continuation of the axionic scalars, and Wick rotation combined with Hodge dualization. We investigate the relation between these formulations for a class of theories which contains the sigma models of N = 2 vector multiplets as a special case. It is shown that semi-classical amplitudes can be expressed equivalently using the two types of axionic actions, while the Hodge-dualized version gives a different value for the instanton action unless the integration constants associated with the axion fields are chosen in a particular way. With this choice the instanton action is equal to the mass of the soliton or black hole obtained by dimensional lifting with respect to time. For supersymmetric models we use the Euclidean supersymmetry algebra to derive a Euclidean BPS condition, and identify a geometrical criterion which distinguishes BPS from non-BPS extremal solutions.

We associate stochastic Lagrangian flows with Navier–Stokes (deterministic) velocities and show their unstable behaviour.

The Journal of Physics A: Mathematical and Theoretical publishing team would like to apologise to the authors of the above paper. Due to an oversight, the article was published with an error in the last line of page 7: 'Ω(a, n)' should read 'Γ(a, n)'.