Table of contents

Ordinary differential equations having a first integral may be solved numerically using one of several methods, with the integral preserved to machine accuracy. One such method is the discrete gradient method. It is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy. The method is illustrated using the Henon–Heiles system.

Thermodynamical perturbation theory provides a method for calculating the partition function or the free energy of a system from the properties of another system. The first-order perturbation takes advantage of inequalities such as the Gibbs–Bogoliubov inequality in classical mechanics and the Peierls and Bogoliubov inequalities in quantum mechanics, which are used in variational calculations. We present here sequences of inequalities which generalize the former ones; they can be presented as rearrangements of perturbation expansions, which provide exact bounds. As an example, the free energy of an anharmonic oscillator is calculated with the first two variational principles.

We consider a generalized Harper equation at quadratic irrational flux, showing, in the strong coupling limit, the fluctuations of the exponentially decaying eigenfunctions are governed by the dynamics of a renormalization operator on a renormalization strange set. This work generalizes previous analyses which have considered only the golden mean case. Projections of the renormalization strange sets are illustrated analogous to the 'orchid' present in the golden mean case.

We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and the spin dynamics is governed by a synchronous execution of the microscopic update rule (Little–Hopfield model). We solve this model within replica symmetry, and by using bifurcation analysis we prove that the spin-glass/paramagnetic and the retrieval/paramagnetic transition lines of our phase diagram are identical to those of sequential dynamics. The first-order retrieval/spin-glass transition line follows by direct evaluation of our observables using population dynamics. Within the accuracy of numerical precision and for sufficiently small values of the connectivity parameter we find that this line coincides with the corresponding sequential one. Comparison with simulation experiments shows excellent agreement.

In the context of nonlinear scattering, a continuous wave incident onto a nonlinear discrete molecular chain of coupled oscillators can be partially absorbed as a result of a three-wave resonant interaction that couples two HF-waves of frequencies close to the edge of the Brillouin zone. Hence both nonlinearity and discreteness are necessary for generating this new absorption process which manifests itself by soliton generation in the medium. As a paradigm of this nonlinear absorption we consider here the Davydov model that describes exciton–phonon coupling in hydrogen-bonded molecular chains.

The problem of two gravitational (or Coulombian) fixed centres is a classical integrable problem, stated and integrated by Euler in 1760. The integrability is due to the unexpected first integral G. We introduce some straightforward generalizations of the problem that still have the generalization of G as a first integral, but do not possess the energy integral. We present some numerical integrations showing the main features of their dynamics. In the domain of bounded orbits the behaviour of these a priori non-Hamiltonian systems is very similar to the behaviour of usual near-integrable systems.

We generalize some widely used mother wavelets by means of the q-exponential function e^x_q ≡ [1 + (1 − q)x]^1/(1−q) that emerges from nonextensive statistical mechanics. In particular, we define extended versions of the Mexican hat and the Morlet wavelets. We also introduce new wavelets that are q-generalizations of the trigonometric functions. All cases reduce to the usual ones as q → 1. Within nonextensive statistical mechanics, departures from unity of the entropic index q are expected in the presence of long-range interactions, long-term memory, multi-fractal structures, among others. Consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q. We exemplify with the wavelet transform modulus-maxima method for mono- and multi-fractal self-affine signals.

The self-contained derivation of the inverse eigenvalue problem is given using a discrete approximation of the Sturm–Liouville operator on a bounded interval. Within this approximation, the Hamiltonian is treated as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that the inverse problem procedure is nothing but the well-known Gram–Schmidt orthonormalization in Euclidean space for special vectors numbered by the space coordinate index. All the results of the usual inverse problem with continuous coordinate are reobtained by employing a limiting procedure, including the Goursat problem—the equation in partial derivatives for the solutions of the inversion integral equation.

We study a system of N-bosons in the plane interacting with delta function potentials. After a coupling constant renormalization we show that the Hamiltonian defines a self-adjoint operator and obtain a lower bound for the energy. The same results hold if one includes a regular inter-particle potential.

GL_q(N)- and SO_q(N)-covariant deformations of the completely symmetric/antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices. The precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined. Adopting the GL_q(N)- and SO_q(N)-covariant differential calculi on the corresponding quantum group covariant noncommutative spaces , we introduce a generalized notion of vielbein basis (or 'frame'), based on differential-operator-valued 1-forms. We then give a thorough definition of a SO_q(N)-covariant -bilinear Hodge map acting on the bimodule of differential forms on , introduce the exterior coderivative and show that the Laplacian acts on differential forms exactly as in the undeformed case, namely it acts on each component as it does on functions.

For a family of Poisson algebras, parametrized by , and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context, we rederive the recently found rth dispersionless modified KP hierarchies and rth dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the rth dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless KP, rth dispersionless Dym and rth dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless Dym equation, for which a complete integral is presented and, therefore, an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the rth dispersionless modified KP equation.

We prove the Hawking effect for gravitational collapse of a charged star in an expanding universe or not, stationary in the past and collapsing to a black hole in the future. In the past, the quantum initial state of the Dirac fields was given by a KMS state with unspecified temperature. With the same physical model, this paper generalizes our previous work to the case of a quantum initial state of KMS type rather than a Boulware vacuum.

We propose an explicit formula for a measure of entanglement of pure multipartite quantum states. We discuss the mathematical structure of the measure and give a brief explanation of its physical motivation. We apply the measure on some pure, tripartite, qubit states and demonstrate that, in general, the entanglement can depend on what actions are performed on the various subsystems, and specifically if the parties in possession of the subsystems cooperate or not. We also give some simple but illustrative examples of the entanglement of four-qubit and m-qubit states.

We study the finite-temperature free energy and fermion number for Dirac fields in a one-dimensional spatial segment, under two different members of the family of local boundary conditions defining a self-adjoint Euclidean Dirac operator in two dimensions. For one of such boundary conditions, compatible with the presence of a spectral asymmetry, we discuss in detail the contribution of this part of the spectrum to the zeta-regularized determinant of the Dirac operator and, thus, to the finite-temperature properties of the theory.

The quantum quartic oscillator is investigated in order to test the many-body technique of the continuous unitary transformations. The quartic oscillator is sufficiently simple to allow a detailed study and comparison of various approximation schemes. Due to its simplicity, it can be used as a pedagogical introduction to the unitary transformations. Both the spectrum and the spectral weights are discussed.

We show that Green functions of second-order differential operators with singular or unbounded coefficients can have an anomalous behaviour in comparison to the well-known properties of the Green functions of operators with bounded coefficients. We discuss some consequences of such an anomalous short or long distance behaviour for a diffusion and wave propagation in an inhomogeneous medium.

We show that the massive noncommutative U(1) theory is embedded in a gauge theory using an alternative systematic way [1], which is based on the symplectic framework. The embedded Hamiltonian density is obtained after a finite number of steps in the iterative symplectic process as opposed to the result proposed using the BFFT formalism [2]. This alternative formalism of embedding shows how to get a set of dynamically equivalent embedded Hamiltonian densities.

While the Hall magnetohydrodynamics (MHD) model has been explored in depth in connection with the dispersive waves relevant in magnetic reconnection, a theoretical study of the mathematical features of this system is lacking. We consider here the boundedness of the solutions of the Hall MHD equations. With Dirichlet boundary conditions the total energy of the system is maintained, and dissipated by diffusion, but the behaviour of the higher moments of the magnetic field is more complicated. It is found that certain unusual geometries of the initial condition may lead to a blow-up of the L³-norm of the field. Nevertheless, reasonable assumptions upon the correlation between the size of the magnetic field and the curvature of field lines imply that the magnetic field remains uniformly bounded.