Table of contents

Type-I matrices were introduced recently as finite-dimensional prototypes of quantum integrable systems. These matrices are linearly dependent on an 'interaction' type parameter, and possess interesting properties such as commuting partner matrices and generically violate the von Neumann Wigner noncrossing rule. The important role of Plücker relations in this construction is noted. Type-I matrices are given a transparent formulation in terms of Fermi or Bose-type particle operators; they represent a quantum glass model with either Fermi or Bose statistics, with several free parameters that may be chosen at will.

The equations we investigate describe the nonlinear and nonlocal interaction of waves in 1+1 dimensions with potential applicability. Although these equations are integrable, the solution of the initial and boundary value problem requires a judicious use of spectral methods. In particular, we show that spectral data evolve linearly only for special boundary values.

Numerous lattice Boltzmann (LB) methods have been proposed for solution of the convection–diffusion equations (CDE). For the 2D problem, D2Q9, D2Q5 or D2Q4 velocity models are usually used. When LB convection–diffusion models are used to solve a CDE coupled with Navier–Stokes equations, boundary conditions are found to be critically important for accurately solving the coupled simulations. Following the idea of a regularized scheme (Latt et al 2008 Phys. Rev. E 77 056703), a regularized boundary condition for solving a CDE is proposed. A simple extrapolation scheme is also proposed for the Neumann boundary condition. Spatial accuracies of three existing and the proposed boundary conditions are discussed in details. The numerical evaluations are based on simulations of steady and unsteady natural convection flows in a cavity and an unsteady Taylor–Couette flow. Our studies show that the simplest D2Q4 model with terms of O(u) in the equilibrium distribution function is capable of obtaining results of equal accuracy as D2Q5 or D2Q9 models for the CDE. A slightly revised LB equation for solving a CDE that is used to cancel some unwanted terms does not seem to be necessary for incompressible flows. The regularized boundary condition for solving the CDE has second-order spatial accuracy and it is the best one in terms of the spatial accuracy. The regularized scheme and non-equilibrium extrapolation scheme are applicable to handle both the Dirichlet and Neumann boundary conditions. For the Neumann boundary condition with zero flux, all the five boundary conditions are applicable to give accurate results and the bounce-back scheme is the simplest one.

Motivated by recent research on complex networks, we study enhancing complex communication networks against intentional attack which takes down network nodes in a decreasing order of their degrees. Specifically, we evaluate an effect which has been largely ignored in existing studies; many real-life systems, especially communication systems, have protection mechanisms for their important components. Due to the existence of such protection, it is generally quite difficult to totally crash a protected node, though partially paralyzing it may still be feasible. Our analytical and simulation results show that such 'imperfect' protections generally speaking still help significantly enhance network robustness. Such insight may be helpful for the future developments of efficient network attack and protection schemes.

The equations for strands of rigid charge configurations interacting nonlocally are formulated on the special Euclidean group, SE(3), which naturally generates helical conformations. Helical stationary shapes are found by minimizing the energy for rigid charge configurations positioned along an infinitely long molecule with charges that are off-axis. The classical energy landscape for such a molecule is complex with a large number of energy minima, even when limited to helical shapes. The question of linear stability and selection of stationary shapes is studied using an SE(3) method that naturally accounts for the helical geometry. We investigate the linear stability of a general helical polymer that possesses torque-inducing nonlocal self-interactions and find the exact dispersion relation for the stability of the helical shapes with an arbitrary interaction potential. We explicitly determine the linearization operators and compute the numerical stability for the particular example of a linear polymer comprising a flexible rod with a repeated configuration of two equal and opposite off-axis charges, thereby showing that even in this simple case the nonlocal terms can induce instability that leads to the rod assuming helical shapes.

In the paper, we mainly study the existence and uniqueness of global weak solutions for the Novikov equation. We first recall some results and definitions on strong solutions and weak solutions for the equation. Then, we show that the equation has smooth solutions which exist globally in time, provided the initial data satisfy certain sign conditions. Finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions.

We establish that, under some simple integral conditions regarding the nonlinearity, the (1 + α)-order fractional differential equation ₀D^α_t(x') + f(t, x) = 0, t > 0, has a solution , with , which can be expanded asymptotically as a + bt^α + O(t^{α − 1}) when t → +∞ for given real numbers a, b. Our arguments are based on fixed point theory. Here, ₀D^α_t designates the Riemann–Liouville derivative of order α ∊ (0, 1).

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group. Excited quasiparticle/quasihole states are members of multiplets under the rotation group and generically there is a nontrivial highest weight member of the multiplet from which all states can be constructed. Some of the trial states proposed in the literature belong to classical families of symmetric polynomials. In this paper we study Macdonald and Jack polynomials that are highest weight states. For Macdonald polynomials, it is a (q, t)-deformation of the raising angular momentum operator that defines the highest weight condition. By specialization of the parameters we obtain a classification of the highest weight Jack polynomials. Our results are valid in the case of staircase and rectangular partition indexing the polynomials.

A direct relation is established between the constants of motion for conformal mechanics and those for its spherical part. In this way, we find the complete set of functionally independent constants of motion for the so-called cuboctahedric Higgs oscillator, which is just the spherical part of the rational A₃ Calogero model (describing four Calogero particles after decoupling their center of mass).

Renormalization group (RG) methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings. However, in cases when the β-function depends only on the coupling, the RG technology does include a part—which is related to the equation for the running coupling and which we will call the 'single variable RG' (SVRG)—that turns out to be a purely formal method for improving the convergence properties of series expansions. In fact, we show that SVRG is an integral Hermite–Padé approximation preceded by a series reversion. A complementary viewpoint on what SVRG does to an expansion is that it performs a 'minimal upgrade to a self-similar form', the minimality being crucial. In applications, SVRG is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable, but also the exponent (but not the prefactor) of the scaling law for large values of the variable. As a practical example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb–Liniger gas. Using a new result for the fourth-order term in the short-distance expansion, we find remarkable agreement with known quantum diffusion Monte Carlo numerical results.

A new approach to constructing coherent states (CS) and semiclassical states (SS) in a magnetic-solenoid field is proposed. The main idea is based on the fact that the AB solenoid breaks the translational symmetry in the xy-plane; this has a topological effect such that there appear two types of trajectories which embrace and do not embrace the solenoid. Due to this fact, one has to construct two different kinds of CS/SS which correspond to such trajectories in the semiclassical limit. Following this idea, we construct CS in two steps, first the instantaneous CS (ICS) and then the time-dependent CS/SS as an evolution of the ICS. The construction is realized for nonrelativistic and relativistic spinning particles both in (2 + 1) and (3 + 1) dimensions and gives a non-trivial example of SS/CS for systems with a nonquadratic Hamiltonian. It is stressed that CS depending on their parameters (quantum numbers) describe both pure quantum and semiclassical states. An analysis is represented that classifies parameters of the CS in such respect. Such a classification is used for the semiclassical decompositions of various physical quantities.

We found that the well-established Fowler–Nordheim–Schottky field emission theory needs to be revisited for strong electric fields F. The classical derivation of the electron tunneling probability through the triangular potential barrier is re-examined and specified. This probability is studied in the fields of arbitrary strength and found as a function with a maximum at some value of the field and decaying when F goes to zero and to ∞. The location and height of this maximum depend only on the ratio of the electron kinetic energy to the work function, but the maximum cannot be realized for real materials. A simple interpolation formula for all possible electric fields is given. The domain of validity of the standard Fowler–Nordheim approximation is shown to be very wide and evaluated in detail. By solving the Schrödinger equation with the help of a power series expansion of the electron wavefunction, the standard Schottky potential is shown to make the tunneling impossible. This can be fixed tentatively by replacing the image potential with a more realistic modification which eliminates its non-physical singularity. The power series method promises to find a wider application in the field emission theory.

The manifestation of measurements, randomly distributed in time, on the evolution of quantum systems are analyzed in detail. The set of randomly distributed measurements (RDM) is modeled within the renewal theory, in which the distribution is characterized by the probability density function (PDF) W(t) of times t between successive events (measurements). The evolution of the quantum system affected by the RDM is shown to be described by the density matrix satisfying the stochastic Liouville equation. This equation is applied to the analysis of the RDM effect on the evolution of a two-level system for different types of RDM statistics, corresponding to different PDFs W(t). Obtained general results are illustrated as applied to the cases of the Poissonian () and anomalous (W(t) ∼ 1/t^{1 + α}, α ⩽ 1) RDM statistics. In particular, specific features of the quantum and inverse Zeno effects, resulting from the RDM, are thoroughly discussed.

In this paper, we investigate various dynamical properties of the Jaynes–Cummings (JC) model beyond the rotating wave approximation (RWA). We first show that in the absence of the RWA, the JC Hamiltonian can be transformed into an intensity-dependent Hamiltonian. Corrections produced by the counter-rotating terms (CRTs) appear in the first order as the intensity-dependent detuning, i.e. the dynamical Stark shift and in the second order as the intensity-dependent atom–field coupling. Then, by determining the atom–field wavefunction evolution, we study the effects of CRTs on various dynamical properties of the JC model, including the atomic population inversion, atomic dipole squeezing, atomic entropy squeezing, photon counting statistics, quadrature field squeezing, field entropy squeezing and quantum phase properties of the cavity field.

Dynamics of coded information over Bloch channels is investigated. We show that the coded information is sent with high accuracy over the Bloch channel by increasing the absolute equilibrium value of the information carrier or decreasing the ratio of relaxation time. The robustness of coded information in maximum and partial entangled states is discussed. It is shown that, the maximum entangled states are more robust than the partial entangled state over these types of channels. The dynamics of the local and the non-local information is investigated for different values of the channel's parameters and the initial state setting. It is found that by increasing the absolute equilibrium values for both qubits, the local information of both qubits decreases faster and consequently the information gained by the eavesdropper increases.

We investigate symmetries of the scalar field theory with a harmonic term on the Moyal space with the Euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can also be restored at the quantum level by restricting the symplectic structures to a particular orbit.

The question of Lorentz invariance in the membrane matrix model is addressed.

We give an explicit differential equation which is expected to determine the instanton partition function in the presence of the full surface operator in SU(N) gauge theory. The differential equation arises as a quantization of a certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and Tsuda.

We study the Laplacian on Stenzel spaces (generalized deformed conifolds), which are tangent bundles of spheres endowed with Ricci flat metrics. The (2d − 2)-dimensional Stenzel space has SO(d) symmetry and can be embedded in through the equation ∑^d_{i = 1}z_i² = ². We discuss Green's function with a source at a point on the S^{d − 1} zero section of TS^{d − 1}. Its calculation is complicated by mixing between different harmonics with the same SO(d) quantum numbers due to the explicit breaking by the -deformation of the U(1) symmetry that rotates z_i by a phase. A similar mixing affects the spectrum of normal modes of warped deformed conifolds that appear in gauge/gravity duality. We solve the mixing problem numerically to determine certain bound state spectra in various representations of SO(d) for the d = 4 and d = 5 examples.

Classical mechanics has only been invoked to account for Landau damping in a rather half-hearted way, alongside plasma perturbation theory. In particular this invocation is essential for the study of the saturation, or post-linear (or 'nonlinear') regime of the damping initiated by Dawson and O'Neill. By embracing mechanics wholeheartedly here, with its attendant phase space, one can access results, old and new, cleanly and directly, and with one fewer numerical integration for the post-linear regime. By using a summation technique familiar in semiclassical quantum mechanics (Poisson summation), the one remaining numerical integration can be much improved in accuracy. Also accessible from mechanics is the ultimate entropy gain. Though zero for any finite time (in the absence of coarse graining), the entropy gain is ultimately non-zero (at infinite time the required coarse graining is zero). It is calculated analytically by using the appropriate asymptotics, hitherto not fully exploited.