Table of contents

Recently the 'toolkit' discretization introduced to accelerate the numerical resolution of the time-dependent Schrödinger equation arising in quantum optimal control problems demonstrated good results on a large range of models. However, when coupling this class of methods with the so-called monotonically convergent algorithms, numerical instabilities affect the convergence of the discretized scheme. We present an adaptation of the 'toolkit' method which preserves the monotonicity of the procedure. The theoretical properties of the new algorithm are illustrated by numerical simulations.

Generalizing results by Bryant and Griffiths (1995 Duke Math. J.78 531), we completely describe local conservation laws of second-order (1 + 1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.

The paper On the multifractal nature of fully developed turbulence and chaotic systems, by Benzi et al (1984 J. Phys. A: Math. Gen.17 3521) has been a starting point of many investigations on the different faces of self-similarity and intermittency in turbulent phenomena. Since then, the multifractal model has become a useful tool for the study of small-scale turbulence, in particular for detailed predictions of different Eulerian and Lagrangian statistical properties. On the occasion of the 50th birthday of our unforgettable friend and colleague Giovanni Paladin (1958–1996), we review here the basic concepts and some applications of the multifractal model for turbulence.

The steady state of a generalized coagulation–decoagulation model on a one-dimensional lattice with reflecting boundaries is studied using a matrix-product approach. It is shown that the quadratic algebra of the model has a four-dimensional representation provided that some constraints on the microscopic reaction rates are fulfilled. The dynamics of a product shock measure with two shock fronts, generated by the Hamiltonian of this model, is also studied. It turns out that the shock fronts move on the lattice as two simple random walkers which repel each other provided that the same constraints on the microscopic reaction rates are satisfied.

We solve the dynamics of large spherical minority games (MG) in the presence of non-negligible time-dependent external contributions to the overall market bid. The latter represent the actions of market regulators or other major natural or political events that impact on the market. In contrast to non-spherical MGs, the spherical formulation allows one to derive closed dynamical order parameter equations in an explicit form and work out the market's response to such events fully analytically. We focus on a comparison between the response to stationary versus oscillating market interventions, and reveal profound and partially unexpected differences in terms of transition lines and the volatility.

We use functional Bethe ansatz equations to calculate the cumulants of the total current in the partially asymmetric exclusion process. We recover known formulae for the first two cumulants (mean value of the current and diffusion constant) and obtain an explicit finite-size formula for the third cumulant. The expression for the third cumulant takes a simple integral form in the limit where the asymmetry scales as the inverse of the square root of the size of the system, which corresponds to a natural separation between weak and strong asymmetry.

A K-user direct-sequence spread-spectrum code-division multiple-access (CDMA) system with (q ≪ log₂K)-bit baseband signal quantization at the demodulator is considered. It is shown that additionally quantizing the K + 1 level output signal of the CDMA modulator into q bits improves significantly the average bit-error performance in a non-negligible regime of noise variance, σ², and user load, β, under various system settings, like additive white Gaussian noise (AWGN), Rayleigh fading, single-user detection, multi-user detection, random and orthogonal spreading codes. For the case of single-user detection in random spreading AWGN-CDMA, this regime is identified explicitly as , where γ(q) is a certain pre-factor depending on q, and the associated BER improvement is derived analytically for q = 1, 2. For the other examined system settings, computer simulations are provided, corroborating this interesting behavior.

We derive P(M, t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved, for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path-integral technique; however, an alternative approach is also outlined which is founded on certain 'agreement formulae' that are encountered more generally in probabilistic studies of Brownian motion processes.

The multi-dimensional q-deformed bosonic Newton oscillator algebra with U(d)-symmetry is considered. The high- and low-temperature thermostatistical properties of a gas of the q-deformed bosonic Newton oscillators are obtained in the thermodynamical limit. It is shown that the Bose–Einstein condensation occurs in such a gas for values of the real deformation parameter q smaller than 1. However, the ordinary boson gas results can be recovered in the limit q = 1.

We investigate the chaotic lowest energy region of the simplified breathing circular billiard, a two-dimensional generalization of the Fermi model. When the oscillation amplitude of the breathing boundary is small and we are near the integrable to non-integrable transition, we obtain numerically that average quantities can be described by scaling functions. We also show that the map that describes this model is locally equivalent to Chirikov's standard map in the region of the phase space near the first invariant spanning curve.

We present a trajectory-based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities, in the regime of many open channels. Our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories, therefore generalizing previous works on conductance and shot noise. The semiclassical results agree, for all values of m, with the corresponding predictions from random matrix theory.

The propagation of coherent, polarized light in a nematic liquid crystal, governed by the nematicon equations, is considered. It is found that in the special case of 1 + 1 dimensions and the highly nonlocal limit, the nematicon equations have an asymptotic bulk solitary wave solution, termed a nematicon, which is given in terms of Bessel functions. This asymptotic solution gives both the ground state and the symmetric and antisymmetric excited states, which have multiple peaks. Numerical simulations of nematicon evolution, for parameters corresponding to experimental scenarios, are presented. It is found, for experimentally reasonable parameter choices, that the validity of the nonlocal approximation depends on the type of nematicon, as in some cases the asymptotic nematicon undergoes large amplitude oscillations. The magnitude of the nonlocality parameter for the asymptotic nematicon amplitude to be constant over a typical experimental propagation distance is also determined.

We present a novel approach for performing multi-dimensional integration of arbitrary functions. The method starts with Smolyak-type sparse grids as cubature formulae on the unit cube and uses a transformation of coordinates based on the conditional distribution method to adapt those formulae to real space. Our method is tested on integrals in one, two, three and six dimensions. The three dimensional integration formulae are used to evaluate atomic interaction energies via the Gordon–Kim model. The six dimensional integration formulae are tested in conjunction with the nonlocal exchange-correlation energy functional proposed by Lee and Parr. This methodology is versatile and powerful; we contemplate application to frozen-density embedding, next-generation molecular-mechanics force fields, 'kernel-type' exchange-correlation energy functionals and pair-density functional theory.

Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form. The open-system density operator can be represented as a statistical mixture over non-unitarily evolving pure states, driven by the stochastic nonlinear Schrödinger equation (sNLSE). The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system. This solution is a generalized coherent state (GCS), associated with the algebra. The result is based on proving that the GCS minimizes the trace-norm of the covariance matrix, associated with the spectrum-generating algebra.

We classify regular subalgebras of Kac–Moody algebras in terms of their root systems. In the process, we establish that a root system of a subalgebra is always an intersection of the root system of the algebra with a sublattice of its root lattice. We also discuss applications to investigations of regular subalgebras of hyperbolic Kac–Moody algebras and conformally invariant subalgebras of affine Kac–Moody algebras. In particular, we provide explicit formulae for determining all Virasoro charges in coset constructions that involve regular subalgebras.

We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear 2 × 2 system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the Schlesinger system and the sixth Painlevé equation is given from this linear system. A unified approach to treat the other Painlevé equations from the usual two-component KP hierarchy is also considered.

In this paper, we develop the asymptotic formulae, obtained in my previous papers, for the band functions and the Bloch functions of the Schrödinger operator with the smooth periodic potentials. Then using these formulae, we determine constructively a family of spectral invariants of this operator from the given band functions. Some of these invariants generalize the well-known invariants and others are entirely new. The new invariants are explicitly expressed by Fourier coefficients of the potential which present the possibility of determining the potential constructively by using the band functions as input data.

It is shown that for inhomogeneous Lie algebras with only one Casimir operator, the latter can be explicitly constructed from the Maurer–Cartan equations by means of wedge products. It is further proved that this constraint imposes sharp bounds for the dimension of the representation R defining the semidirect product. The procedure is generalized to compute also the rational invariant of some Lie algebras.

We propose a concrete model which exhibits ferromagnetism and electron-pair condensation simultaneously. The model is defined on two chains and consists of the electron hopping term, the on-site Coulomb repulsion and a ferromagnetic interaction which describes ferromagnetic coupling between two electrons, one on a bond in a chain and the other on a site in the other chain. It is rigorously shown that the model has fully-polarized ferromagnetic pairing ground states. The higher dimensional version of the model is also presented.

We consider a matrix-valued version of the bispectral problem, that is, find differential operators and with matrix coefficients such that there exists a family of matrix-valued common eigenfunctions ψ(x, z): where f and Θ are matrix-valued functions. Using quasideterminants, we prove that the operators L obtained by non-degenerated rational matrix Darboux transformations from are bispectral operators, where and D is a diagonal matrix. We also give a procedure to find an explicit formula for the operator B extending previous results in the scalar case.

We consider tripartite entangled states for continuous variable systems of EPR type, which generalize the famous bipartite CV EPR states (eigenvectors of conjugate choices X₁ − X₂, P₁ + P₂, of the systems' relative position and total momentum variables). We give the regularized forms of such tripartite EPR states in second-quantized formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r → ∞, or tanh r → 1⁻ (approaching the EPR states in the bipartite case), our regularized tripartite EPR states show singular behaviour not only in the approach to the EPR-type region (s → 1 in our notation), but also for an additional, auxiliary regime of the regulator . While the s → 1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalized CHSH inequalities. Violations of the tripartite CHSH bound (B₃ ⩽ 2) are established, with in the canonical regime (s → 1⁺), as well as in the auxiliary regime .

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.

Hertz potentials are used as an alternative to Fresnel's equation of wave normals to analyse harmonic plane wave propagation in uniaxially anisotropic media. Wave vector and amplitudes of ordinary and extraordinary waves are explicitly given. Refraction of a TM field at the plane face of a uniaxial medium is discussed and it is shown that in this particular situation, the refracted wave is identified with the extraordinary wave. Hertz potentials are also a powerful tool to tackle the same problems when harmonic plane waves are changed into Gaussian beams.

We consider the possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants. We find relations between the directions of the vector field and the eigenvectors of the derivative of the back-to-label map near the singularity. We also find an invariant when we follow the motion of the integral curves of the vector field. For the 3D incompressible Euler equations these results have immediate implications about the directions of the vortex stretching and the material stretching near the possible singularities. We also have similar applications to other inviscid fluid equations such as the 2D quasi-geostrophic equation and the 3D magnetohydrodynamics equations.

The first sentence in the caption of figure 15, 'Fragments of figure 10(d) in larger scales in the left (figures left) and right lanes (right)' should be replaced by 'Simulated single-vehicle speed data within a mega-jam presented in space and time by regions with variable darkness (the lower the speed, the darker the region). Left and right figures are related to the left and right lanes, respectively'.