On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.     

On the phase diagram of a two-dimensional electron–hole system

The phase diagram for a system of spatially separated electrons and holes in coupled quantum wells or graphene double layers is studied in the framework of a BCS-like mean-field approach and a Landau expansion in terms of the pairing order parameter. We find a second order transition between an electron–hole plasma and a BCS phase, as well as a first-order transition between the BCS phase and a bosonic Mott phase of tightly bound electron–hole pairs without phase coherence. The electron–hole plasma exists at low and at high densities for weak interaction, the BCS phase at moderate density and the Mott phase at high density and strong interaction.     

On the ζ-function of a one-dimensional classical system of hard-rods

The -function of a one-dimensional classical hard-rod system with exponential pair interaction is defined as the generating function for the partition function of the system with periodic boundary conditions. It is shown, here, that the -function for this system is simply related to the traces of the restrictions of the Ruelle's transfer matrix, and related operators to a suitable function space. This -function does not, in general, extend to a meromorphic function.     

On a q-analog of the Wallach–Okounkov Formula

We obtain a q-analog of the well-known result on a joint spectrum of invariant differential operators with polynomial coefficients on a prehomogeneous vector space of complex n × n-matrices. We are motivated by applications to the problems of harmonic analysis in the quantum matrix ball: our main theorem can be used to prove the Plancherel formula (to be published).     

On a relation of the angular frequency to the Aharonov–Casher geometric phase in a quantum dot

By analysing the behaviour of a neutral particle with permanent magnetic dipole moment confined to a quantum dot in the presence of a radial electric field, Coulomb-type and linear confining potentials, then, an Aharonov–Bohm-type effect for bound states and a dependence of the angular frequency of the system on the Aharonov–Casher geometric phase and the quantum numbers associated with the radial modes, the angular momentum and the spin are obtained. In particular, the possible values of the angular frequency and the persistent spin currents associated with the ground state are investigated in two different cases.     

On a Local Version of the Bak–Sneppen Model

A major difficulty in studying the Bak–Sneppen model is in effectively comparing it with well-understood models. This stems from the use of two geometries: complete graph geometry to locate the global fitness minimizer, and graph geometry to replace the species in the neighborhood of the minimizer. Over the years a number of models inspired by Bak–Sneppen were studied, usually by introducing different or new features (e.g. discretizing fitness, randomized neighbors or population size). We present a variant that only uses features present in Bak–Sneppen, and whose difference from the Bak–Sneppen is that only the graph geometry is used for the evolution. This allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model. We use this to show that for constant-degree graphs, the stationary fitness distribution converges to an IID law as the number of vertices tends to infinity. We also discuss exponential ergodicity through coupling, and avalanches for the model.     

On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations

In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s ? 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s ? 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, &; Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).     

On the Klein–Gordon oscillator subject to a Coulomb-type potential

By introducing the scalar potential as modification in the mass term of the Klein–Gordon equation, the influence of a Coulomb-type potential on the Klein–Gordon oscillator is investigated. Relativistic bound states solutions are achieved to both attractive and repulsive Coulomb-type potentials and the arising of a quantum effect characterized by the dependence of angular frequency of the Klein–Gordon oscillator on the quantum numbers of the system is shown.     

On the possibility of generation of high-power low-frequency radiation as a result of evolution of explosive instability in the flow–nonisothermal plasma system

We discuss the possibility of realization of explosive instability in a resonance triplet of longitudinal waves, specifically, an ion Langmuir mode and two beam modes (slow and fast ones) in the system of a beam of charged particles and nonisothermal plasma. Estimates for the plasmasphere of the Earth are given.     

On the dressing method for the generalised Zakharov–Shabat system

The dressing procedure for the generalised Zakharov–Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ‘dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.     

On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system

In this paper, we deal with the computation of ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system. To this end, backward Euler and time-splitting pseudospectral methods are proposed for the nonlinear Schrödinger equation with the nonlocal Hartree potential approximated by solving a Poisson equation. The approximation approaches for the Hartree potential include fast convolution algorithms, which are accelerated by using FFT in 1D and fast multipole method (FMM) in 2D and 3D, and sine/Fourier pseudospectral methods. The inconsistency in 0-mode in Fourier pseudospectral approach is pointed out, which results in a significant loss of high-order of accuracy as expected for spectral methods. Numerical comparisons show that in 1D the fast convolution and sine pseudospectral approaches are compatible. While, in 3D the fast convolution approach based on FMM is second-order accurate and the Fourier pseudospectral approach is better than it from both efficiency and accuracy point of view. Among all these approaches, the sine pseudospectral one is the best candidate in the numerics of the SPS system. Finally, we apply the backward Euler and time-splitting sine pseudospectral methods to study the ground state and dynamics of 3D SPS system in different setups.     

On colour non-singlet representations of the quark–gluon system at finite temperature

We use a group theoretical technique to project out the partition function for a system of quarks, antiquarks and gluons onto a particular representation of the internal symmetry group SU(3): the colour singlet, colour octet and colour 27-plet, at finite temperature. We do this to calculate the thermodynamic quantities for those representations. We also calculate the change in free energy of the plasma droplet formed from the hot hadronic gas. We find that the size of the droplet in the colour-octet representation is smaller than that in the colour-singlet representations at different temperatures in the vicinity of the critical temperatures of the phase transitions. Received: 1 February 1999 / Revised version: 24 February 2000 / Published online: 18 May 2000     

On the Theory of the Mandelstam–Brillouin Scattering in a Plasma Layer

Optics and Spectroscopy - The evolution of a perturbation from a local source upon the Mandelstam–Brillouin scattering in a plasma layer of a finite thickness and infinite length is examined...     

On the effect of lead concentration on the kinetics of contact melting in the Sn–Pb–Bi system in the presence of electromigration

It is determined by electron microscopy that the contact layer in the Sn–(Bi + 30 wt % Pb) system contains separate solid inclusions, while the contact layer in the Sn–(Bi + 40 wt % Pb) system is microheterogeneous. The observed structural states of alloys in the contact layers are explained by a change in the concentration of the initial contacting samples. The effect of the alloy structure and electromigration on the kinetics of contact melting is found.     

On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds

In this paper, we study the influence of the Aharonov–Casher effect [Y. Aharonov, A. Casher, Phys. Rev. Lett. 53 (1984) 319] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov–Casher geometric phase. We also discuss the influence of curvature and torsion on the relativistic energy levels and the Dirac spinors due to the topology of the cosmic string and cosmic dislocation spacetimes.     

Enhancement of the shot noise of a quantum dot–Luttinger lead system

We investigate the joint effects of the intralead electron interaction and Coulombic dot–lead interaction on the shot noise of a quantum dot coupled to Luttinger liquid leads. A formula of the shot noise is derived by applying the nonequilibrium Green function technique. The shot noise is enhanced by the dot–lead interaction. For a weak or moderately strong interaction the differential shot noise demonstrates resonant-like behavior as a function of bias and gate voltages. In the limit of strong interaction resonant behavior disappears and the differential shot noise and Fano factor scale as a power law in bias voltage. Under some parameters, the differential shot noise may become negative around resonant peaks, and the physical reason is analyzed.     

On the possibility of dark energy from corrections to the Wheeler–DeWitt equation

We present a method for approximating the effective consequence of generic quantum gravity corrections to the Wheeler–DeWitt equation. We show that in many cases these corrections can produce departures from classical physics at large scales and that this behaviour can be interpreted as additional matter components. This opens up the possibility that dark energy (and possible dark matter) could be large scale manifestations of quantum gravity corrections to classical general relativity. As a specific example we examine the first order corrections to the Wheeler–DeWitt equation arising from loop quantum cosmology in the absence of lattice refinement and show how the ultimate breakdown in large scale physics occurs.     

On the Acid–Base State of the Surface of Semiconductor Components of the ZnSe–CdS System Exposed to Different Media

Using the methods of hydrolytic adsorption, mechanochemistry, and nonaqueous conductometric titration, the acid–base properties of the surfaces of binary semiconductors and solid solutions of the ZnSe?CdS system are studied. The nature of acid sites, and the extent, character, and mechanism of the surface (absorption) interaction are established. The interrelationships in the changes of the acid–base and bulk physical properties upon changing the chemical composition are determined, which make it possible to draw conclusions that are fundamentally important in scientific and practical aspects.