On a Generalized Frobenius–Stickelberger Addition Formula

In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) -function of a genus 2 curve in the case of three vector-valued variables. The result is given explicitly in the form of a polynomial in Kleinian -functions.     

On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization

We give a proof of (a slightly refined version of) a graph theoretic formula due to Gammelgaard, Karabegov and Schlichenmaier for Berezin–Toeplitz quantization on Kähler manifolds. We obtain the formula by inverting the Berezin transform using a composition formula for the ring of differential operators encoded by linear combinations of strongly connected graphs. The same method is also used to identify the dual Karabegov–Bordemann–Waldmann star product. Our proof has the merit of giving more insight into Karabegov–Schlichenmaier’s identification theorem (Karabegov in J Reine Angew Math 540:49–76, 2001) that the Karabegov classifying form of the Berezin and Berezin–Toeplitz star products are, respectively, obtained by deforming the Kähler metric along the Ricci curvature and the logarithm of the Bergman kernel.     

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 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.     

Extension of the Arrowsmith–Essam Formula to the Domany–Kinzel Model

Arrowsmith and Essam gave an expansion formula for point-to-point connectedness functions of the mixed site-bond percolation model on oriented lattices, in which each term is characterized by a graph. We extend this formula to general k-point correlation functions, which are point-to-set (with k points) connectivities in the context of percolation, of the two-neighbor discrete-time Markov process (stochastic cellular automata with two parameters) in one dimension called the Domany–Kinzel model, which includes the mixed site-bond oriented percolation model on a square lattice as a special case. Our proof of the formula is elementary and based on induction with respect to time-step, which is different from the original graph-theoretical one given by Arrowsmith and Essam. We introduce a system of m interacting random walkers called m friendly walkers (m FW) with two parameters. Following the argument of Cardy and Colaiori, it is shown that our formula is useful to derive a theorem that the correlation functions of the Domany–Kinzel model are obtained as an m0 limit of the generating functions of the m FW.     

On the Leontiev Formula for the Grüneisen Parameter of Solid States

Physics of the Solid State - It has been established that the Leontiev formula is in satisfactory agreement with the Grüneisen equation. A comment on the derivation of this formula has been...     

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 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 Darboux integrability of a three–dimensional forced–damped differential system

In 2011 Pehlivan proposed a three–dimensional forced–damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.     

On Gammelgaard’s Formula for a Star Product with Separation of Variables

We show that Gammelgaard’s formula expressing a star product with separation of variables on a pseudo-Kähler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard’s formula which gives more insight into the question why the directed graphs in his formula have no cycles.     

On the mechanical behaviours of a craze in particulate–polymer composites

In polymeric composites, well-defined inclusions are incorporated into the polymer matrix to alleviate the brittleness of polymers. When a craze is initiated in such a composite, the interaction between the craze and the surrounding inclusions will greatly affect the composite’s mechanical behaviours and toughness. To the best knowledge of the authors, only little research work has been found so far on the interaction between a craze and the near-by inclusions in particulate–polymer composites. In the current study, the first time, the influences of the surrounding inclusions on the craze are investigated in particulate–polymer composites. The three-phase model is adopted to study the fracture behaviours of the craze affected by multiple inclusions. An iterative procedure is proposed to solve the stress intensity factors. Parametric studies are performed to investigate the influences of the reinforcing particle volume fraction and the shear modulus ratio on fracture behaviours of particulate–polymer composites.     

On the Thermal Explosion Theory of a Heterogeneous Liquid–Liquid System

In this paper, the regularities of a thermal explosion of a heterogeneous system consisting of two immiscible liquids have been studied. Each phase is a solution of A and B reagents. Reagent B is extracted into a solution of reagent A, where the bimolecular exothermic reaction A + B → Products takes place. It has been shown that an exothermic reaction (combustion regime) continues to proceed in the system at high mass-exchange rates between phases after a thermal explosion. As a result, the maximal temperature may significantly exceed the temperature of the thermal explosion. The critical value of the Semenov parameter decreases with an increase in the mass-exchange rate between phases. In the limited range of values of the distribution coefficient of reagent B between phases, the increase of this coefficient is also accompanied by a decrease in the critical value of the Semenov parameter. The concentration of reagent B in the initial phase decreases monotonically due to its extraction into another phase. However, the equilibrium of the extraction of reagent B can shift, due to the temperature dependence of the distribution coefficient during the reaction. Thus, the time dependence of the concentration of reagent B on may be more complex and can pass through a minimum.     

On the stability of the shear–free condition

The evolution equation for the shear is reobtained for a spherically symmetric anisotropic, viscous dissipative fluid distribution, which allows us to investigate conditions for the stability of the shear–free condition. The specific case of geodesic fluids is considered in detail, showing that the shear–free condition, in this particular case, may be unstable, the departure from the shear–free condition being controlled by the expansion scalar and a single scalar function defined in terms of the anisotropy of the pressure, the shear viscosity and the Weyl tensor or, alternatively, in terms of the anisotropy of the pressure, the dissipative variables and the energy density inhomogeneity.     

On the asymptotic character of electromagnetic waves in a Friedmann–Robertson–Walker universe

Asymptotic properties of electromagnetic waves are studied within the context of Friedmann–Robertson–Walker (FRW) cosmology. Electromagnetic fields are considered as small perturbations on the background spacetime and Maxwells equations are solved for all three cases of flat, closed and open FRW universes. The asymptotic character of these solutions is investigated and their relevance to the problem of cosmological tails of electromagnetic waves is discussed.     

On the motion of a quantum particle in the spinning cosmic string space–time

We analyze the energy spectrum and the wave function of a particle subjected to magnetic field in the spinning cosmic string space–time and investigate the influence of the spinning reference frame and topological defect on the system. To do this we solve Schrödinger equation in the spinning cosmic string background. In our work, instead of using an approximation in the calculations, we use the quasi-exact ansatz approach which gives the exact solutions for some primary levels.     

On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U _x (y)=ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f(r,v)=f(V), with Vv–U(r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K()lim ₀ ^–2 (–), where is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker–Planck operator. We have found analytically that for shear rates larger than a certain threshold value a _th0.3520 (where is an average collision frequency and a _th/ is the real root of the cubic equation 64x ³+16x ²+12x–9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f(V;a)|V|^–4–(a) (;a), where tan V _y /V _x and the angular distribution function (;a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition (;a)=(+;a) allows one to obtain the exponent (a) as a function of the shear rate. It diverges when aa _th and tends to a minimum value _min1.252 in the limit a. As a consequence of this power-law decay for a>a _th, all the velocity moments of a degree equal to or larger than 2+(a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~(a), which rotates from ~=–/4,3/4 when aa _th to ~=0, in the limit a.     

Towards a Skyrme–Hartree–Fock–Bogolyubov Mass Formula

In this work, we explain our astrophysical motivations for deriving a mass formula based on HFB calculations with a Skyrme interaction. We give an overview of existing mass formulae and present briefly the last HF+BCS mass formula [1]. The Skyrme force MSk7 [1] is considered in the study of shell effects at N=82, in the neutron-rich region far from stability, within the HFB and HF+BCS theories, and compared with results obtained using the forces SkP^δ and SkP^δρ [2]. This revised version was published online in July 2006 with corrections to the Cover Date.