Table of contents

Warped products provide a rich class of physically significant geometric objects. The existence of compact Einstein warped products was questioned in Besse (1987 Einstein Manifolds, section 9.103). It is shown that there exists a metric on every compact manifold B such that (non-trivial) Einstein warped products, with base B, cannot be constructed.

We use Monte Carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity. We investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes. In addition, we examine the effect of stiffness on the writhe of the polygons.

The goal of this letter is to give an elementary approach to the solution of Euler–Frahm equations for the Manakov four-dimensional case. For this, we use the Kötter approach and some results from the original papers by Schottky, Weber and Caspary. We hope that such an approach will be useful for the solution of the problem of an n-dimensional top.

This paper introduces a new theoretical formulation based on a composition method and a statistical discretization approach by matrical block. Anisotropic properties and boundary conditions are considered, introducing the analytical bounds expressions of the effective dielectric constants in the limit of the long-wavelength regime for an idealized superlattice (SL) possessing two directions of periodicity (2D-SL). Such a SL can be described as a multilayer array of alternating cells, (N × M) rectangular dielectric bars, allowing the structure to be shaped as a function of the dielectric constants of each of the anisotropic constituents. It is worth noting that in the simplified case of a 2D-SL made of only two different isotropic materials showing off the same periodicity in both directions, our general matrix formulation, due to the alternative composition laws, leads to the well-established results called, respectively, 'Wiener's and Lichtenecker's bounds' regarding the dielectric constant. This new formalism refashions the concept of bounds of effective dielectric tensors and the notion of form birefringence applied to 2D-SL, with relevant (N × M) rectangular anisotropic columns for arbitrary symmetries in the low-frequency model.

We show the conditions under which nonlinear time-delayed dynamical systems with multiplicative noise sources can be transformed into linear time-delayed systems with additive noise sources. We show that, for such reducible systems, analytical expressions for stationary distributions can be obtained. We demonstrate that fluctuation–dissipation relations of reducible systems become trivial and we show that reducible systems may exhibit delay- and noise-induced transitions to bistability and secondary transitions to non-stationarity. Our general findings are exemplified for three models: a Gompertz model, a Hongler model and a model involving a 1 − x² noise amplitude.

In generic Hamiltonian systems with a mixed phase-space, chaotic transport may be directed and ballistic rather than diffusive. We investigate one particular model showing this behaviour, namely a spatially periodic billiard chain in which electrons move under the influence of a perpendicular magnetic field. We analyse the phase-space structure and derive an explicit expression for the chaotic transport velocity. Unlike previous studies of directed chaos our model has a parameter regime in which the dispersion of an ensemble of chaotic trajectories around its moving centre of mass is essentially diffusive. We explain how in this limit the deterministic chaos reduces to a biased random walk in a billiard with a rough surface. The diffusion constant for this simplified model is calculated analytically.

In this paper, we compute, for large n, the determinant of a class of n × n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the unknown determinant D_n[w_α,βh] is compared with the known determinant D_n[w_α,β]. Here w_α,β is the Jacobi weight and w_α,βh, where h = h(x), x ∊ [−1, 1], is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight w_α,β(x) := (1 − x)^α(1 + x)^β. Applying a previously known formula on the distribution function of linear statistics, we compute the large-n asymptotics of D_n[w_α,βh] and supply a missing constant of the expansion.

Two formulae expressing explicitly the derivatives and moments of Al-Salam–Carlitz I polynomials of any degree and for any order in terms of Al-Salam–Carlitz I themselves are proved. Two other formulae for the expansion coefficients of general-order derivatives D^p_qf(x), and for the moments x^ℓD^p_qf(x), of an arbitrary function f(x) in terms of its original expansion coefficients are also obtained. Application of these formulae for solving q-difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Al-Salam–Carlitz I polynomials and any system of basic hypergeometric orthogonal polynomials, belonging to the q-Hahn class, is described.

The Drinfeld twist for the opposite quasi-Hopf algebra, H^cop, is determined and is shown to be related to the (second) Drinfeld twist on a quasi-Hopf algebra. The twisted form of the Drinfeld twist is investigated. In the quasi-triangular case, it is shown that the Drinfeld u-operator arises from the equivalence of H^cop to the quasi-Hopf algebra induced by twisting H with the R-matrix. The Altschuler–Coste u-operator arises in a similar way and is shown to be closely related to the Drinfeld u-operator. The quasi-cocycle condition is introduced and is shown to play a central role in the uniqueness of twisted structures on quasi-Hopf algebras. A generalization of the dynamical quantum Yang–Baxter equation, called the quasi-dynamical quantum Yang–Baxter equation, is introduced.

In this paper a novel analytical method is applied to the problem of transient heat conduction in a one-dimensional hollow composite cylinder with a time-dependent boundary temperature. It is known that for such problems in general, the underlying eigenvalue and residue calculations pose a challenge in practice because of the computational requirements especially for a cylinder with many layers. A new approximated analytical solution is derived by a novel application of the Laplace transformation. As a result, the problem of eigenvalue or residue computation is avoided. A closed-form solution is presented. A further comparison of analytical results with numerical models demonstrates a high accuracy of the developed analytical solution.

The Lagrange–d'Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange–d'Alembert–Poincaré equations. In a previous contribution we have shown that both sets of equations fall in the category of the so-called 'Lagrangian systems on a subbundle of a Lie algebroid'. In this paper, we investigate the special case when the reduced system is again invariant under a new symmetry group (and so forth). Via Lie algebroid theory, we develop a geometric context in which successive reduction can be performed in an intrinsic way. We prove that, at each stage of the reduction, the reduced systems are part of the above mentioned category, and that the Lie algebroid structure in each new step is the quotient Lie algebroid of the previous step. We further show that that reduction in two stages is equivalent with direct reduction.

Explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi-dimensional spaces in a continuous sequence of orthonormal bases for unitary SU(1, 1) irreps. Explicit expressions are also given for SO(N)-reduced matrix elements of basic orbital observables. These developments make it possible to determine the matrix elements of polynomial and other Hamiltonians analytically, to within SO(N) Clebsch–Gordan coefficients, and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent.

We consider a free charged particle interacting with an electromagnetic bath at zero temperature. The dipole approximation is used to treat the bath wavelengths larger than the width of the particle wave packet. The effect of these wavelengths is then described by a linear Hamiltonian whose form is analogous to the phenomenological Hamiltonians previously adopted to describe the free particle–bath interaction. We study how the time dependence of decoherence evolution is related with initial particle–bath correlations. We show that decoherence is related to the time dependent dressing of the particle. Moreover, because decoherence induced by the T = 0 bath is very rapid, we make some considerations on the conditions under which interference may be experimentally observed.

Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann–Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space of density states on an n-dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank-k states, k = 1, ..., n, is a smooth manifold of (real) dimension 2nk − k² − 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.

In introducing second quantization for fermions, Jordan and Wigner (1927, 1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov–Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.

In the presence of an external field, the imposition of specific boundary conditions can lead to interesting new manifestations of the Casimir effect. In particular, it is shown here that even a single conducting plate may experience a non-zero force due to vacuum fluctuations. The origins of this force lie in the change induced by the external potential in the density of available quantum states.

Granular gases are composed of macroscopic bodies kept in motion by an external energy source such as a violent shaking. The behaviour of such systems is quantitatively different from that of ordinary molecular gases: due to the size of the constituents, external fields have a stronger effect on the dynamics and, more importantly, the kinetic energy of the gas is no longer a conserved quantity. The key role of the inelasticity of collisions has been correctly appreciated for about fifteen years, and the ensuing consequences in terms of phase behaviour or transport properties studied in an increasing and now vast body of literature.

The purpose of this book is to help the newcomer to the field in acquiring the essential theoretical tools together with some numerical techniques. As emphasized by the authors—who were among the pioneers in the domain— the content could be covered in a one semester course for advanced undergraduates, or it could be incorporated in a more general course dealing with the statistical mechanics of dissipative systems.

The book is self-contained, clear, and avoids mathematical complications. In order to elucidate the main physical ideas, heuristic points of views are sometimes preferred to a more rigorous route that would lead to a longer discussion. The 28 chapters are short; they offer exercises and worked examples, solved at the end of the book. Each part is supplemented with a relevant foreword and a useful summary including take-home messages. The editorial work is of good quality, with very few typographical errors.

In spite of the title, kinetic theory stricto sensu is not the crux of the matter covered. The authors discuss the consequences of the molecular chaos assumption both at the individual particle level and in terms of collective behaviour. The first part of the book addresses the mechanics of grain collisions. It is emphasized that considering the coefficient of restitution ε —a central quantity governing the inelasticity of inter-grain encounters—as velocity independent is inconsistent with the mechanical point of view. An asymptotic expression for the impact velocity dependence of ε is therefore derived for visco-elastic spheres. The important inelastic Boltzmann equation is introduced in part II and the associated velocity distribution characterized for a force-free medium (so-called free cooling regime). Transport processes can then be analyzed in part III at the single particle level, and part IV from a more macroscopic viewpoint. The corresponding Chapman–Enskog-like hydrodynamic approach is worked out in detail, in a clear fashion. Finally, the tendency of granular gases to develop instabilities is illustrated in part V where the hydrodynamic picture plays a pivotal role.

This book clearly sets the stage. For the sake of simplicity, the authors have discarded some subtle points, such as the open questions underlying the hydrodynamic description (why include the temperature among the hydrodynamic modes, and what about the separation of space and time scales between kinetic and hydrodynamic excitations?). Such omissions are understandable. To a certain extent however, the scope of the book is centered on previous work by the authors, and I have a few regrets. Special emphasis is put on the (variable ε) visco-elastic model, which enhances the technical difficulty of the presentation. On the other hand, the important physical effects including scaling laws, hydrodynamic behaviour and structure formation, can be understood in two steps, from the results derived within the much simpler constant ε model, allowing subsequently $\varepsilon$ to depend on the granular temperature. The authors justify their choice with the inconsistency of the constant ε route. The improvements brought by the visco-elastic model remain to be assessed, since the rotational degrees of freedom, discarded in the book, play an important role and require due consideration of both tangential and normal restitution coefficients, that are again velocity dependent. This seems to be the price of a consistent approach, which does not lend itself to much insight. In addition, the behaviour of driven systems is not addressed, whereas in the realm of granular media, force-free systems are the exception rather than the rule. The differences between constant ε and visco-elastic models is presumably less pronounced in the driven case. Study of driven systems also reveals that the rheology of granular gases is intrinsically non-Newtonian, which is a key feature. Finally, the powerful direct simulation Monte Carlo technique is not described, whereas it is an important tool, particularly relevant for the physics of the Boltzmann equation, and straightforward to implement in its simplest version. N Brilliantov and T Pöschel concentrate on the (equally relevant) molecular dynamics method instead.

In conclusion, the book fills a gap in the field. The companion webpage from where molecular dynamics and symbolic algebra programs can be downloaded is also useful.