Recent zbMATH articles in MSC 37Jhttps://www.zbmath.org/atom/cc/37J2022-04-28T18:15:44.554837ZWerkzeugOn limit cycles near two centres and a double homoclinic loop in Liénard differential systemhttps://www.zbmath.org/1482.340982022-04-28T18:15:44.554837Z"Wei, Lijun"https://www.zbmath.org/authors/?q=ai:wei.lijun"Zhang, Qingjing"https://www.zbmath.org/authors/?q=ai:zhang.qingjing"Zhang, Xiang"https://www.zbmath.org/authors/?q=ai:zhang.xiangThe number of limit cycles in a Liénard differential system are bifurcate from the families of periodic orbits near two centres and a double homoclinic loop of a Hamiltonian system with an elliptic Hamiltonian function is studied. The lower bounds on the number of limit cycles in different cases of this system are reached as can be shown by means of Maple programs.
Reviewer: Valery A. Gaiko (Minsk)Generalized analytic integrability of a class of polynomial differential systems in \(\mathbb{C}^2\)https://www.zbmath.org/1482.370272022-04-28T18:15:44.554837Z"Llibre, Jaume"https://www.zbmath.org/authors/?q=ai:llibre.jaume"Tian, Yuzhou"https://www.zbmath.org/authors/?q=ai:tian.yuzhouThe paper is focused on two-dimensional autonomous complex polynomial ordinary differential systems with separable variables. Necessary and sufficient conditions for the existence of a first autonomous integral in the classes of generalized analytic and polynomial functions are obtained. Some examples are given and discussed.
Reviewer: Valentine Tyshchenko (Grodno)Corrigendum and addendum to: ``Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation''https://www.zbmath.org/1482.370522022-04-28T18:15:44.554837Z"Bustamante, Adrián P."https://www.zbmath.org/authors/?q=ai:bustamante.adrian-p"Calleja, Renato C."https://www.zbmath.org/authors/?q=ai:calleja.renato-cSummary: We correct some tables and figures in our paper [ibid. 395, 15--23 (2019; Zbl 1451.37080)]. We also report on the new computations that verify the accuracy of the data and extend the results. The new computations have led us to find new patterns in the data that were not noticed before. We formulate some more precise conjectures.Lyapunov functions and asymptotics at infinity of solutions of equations that are close to Hamiltonian equationshttps://www.zbmath.org/1482.370532022-04-28T18:15:44.554837Z"Sultanov, O. A."https://www.zbmath.org/authors/?q=ai:sultanov.oskar-aThe author considers perturbed Hamiltonian systems of second order: \[ \frac{dx}{dt} = \partial_y H(x,y,t)\ ,\ \frac{dy}{dt} = - \partial_x H(x,y,t) + F(x,y,t), \] displaying the asymptotic properties for \(t\rightarrow\infty\) \[ H(x,y) = H_0(x,y) + \sum_{k=1}^\infty t^{-k/q}H_k(x,y), \; F(x,y,t)=\sum_{k=1}^\infty t^{-k/q}F_k(x,y)\ ,\ q\in \mathbb{N}, \] and having \((0,0)\) as center-type fixed point.
The asymptotic properties are studied by considering a Lyapunov function of the form \[ V(x,y,t)=2H(x,y,t) + \sum_{k=1}^p t^{-k/q}v_k(x,y) ,\] where the \(v_k\)'s are subject to \[ (\partial_y H_0)\partial_x v_k - (\partial_x H_0)\partial_y v_k = G_k(x,y) .\] The right hand side of the above equation is defined in terms of the \(v_k\)'s as follows: \[ G_1(x,y) = - (2F_1)\partial_y H_0,\]
\[ G_2(x,y) = - (2F_2)\partial_y H_0 - \partial_y v_1 F_1 - [(\partial_y H_1)\partial_x v_1 - (\partial_x H_1)\partial_y v_1], \]
etc.
If for some \(p\in \mathbb{N}\) the system is solved with \(v_k= O(x^2+y^2)\) and \[ G_p(x,y)=-\gamma(x^2+y^2) + O((x^2+y^2)^{3/2}), \] the Lyapunov function is constructed. Several examples are considered, e.g., the 4-th Painlevé equation, auto-resonance systems, etc.
Reviewer: Vladimir Răsvan (Craiova)On the existence of focus singularities in one model of a Lagrange top with a vibrating suspension pointhttps://www.zbmath.org/1482.370542022-04-28T18:15:44.554837Z"Borisov, A. V."https://www.zbmath.org/authors/?q=ai:borisov.alexey-v"Ryabov, P. E."https://www.zbmath.org/authors/?q=ai:ryabov.p-e"Sokolov, S. V."https://www.zbmath.org/authors/?q=ai:sokolov.sergei-vSummary: We consider a completely integrable Hamiltonian system with two degrees of freedom that describes the dynamics of a Lagrange top with a vibrating suspension point. The results of a stability analysis of equilibrium positions are clearly presented. It turns out that, in the case of a vibrating suspension point, both equilibrium positions can be unstable, which corresponds to the existence of focus singularities in the considered model.Billiards with changing geometry and their connection with the implementation of the Zhukovsky and Kovalevskaya caseshttps://www.zbmath.org/1482.370552022-04-28T18:15:44.554837Z"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-t"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovnaSummary: The paper presents a class of billiards with varying geometry, the so-called force or evolutionary billiards, which enable us to realize, in the sense of Liouville equivalence, the well-known cases of Zhukovsky and Kovalevskaya for certain energy zones. On the corresponding 4-dimensional open phase submanifolds, the indicated systems are implemented for an increase in energy on all successively occurring isoenergy 3-surfaces.Integrability of point-vortex dynamics via symplectic reduction: a surveyhttps://www.zbmath.org/1482.370562022-04-28T18:15:44.554837Z"Modin, Klas"https://www.zbmath.org/authors/?q=ai:modin.klas"Viviani, Milo"https://www.zbmath.org/authors/?q=ai:viviani.miloThe authors consider point-vertex dynamics and their integrability. This dynamics is described via idealized non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. The aim is to provide a unified treatment for proving integrability results for 2-, 3-, or 4-point-vertices. Part of their goal is to show how the symplectic reduction can provide a broader approach for proving integrability results, especially for point-vertex dynamics.
Euler equations on an orientable Riemannian manifold that govern an incompressible inviscid fluid have the form \(\dot{\mathbf{v}} + \nabla _\mathbf{v} \mathbf{v} = - \nabla p\) with div \(\mathbf{v} = 0\), where \(\mathbf{v}\) is a vector field on a manifold \(M\) incorporating the motion of the fluid's particles, \(p\) is the pressure function and \(\nabla _\mathbf{v}\) is the covariant derivative along \(\mathbf{v}\). \textit{H. Helmholtz} [J. Reine Angew. Math. 55, 25--55 (1858; ERAM 055.1448cj)] showed that the two-dimensional Euler equations have special solutions with a finite number of point-vertices. These solutions are not smooth and are characterized by the vorticity \mbox{curl \(\mathbf{v} = \sum_{i=1} ^ {n} \Gamma _i \delta_{\mathbf{r}_i}\)} where non-zero \(\Gamma_i\) is the strength of the vortex \(i\), \(\mathbf{r}_i\) is its position, and \(\delta_{\mathbf{r}_i}\) is a delta function.
The authors describe point-vertex equations and their Hamiltonian structures on the sphere, the plane, the hyperbolic plane, and the flat torus. Each of these cases has a different symmetry group, and the symplectic reduction is treated separately in each case to establish integrability.
The authors also briefly review nonintegrability results.
They conclude with some observations about how their results pertain to long term predictions for the Euler equations. An appendix offers a kind of visual portrait of point-vortex solutions.
Reviewer: William J. Satzer Jr. (St. Paul)New cases of homogeneous integrable systems with dissipation on tangent bundles of two-dimensional manifoldshttps://www.zbmath.org/1482.370572022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of two-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of three-dimensional manifoldshttps://www.zbmath.org/1482.370582022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of three-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of four-dimensional manifoldshttps://www.zbmath.org/1482.370592022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of four-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.Force evolutionary billiards and billiard equivalence of the Euler and Lagrange caseshttps://www.zbmath.org/1482.370602022-04-28T18:15:44.554837Z"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovna"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-tSummary: A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).(Co)isotropic triples and poset representationshttps://www.zbmath.org/1482.370612022-04-28T18:15:44.554837Z"Herrmann, Christian"https://www.zbmath.org/authors/?q=ai:herrmann.christian"Lorand, Jonathan"https://www.zbmath.org/authors/?q=ai:lorand.jonathan"Weinstein, Alan"https://www.zbmath.org/authors/?q=ai:weinstein.alan-d|weinstein.alan-mSummary: We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we assume only to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is ``2 + 2 + 2'' consisting of three independent ordered pairs, with the involution exchanging the members of each pair.
A key feature of the classification is that any indecomposable (co)isotropic triple is either ``split'' or ``non-split.'' The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the ``split'' case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a ``symplectification.''
In the course of the paper we develop the framework of ``symplectic poset representations,'' which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples.Theoretical and numerical results on Birkhoff normal forms and resonances in the Born-Oppenheimer approximationhttps://www.zbmath.org/1482.370622022-04-28T18:15:44.554837Z"Latigui, Nawel"https://www.zbmath.org/authors/?q=ai:latigui.nawel"Ghomari, Kaoutar"https://www.zbmath.org/authors/?q=ai:ghomari.kaoutar"Messirdi, Bekkai"https://www.zbmath.org/authors/?q=ai:messirdi.bekkaiSummary: This paper mainly focuses on the Birkhoff normal form theorem for the Born-Oppenheimer Hamiltonians. Normal forms are accessible via those of the effective Hamiltonian obtained by the Grushin reduction method and the pseudodifferential calculus with operator-valued symbols. Resonance situations are discussed; the theoretical computations of Birkhoff normal form in the 1:1 resonance are written explicitly. Our approach gives compatible numerical results while using a computer program.Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentialshttps://www.zbmath.org/1482.370632022-04-28T18:15:44.554837Z"Sun, Yingte"https://www.zbmath.org/authors/?q=ai:sun.yingteThe author considers the ordinary differential equation
\[
-y''+u(\omega t)y=Ey,
\]
where \(y=y(t)\), \(t\in\mathbb{R}\), \(u\) is a real-analytic quasi-periodic function such that
\[
|\ell \, \omega|\geq M(\alpha/(|\ell|^\tau)),
\]
where \(\alpha>0\), \(\tau>d-1\), \(\ell\in\mathbb{Z}^{d}\backslash 0\). It is proved that for any \(a>0\), \(r>0\) and \(\alpha>0\) there exists \(M^*=M^*(d,r,a,\alpha)\) such that if \(M\geq M^*\) for the interval \([a,\infty)\) there exists a Cantor subset \(\Delta_\alpha\subset\Delta\) such that for any \(\sqrt{E}\in\Delta_\alpha\), the above equation has two linearly independent solutions.
Reviewer: Ekin Uğurlu (Ankara)Minimal periodic problem for brake orbits of first-order Hamiltonian systemshttps://www.zbmath.org/1482.370642022-04-28T18:15:44.554837Z"Zhang, Xiaofei"https://www.zbmath.org/authors/?q=ai:zhang.xiaofei"Liu, Chungen"https://www.zbmath.org/authors/?q=ai:liu.chungen"Lu, Xinnian"https://www.zbmath.org/authors/?q=ai:lu.xinnianA brake orbit for a Hamiltonian system is a periodic solution of Hamilton's equations for which the generalized momenta are zero at two different points. This paper considers the autonomous system defined by \(\dot{z} = J \nabla H(z(t))\), \(z(-t) = Nz(t)\) and \(z(t+\tau) = z(t)\) where \(H \in C^2(\mathbb{R}^{2n}, \mathbb{R})\), \(H(z) = H(N(z)\) for all \(z \in \mathbb{R}^{2n}\), \(t \in \mathbb{R}\) and \(\tau > 0\). Matrices \(N\) and \(J\) are defined by \[N =\left( \begin{matrix} -I_n & 0\\
0 & I_n \end{matrix}\right),\] and \[J =\left( \begin{matrix} 0 & -I_n\\
I_n & 0 \end{matrix}\right),\] where \(I_n\) is the \(n \times n\) identity matrix.
To find a brake orbit solution of this system the authors look at the \(L_0\)-boundary problem defined by \(\dot{z} = J \nabla H(z(t))\) with \(z(0) \in L_0\), \(z(\tau/2) \in L_0\) \(t \in [0, \tau/2]\), and where \(L_0 = \{ 0 \} \times \mathbb{R}^n\).
The main result of the paper is about the existence of a non-trivial brake orbit under a number of technical conditions on the Hamiltonian function \(H\). In particular there are conditions on the Hessian of \(H(z)\) with respect to \(q\) where \(z = (p,q)\) and \(p, q \in \mathbb{R}^n\). With even stricter conditions, the brake orbit has a minimal period of \(\tau\) or \(\tau/2\).
The authors argue that brake orbit problems can be transformed into \(L_0\)-boundary value problems which are amenable to treatment as Lagrangian boundary value problems.
The authors rely on several related papers to provide the necessary background.
Reviewer: William J. Satzer Jr. (St. Paul)Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problemhttps://www.zbmath.org/1482.371012022-04-28T18:15:44.554837Z"Podobryaev, A. V."https://www.zbmath.org/authors/?q=ai:podobryaev.aleksei-vSummary: We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.Duality of gauges and symplectic forms in vector spaceshttps://www.zbmath.org/1482.520102022-04-28T18:15:44.554837Z"Balestro, Vitor"https://www.zbmath.org/authors/?q=ai:balestro.vitor"Martini, Horst"https://www.zbmath.org/authors/?q=ai:martini.horst"Teixeira, Ralph"https://www.zbmath.org/authors/?q=ai:teixeira.ralph-costaLet \(X\) be a finite dimensional vector space endowed with the unique Hausdorff vector topology. By a convex body one understands a compact convex subset \(K\) of \(X\) with \(0\in \operatorname{int}K\). The set of all these convex bodies is denoted by \(\mathcal{K}_0(X)\). The Minkowski functional \(\gamma_K\), given by \(\gamma_K(x)=\inf\{\lambda\ge 0: x\in \lambda K\}\), is called the gauge associated to \(K\). Then \(\gamma_K\) is an asymmetric norm on \(X\), i.e. a positively definite, positively homogeneous and subadditive functional on \(X\). Conversely, every asymmetric norm \(\gamma\) on \(X\) is the gauge of its unit ball \(B_\gamma=\{x\in X:\gamma(x)\le 1\}\). The topology generated by \(\gamma\) agrees with the initial topology of \(X\). The dual gauge of \(\gamma\) is the functional \(\gamma^*\) defined on the dual space \(X^*\) of \(X\) by \(\gamma^*(f)=\max\{f(x): x\in K\}\) and the polar body of \(K\) is \(K^\circ =\{f\in X^*: f(x)\le 1,\, \forall x\in K\}\). It turns out that \(\gamma^*\) is a gauge on \(X^*\), namely the gauge asoociated to the polar body \(K^\circ\).
Suppose now that \(X\) is even-dimensional and let \(\omega:X\times X\to\mathbb{R}\) be a fixed symplectic form on \(X\), i.e. a nondegenerate alternating bilinear form on \(X\). Then \(\omega\) yields an identification of \(X\) and its dual space \(X^*\). Namely, for every \(f\in X^*\) there is a unique \(x_f\in X\) s.t. \(f(\cdot)=\omega(x_f,\cdot)\) and the mapping \(\mathcal I:X^*\to X\) given by \(\mathcal I(f)=x_f\) is an isomorphism. If \(K\in\mathcal{K}_0(X)\), then \(K^\omega:=\mathcal I(K^\circ)\) is called the dual body of \(K\). It follows that \(K^\omega\in\mathcal{K}_0(X)\), the gauge \(\gamma_{K^\omega}\) satisfies the equality \(\gamma_{K^\omega}(x)=\max\{\omega(x,y):y\in K\}\) and its dual gauge is \(\gamma_{-K}\).
As the authors say in the Abstract: ``In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur-Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.''
Reviewer: Stefan Cobzaş (Cluj-Napoca)Two-body problem on a sphere in the presence of a uniform magnetic fieldhttps://www.zbmath.org/1482.700072022-04-28T18:15:44.554837Z"Balabanova, Nataliya A."https://www.zbmath.org/authors/?q=ai:balabanova.nataliya-a"Montaldi, James A."https://www.zbmath.org/authors/?q=ai:montaldi.james-aSummary: We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.Radial kinetic nonholonomic trajectories are Riemannian geodesics!https://www.zbmath.org/1482.700182022-04-28T18:15:44.554837Z"Anahory Simoes, Alexandre"https://www.zbmath.org/authors/?q=ai:anahory-simoes.alexandre"Marrero, Juan Carlos"https://www.zbmath.org/authors/?q=ai:marrero.juan-carlos"Martín de Diego, David"https://www.zbmath.org/authors/?q=ai:martin-de-diego.davidIn this paper, an interesting result that characterize radial nonholonomic geodesics is presented. It is proved that for kinetic nonholonomic systems, the solutions starting from a fixed point \(q\) are true geodesics for a family of Riemannian metrics on the image submanifold \(\mathcal{M}_q^{nh}\) of the nonholonomic exponential map. This fact shows that the kinetic nonholonomic trajectories with starting point \(q\), for sufficiently small times, minimize length in \(\mathcal{M}_q^{nh}\). This result opens the door to new developments in nonholonomic mechanics using Riemannian geometry techniques: Jacobi fields, global minimizing properties of nonholonomic trajectories, construction of variational integrators for nonholonomic mechanics, Hamiltonization or Lagrangianization of nonholonomic systems.
Reviewer: Liviu Popescu (Craiova)Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systemshttps://www.zbmath.org/1482.700202022-04-28T18:15:44.554837Z"Rashkovskiy, Sergey"https://www.zbmath.org/authors/?q=ai:rashkovskiy.sergey-aThe aim of this paper is to describe a generalization of Hamilton-Jacobi theory to arbitrary dynamical systems. The overall approach is to develop a kind of ensemble theory. The author draws upon Newtonian, Lagrangian and Hamiltonian formulations of classical mechanics.
The ensemble, one the author calls the Hamilton-Jacobi ensemble, consists of identical systems of non-interacting classical particles that differ in initial conditions. The particles are assumed to move in the same physical space, a configuration space in which all particles in the ensemble move simultaneously but do not interact with each other. The author also assumes a consistency requirement that all particles at the same point in configuration space at the same time have the same velocity. Each particle is assumed to move according to Newton's law \(m \frac{d\mathbf{v}}{dt} = \mathbf{F} (t, \mathbf{r}, \mathbf{v})\) and the kinematic equation \(\frac{d\mathbf{r}}{dt} = \mathbf{v}\).
The ensemble of non-interacting particles is then defined by an equation of motion \(m(\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \nabla \mathbf{v}) = \mathbf{F} (t, \mathbf{r}, \mathbf{v})\) (where \(\mathbf{r}\) is position and \(\mathbf{v}\) is velocity) and a continuity equation \(\frac{\partial \rho}{\partial t} + \) div \(\rho \mathbf{v} = 0\) where \(\rho\) is a density that can be normalized, for example, as the mass or number of particles per unit volume of the configuration space. The author suggests that this Hamilton-Jacobi ensemble can be thought of a as a compressible continuous medium moving under the influence of an external force field \( \mathbf{F} (t, \mathbf{r}, \mathbf{v})\).
The author claims that the system ensemble described by these last two equations generalize Hamilton-Jacobi theory and can include arbitrary non-Hamiltonian systems. Several examples are provided in a sequence of appendices.
Reviewer: William J. Satzer Jr. (St. Paul)Notes about the KP/BKP correspondencehttps://www.zbmath.org/1482.810132022-04-28T18:15:44.554837Z"Orlov, A. Yu."https://www.zbmath.org/authors/?q=ai:orlov.aleksandr-yuSummary: We present a set of remarks related to previous work. These are remarks on polynomials solutions, the application of the Wick theorem, examples of creation of polynomial solutions with the help of vertex operators, the eigenproblem for polynomials, and a remark on the conjecture by Alexandrov and Mironov, Morozov about the ratios of the projective Schur functions. New results on the bilinear relations between characters of the symmetric group and the Sergeev group and on bilinear relations between skew Schur and projective Schur functions and also between shifted Schur and projective Schur functions are added. Certain new matrix models are discussed.Fixed-time \(\mathcal {H}_{\infty }\) control for port-controlled Hamiltonian systemshttps://www.zbmath.org/1482.931882022-04-28T18:15:44.554837Z"Liu, Xinggui"https://www.zbmath.org/authors/?q=ai:liu.xinggui"Liao, Xiaofeng"https://www.zbmath.org/authors/?q=ai:liao.xiaofengEditorial remark: No review copy delivered.