Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

Thermal post-buckling analysis of square plates resting on elastic foundation: A simple closed-form solutions

Thermal post-buckling paths of homogeneous, isotropic, square plate configurations resting on elastic foundation (Winkler type) subjected to biaxial compressive thermal loads are expressed as simple closed-form solutions by using the Rayleigh–Ritz method based on coupled displacement fields. Geometric non-linearity of von-Karman type is considered. The in-plane displacement field variations used in the formulation of Rayleigh–Ritz method are derived by using the governing in-plane static differential equations of the plate which in turn simplifies the difficulty of assuming an in-plane displacement field variations of the square plate. Accuracy and robustness of the proposed closed-form solutions are demonstrated by using the non-linear finite element formulation results which are obtained from an equilibrium path approach.     

Synthesis and stereochemistry of 3‐acetyl‐5‐aminospiro[1,3,4‐thiadiazoline‐2,4′‐thioflavans]

Under acetylating conditions racemic thioflavanone thiosemicarbazones cyclize into racemic 3‐acetyl‐spiro[1,3,4‐thiadiazoline‐2,4′‐thioflavans] and a racemic 3‐acetylspiro[1,3,4‐oxadiazoline‐2,4′‐thioflavan] with trans O(1) or S(1) and Ph(2′_eq). Hindered rotation of the endocyclic N(3) acetyl group spirothia‐diazolines caused the formation of isomers separable by HPLC. X‐ray diffraction analyses, ¹H‐, ¹³C‐, and ¹⁵N NMR measurements as well as MOPAC QM calculations were performed to reveal the structures of these isomers.     

Ghost circles in lattice Aubry–Mather theory

Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions $x:{\mathbb{Z}}^d\to \mathbb{R}$ are the stationary points of a formal action function $W(x)$. Given any rotation vector $\omega \in {\mathbb{R}}^d$, classical Aubry–Mather theory establishes the existence of a large collection of solutions of $\mathrm{?}W(x)=0$ of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow $\frac{dx}{dt}=?\mathrm{?}W(x)$ and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension $d=1$, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions.As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.     

Elemental labelling combined with liquid chromatography inductively coupled plasma mass spectrometry for quantification of biomolecules: A review

This article reviews novel quantification concepts where elemental labelling is combined with flow injection inductively coupled plasma mass spectrometry (FI-ICP-MS) or liquid chromatography inductively coupled plasma mass spectrometry (LC–ICP-MS), and employed for quantification of biomolecules such as proteins, peptides and related molecules in challenging sample matrices. In the first sections an overview on general aspects of biomolecule quantification, as well as of labelling will be presented emphasizing the potential, which lies in such methodological approaches. In this context, ICP-MS as detector provides high sensitivity, selectivity and robustness in biological samples and offers the capability for multiplexing and isotope dilution mass spectrometry (IDMS). Fundamental methodology of elemental labelling will be highlighted and analytical, as well as biomedical applications will be presented. A special focus will lie on established applications underlining benefits and bottlenecks of such approaches for the implementation in real life analysis. Key research made in this field will be summarized and a perspective for future developments including sophisticated and innovative applications will given.     

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries. Received: 25 July 2000 / Accepted: 20 December 2000     

ESR studies of E′ centres in geological flint

Properties of E′ centres of high geological age in flint from different locations have been studied. ESR spectra of γ-irradiated samples of heated and unheated Silurian and Eocene flint show thermally unstable components. The spectrum of one species could be simulated using the g values: 2.00164, 2.00059, and 2.00046. Two others deliver ESR-signals at g=2.0015 and 2.0012, respectively. These unstable components cannot be identified with any known E′ centres. About 50% of the ESR intensity of the E′ centres in unheated flint induced by artificial γ-rays is thermally stable. This effect is probably due to a steady state and might be useful for dating of unburned flint tools (“skinflint dating”).