The 1:1 resonance in Hamiltonian systems

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances $k:l$, but not the semisimple cases $1:1$ and $1:?1$. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to $S^1$-symmetric systems. The question we wish to address is about the co-dimension of such a system in $1:1$ resonance with respect to left-right-equivalence, where the right action is $S^1$-equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.     

Diffuse scattering of modulated structures and quasicrystals

The method of Fourier transforms is applied to the calculation of sharp (coherent) and diffuse (incoherent) scattering of modulated structures and quasicrystals. It is shown that the fundamental difference between both may best be described by introducing different distribution functions of atoms, before applying modulations of the structure.     

High Activity and Selective Fischer–Tropsch Catalysts for Use in a Microchannel Reactor

Each process configuration for practicing the Fischer–Tropsch synthesis places demands particular to that configuration on the catalyst to be used. We discuss how a particular catalyst, prepared by the OMX (organic matrix combustion) method, when used in conjunction with the Velocys microchannel reactor system, results in a very stable, high performance Fischer–Tropsch synthesis system. With the ability to remove heat far more effectively than a conventional reactor system, this microchannel reactor requires a catalyst with much higher volumetric reactive site density. Further, with such a high volumetric reaction rate, mass transfer effects will be important in both the observed activity and selectivity of the operating catalyst. Nevertheless, the catalyst prepared using the OMX method exhibits an apparent turnover frequency which is considerably higher than reported for other catalysts in the literature. In addition to high activity, an economically useful catalyst must exhibit a stable, high selectivity for liquid products and be able to recover near-fresh performance using a regeneration approach which can be carried out with the catalyst in-place. An example of such a stable, multiply regenerated catalyst is given. Finally, further development has focused on a catalyst with even higher C₅₊ selectivity.     

The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., they have or “almost have” fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover, we present various examples illustrating that our bounds are sharp.     

Measure and integration: The basic extension and representation theorems in terms of new inner and outer envelopes

The work of the author in measure and integration is based on new inner and outer envelope formations, which replace the traditional Carathéodory outer measure and certain simple suprema and infima. The new formations lead to essential improvements in both the extent and the adequacy of the basic results. However, they did not find an entrance into the recent textbook literature. The present paper seeks to demonstrate their power with the examples of the basic inner and outer extension and representation theorems for set functions and functionals.     

Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts.     

The Nature of the Phosphorus Tellurium Bond in Tellurophosphoranes and their Derivatives

Abstract

Tellurophosphoranes, which may be considered as phosphane chalkogenides (I) or as phosphane stabilized tellurium atoms (II), form coordination compounds III and IV. The nature of both the phosphorus tellurium and the phosphorus element bond is discussed from chemical and spectroscopic properties.