New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ with p∈(0,∞], s∈?, τ∈[0,∞), and q∈(0,∞], as well as their preduals, the Besov-Hausdorff spaces $B\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-Hausdorff spaces $F\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ , in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of Rauhut (Stud. Math. 180:237–253, 2007) and discretizations via biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot{F}^{s}_{\infty,q}({\mathbb{R}}^{n})$ for s∈?, q∈(0,∞] (including ?BMO(? ⁿ ) when s=0 and q=2), the Q space Q _α (? ⁿ ), the Hardy-Hausdorff space HH _?α (? ⁿ ) for α∈(0,min{n/2,1}), the Morrey space ${\mathcal{M}}^{u}_{p}({\mathbb{R}}^{n})$ for 1<p≤u<∞, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^{s}_{upq}({\mathbb{R}}^{n})$ for 0<p≤u<∞, s∈? and q∈(0,∞], some of these results are new.     

Laser sources for precision spectroscopy on atomic strontium

We present a new laser setup designed for high-precision spectroscopy on laser cooled atomic strontium. The system, which is entirely based on semiconductor laser sources, delivers 200 mW at 461 nm for cooling and trapping atomic strontium from a thermal source, 4 mW at 497 nm for optical pumping from the metastable P23 state, 12 mW at 689 nm on linewidth less than 1 kHz for second-stage cooling of the atomic sample down to the recoil limit, 1.2 W at 922 nm for optical trapping close to the "magic wavelength" for the 0-1 intercombination line at 689 nm. The 689 nm laser was already employed to perform a frequency measurement of the 0-1 intercombination line with a relative accuracy of 2.3 x 10(-11), and the ensemble of laser sources allowed the loading in a conservative dipole trap of multi-isotopes strontium mixtures. The simple and compact setup developed represents one of the first steps towards the realization of a transportable optical standards referenced to atomic strontium.     

Matrix‐free single‐cell LDI‐MS investigations of the diatoms Coscinodiscus granii and Thalassiosira pseudonana

Single‐cell investigations of the diatoms Coscinodsicus granii and Thalassiosira pseudonana were performed using laser desorption/ionization (LDI)‐MS without the addition of chemical matrices. The unique cell wall architecture of these microalgae, more precisely the biomineralized nanostructured surface, supported the ionization of cellular as well as surface‐related metabolites. In model experiments with purified diatom cell walls of eight species C. granii and T. pseudonana proved to promote the ionization of the polymer polyethylene glycol most efficiently. These species were therefore chosen for further experiments. Without any additional workup, living diatom cells can be washed, can be placed on the LDI target and can immediately be profiled using LDI‐MS. Characteristic signals arising from the two species were assigned to common metabolites known from diatom metabolism. Among others, chlorophyll, phospholipids and amino acids were detected. Using these fingerprint signals, we were able to perform species‐specific MS imaging down to a single‐cell resolution of 20 by 20 µm. The larger C. granii cells can be directly visualized, while more than one of the smaller T. pseudonana cells is needed to generate high‐quality images. The introduced technique will pave the way toward a chemotyping of phytoplankton that will enable the automated annotation of microalgal species. But also, an assignment of metabolic plasticity on a single‐cell level that could answer fundamental questions about plankton diversity is now in reach. Copyright © 2013 John Wiley & Sons, Ltd.     

A branch-and-cut algorithm for the multi-compartment vehicle routing problem with flexible compartment sizes

Multi-compartment vehicle routing problems arise in a variety of problem settings in which different product types have to be transported separated from each other. In this paper, a problem variant which occurs in the context of glass waste recycling is considered. In this problem, a set of locations exists, each of which offering a number of containers for the collection of different types of glass waste (e.g. colorless, green, brown glass). In order to pick up the contents from the containers, a fleet of homogeneous disposal vehicles is available. Individually for each disposal vehicle, the capacity can be discretely separated into a limited number of compartments to which different glass waste types are assigned. The objective of the problem is to minimize the total distance to be travelled by the disposal vehicles. For solving this problem to optimality, a branch-and-cut algorithm has been developed and implemented. Extensive numerical experiments have been conducted in order to evaluate the algorithm and to gain insights into the problem structure. The corresponding results show that the algorithm is able to solve instances with up to 50 locations to optimality and that it reduces the computing time by 87% compared to instances from the literature. Additional experiments give managerial insights into the use of different variants of compartments with flexible sizes.

     

2 μm semiconductor disk laser with a heterodyne linewidth below 10 kHz

We demonstrate a 2 μm semiconductor disk laser emitting in a single longitudinal mode with a linewidth in the <10 kHz range. A heterodyne detection scheme was used for precise linewidth measurements. In these experiments, the output beams of two identical laser cavities were superposed in order to generate a beat note signal on a photodiode. In the absence of active frequency stabilization, a linewidth of 45 kHz was measured at an output power of 100 mW. When using a frequency stabilization consisting of a feedback loop with a Fabry-Perot interferometer as wavelength reference, the linewidth could be further reduced to 9 kHz.     

Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Using this we derive several useful characterizations from the one-dimensional case to the d-dimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.     

Spectres de masse des composés organiques. 4e communication. Fragmentation de l'hexane

With the help of metastable peaks and high resolution, and by making extensive use of deuterated species we determined the main paths of fragmentation of hexane. Beside the simple splitting of a C? C-bond there are a series of internal rearrangement reactions. The loss of neutral fragments from alkyl ions is often, but not always, statistical. A small primary (0.96) and an even smaller (0.99) secondary isotope effect for a hydrogen transfer can be observed.     

On the orthogonality of the Chebyshev–Frolov lattice and applications

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a power of two, i.e. \(d=2^m, m \in \mathbb {N}\), the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube \([-1/2,1/2]^d\) up to dimension \(d=16\).