Low-pass filters and representations of the Baumslag Solitar group

We analyze representations of the Baumslag Solitar group

that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

     

Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

The changing of the guard: Molecular diversity and rapid evolution of β-defensins

Summary Defensins are small cationic peptides involved in innate immunity and are components of the first line of defence against invading pathogens. β-defensins are a subgroup of the defensin family that display a particular cysteine spacing and pattern of intramolecular bonding. These molecules are produced mostly by epithelia lining exposed surfaces and appear to have both antimicrobial and cell signalling functions. The unusually high degree of sequence variation in the mature peptide produced by the paralogous and in some cases orthologous genes implies extensive specialisation and species specific adaptation. Here we review recent functional data that are an important addition to our knowledge of the innate immune response and novel antibiotic design. We also consider the organisation and evolution of the genomic loci harbouring these genes where radical and rapid changes in β-defensin sequences have been shown to result from the interplay of both positive and negative selection. Consequently these genes provide some unusually clear glimpses of the processes of duplication and specialisation that have shaped the mammalian genome.     

Quantitative description of the T1 formation kinetics in an Al–Cu–Li alloy using differential scanning calorimetry,small-angle X-ray scattering and transmission electron microscopy

The object of the present study is to design a methodology to follow the kinetics of T₁ precipitation, in an AA2198 alloy, in terms of precipitate size, morphology (thickness, diameter) and volume fraction, during a two-temperature isothermal heat treatment. We used in situ small-angle X-ray scattering (SAXS) as a way to measure the evolution of the T₁ mean thickness and diameter during the heat treatment. Transmission electron microscopy (TEM) was then performed in order to calibrate these evolutions. Furthermore, we demonstrate that the volume fraction evolution can be described successfully using a simple analysis of the differential scanning calorimetry (DSC) thermograms. The latter was calibrated by selected observations in high angular annular dark field scanning transmission electron microscopy (HAADF-STEM). Microstructure evolution during DSC heating ramps was analysed using in situ SAXS: the T₁ phase transformation is found to consist in a two-step thickening process explained by two consecutive diffusion stages. The enthalpy of formation of the T₁ phase is deduced from the DSC measurements.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.