Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.     

On the meaning of mean shape: manifold stability, locus and the two sample test

Various concepts of mean shape previously unrelated in the literature are brought into relation. In particular, for non-manifolds, such as Kendall’s 3D shape space, this paper answers the question, for which means one may apply a two-sample test. The answer is positive if intrinsic or Ziezold means are used. The underlying general result of manifold stability of a mean on a shape space, the quotient due to an proper and isometric action of a Lie group on a Riemannian manifold, blends the slice theorem from differential geometry with the statistics of shape. For 3D Procrustes means, however, a counterexample is given. To further elucidate on subtleties of means, for spheres and Kendall’s shape spaces, a first-order relationship between intrinsic, residual/Procrustean and extrinsic/Ziezold means is derived stating that for high concentration the latter approximately divides the (generalized) geodesic segment between the former two by the ratio 1:3. This fact, consequences of coordinate choices for the power of tests and other details, e.g. that extrinsic Schoenberg means may increase dimension are discussed and illustrated by simulations and exemplary datasets.     

On one-sided versus two-sided classification

One-sided classifiers are computable devices which read the characteristic function of a set and output a sequence of guesses which converges to 1 iff the set on the input belongs to the gven class. Such a classifier istwo-sided if the sequence of its output in addition converges to 0 on setsnot belonging to the class. The present work obtains the below mentionedresults for one-sided classes (= Σ⁰ ₂ classes) with respect to four areas: Turing complexity, 1-reductions, index sets and measure. There are one-sided classes which are not two-sided. This can have two reasons: (1) the class has only high Turing complexity. Then there are some oracles which allow to construct noncomputale two-sided classifiers. (2) The class is difficult because of some topological constraints and then there are also no nonrecursive two-sided classifiers. For case (1), several results are obtainedto localize the Turing complexity of certain types of one-sided classes. The concepts of 1-reduction, 1-completeness and simple sets is transferred to one-sided classes: There are 1-complete classes and simple classes, but no class is at the same time 1-complete nd simple. The one-sided classes have a natural numbering. Most of the common index sets relative to this numbering have the high complexity Π¹ ₁: the index set of the class {0,1}^∞, the index set of the equality problem and the index set of all two-sided classes. On the other side the index set of the empty class has complexity Π⁰ ₂; Π⁰ ₂ and Σ⁰ ₂ are the least complexities any nontrivial index set can have. Lusin showed that any one-sided class is measurable. Concerning the effectiveness of this measure, it is shown that a one-sided class has recursive measure 0 if it has measure 0, but that thre are one-sided classes having measure 1 without having measure 1 effectively. The measure of a two-sided class can be computed in the limit. Received: 2 December 1999 / Revised version: 28 February 2000 / Published online: 15 June 2001     

Modeling and Simulation of Damage Processes based on a Gradient-Enhanced Free Energy Function

Taking into account softening effects in connection with conventional inelastic material models can cause ill-posed boundary value problems. These problems can be established by obtaining no unique solution for the resulting algebraic system or by having a strong mesh dependence of the numerical results. This is the consequence of losing ellipticity of the governing field equations. A possible approach to solve these problems is to introduce a non-local field function in the model which includes an internal material length scale. For this purpose a gradient-enhanced free energy function is used for the current continuum damage model from which two variational equations are resulting. Calculations with less effort can be achieved due to the enhancement of the free energy function in comparison to other approaches. The mentioned model is applied to a material with locally varying damage properties (yield limits). Furthermore, the model is able to describe crack propagation in cases of completely damaged material. Therewith, a matrix material including precipitates, such as carbides, is modeled. This allows to investigate ship screws, which usually exhibit the mentioned composition, with regard to the influence of cavitation. Cavitation describes the implosion of risen vapor bubbles, whereby the impact on screws causes heavy damages which can lead to a complete destruction. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectroscopic characterization of molybdenum dinitrogen complexes containing a combination of di- and triphosphine coligands: 31P NMR analysis of five-spin systems

The three molybdenum-N2 complexes [Mo(N2)(dpepp)(depe)] (1), [Mo(N2)(dpepp)(dppe)] (2), and [Mo(N2)(dpepp)(1,2-dppp)] (3), all of which contain a combination of a bi- and a tridentate phosphine ligand, were prepared and investigated by vibrational and (31)P NMR spectroscopy. As a tridentate ligand bis(2-diphenylphosphinoethyl)phenylphosphine (dpepp) has been employed. The three different bidentate ligands are 1,2-bis(diethylphosphino)ethane (depe), 1,2-bis(diphenylphosphino)ethane (dppe), and R-(+)-1,2-bis(diphenylphosphino)propane (1,2-dppp). N-N as well as metal-N vibrations of 1-3 are identified and interpreted in terms of the geometric and electronic structures of the complexes. (31)P NMR spectra are recorded and fully analyzed. Moreover, correlation spectroscopy (COSY)-45 measurements are performed to determine the relative signs of coupling constants. Special attention is directed to a detection of different isomers and their (31)P NMR, as well as vibrational spectroscopic properties. The implications of the results for the area of synthetic nitrogen fixation with phosphine complexes are discussed.     

A Chemical Proteomic Strategy Reveals Inhibitors of Lipoate Salvage in Bacteria and Parasites

The development of novel anti-infectives requires unprecedented strategies targeting pathways which are solely present in pathogens but absent in humans. Following this principle, we developed inhibitors of lipoic acid ( LA ) salvage, a crucial pathway for the survival of LA auxotrophic bacteria and parasites but non-essential in human cells. An LA -based probe was selectively transferred onto substrate proteins via lipoate protein ligase (LPL) in intact cells, and their binding sites were determined by mass spectrometry. Probe labeling served as a proxy of LPL activity, enabling in situ screenings for cell-permeable LPL inhibitors. Profiling a focused compound library revealed two substrate analogs ( LAMe and C3 ) as inhibitors, which were further validated by binding studies and co-crystallography. Importantly, LAMe exhibited low toxicity in human cells and achieved killing of Plasmodium falciparum in erythrocytes with an EC₅₀ value of 15 μM, making it the most effective LPL inhibitor reported to date.     

Phosphino-Phosphination Reactions: Frustrated Lewis Pair Reactivity of Phosphino-Phosphonium Cations with Alkynes

The phosphino-phosphonium cations of the form [R₃PPR′₂]⁺ are labile and provide access to the constituent Lewis acidic and Lewis basic fragments. This permits frustrated Lewis pair-type addition reactions to alkynes, affording unprecedented phosphino-phosphination reactions and giving cations of the form [cis-R₃PCHC(R′′)PR′₂]⁺. This reactivity is further adapted to prepare several examples of a rare class of dissymmetric cis-olefin-linked bidentate phosphines.     

Non-affine structural evolution of soft colloidal crystalline latex films under stretching as observed via synchrotron X-ray scattering

A polymer dispersion consisting of soft latex spheres with a diameter of 135 nm was used to produce a crystalline film with face-centered cubic (fcc) packing of the spheres. Different from conventional small-molecule and hard-sphere colloidal crystals, the crystalline latex film in the present case is soft (i.e., easily deformable). The structural evolution of this soft colloidal latex film under stretching was investigated by in-situ synchrotron ultra-small-angle X-ray scattering. The film exhibits polycrystalline scattering behavior corresponding to fcc structure. Stretching results not only in a large deformation of the crystallographic structure but also in considerable nonaffine deformation at high draw ratios. The unexpected nonaffine deformation was attributed to slippage between rows of particles and crystalline grain boundaries. The crystalline structure remains intact even at high deformation, suggesting that directional anisotropic colloidal crystallites can be easily produced.     

PLD-assisted VLS growth of aligned ferrite nanorods, nanowires, and nanobelts-synthesis, and properties

We report here a systematic synthesis and characterization of aligned alpha-Fe2O3 (hematite), epsilon-Fe2O3, and Fe3O4 (magnetite) nanorods, nanobelts, and nanowires on alumina substrates using a pulsed laser deposition (PLD) method. The presence of spherical gold catalyst particles at the tips of the nanostructures indicates selective growth via the vapor-liquid-solid (VLS) mechanism. Through a series of experiments, we have produced a primitive "phase diagram" for growing these structures based on several designed pressure and temperature parameters. Transmission electron microscopy (TEM) analysis has shown that the rods, wires, and belts are single-crystalline and grow along <111>m or <110>h directions. X-ray diffraction (XRD) measurements confirm phase and structural analysis. Superconducting quantum interference device (SQUID) measurements show that the iron oxide structures exhibit interesting magnetic behavior, particularly at room temperature. This work is the first known report of magnetite 1D nanostructure growth via the vapor-liquid-solid (VLS) mechanism without using a template, as well as the first known synthesis of long epsilon-Fe2O3 nanobelts and nanowires.