Transfers of K-types on local theta lifts of characters and unitary lowest weight modules

In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one-dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally, we will use these results to study subquotients of some cohomologically induced modules.     

On the commuting probability and supersolvability of finite groups

For a finite group \(G\) , let \(d(G)\) denote the probability that a randomly chosen pair of elements of \(G\) commute. We prove that if \(d(G)>1/s\) for some integer \(s>1\) and \(G\) splits over an abelian normal nontrivial subgroup \(N\) , then \(G\) has a nontrivial conjugacy class inside \(N\) of size at most \(s-1\) . We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if \(d(G)>5/16\) then either \(G\) is supersolvable, or \(G\) isoclinic to \(A_4\) , or \(G/\mathbf{Z}(G)\) is isoclinic to \(A_4\) .     

Characterization of Ir(ppy)3 and [Ir(ppy)2 bpy]+ by infrared,Raman spectra and surface‐enhanced Raman scattering

The Raman and infrared spectra of fac ‐tris(2‐phenylpyridinato‐N,C2′)iridium(III), Ir(ppy)₃ and surface‐enhanced resonance Raman spectra of bis(2‐phenyl pyridinato‐) (2,2′bipyridine) iridium (III), [Ir(ppy)₂ (bpy)]⁺ cation were recorded in the wavenumber range 150–1700 cm⁻¹, and complete vibrational analyses of Ir(ppy)₃ and [Ir(ppy)₂ (bpy)]⁺ were performed. Most of the vibrational wavenumbers were calculated with density‐functional theory agree with experimental data. On the basis of the results of calculation and comparison of the spectra of both complexes and their analogue [Ru(bpy)₃]²⁺, we assign the vibrational wavenumbers for metal–ligand modes; metal–ligand stretching wavenumbers are 277/307 and 261/236 cm⁻¹ for Ir(ppy)₃, and 311/324, 257/270, 199/245 cm⁻¹ for [Ir(ppy)₂ bpy]⁺. Surface‐enhanced Raman scattering spectra of [Ir(ppy)₂ bpy]²⁺ were measured at two wavelengths on the red and blue edges of the low‐energy metal‐to‐ligand charge‐transfer band. According to the enhanced Raman intensities for the vibrational modes of both ligands ppy and bpy, the unresolved charge‐transfer band is deduced to consist of charge‐transfer transitions from the triplet metal to both ligands ppy and bpy. Copyright © 2010 John Wiley & Sons, Ltd.     

Capillary electrochromatography with contactless conductivity detection for the determination of some inorganic and organic cations using monolithic octadecylsilica columns

A fast separation of alkali and alkaline earth metal cations and ammonium was carried out by capillary electrochromatography on monolithic octadecylsilica columns of 15 cm length and 100 μm inner diameter using water/methanol mixtures containing acetic acid as mobile phase. On-column contactless conductivity detection was used for quantification of these non-UV-absorbing species. The method was also extended successfully to the determination of small amines as well as of amino acids, and the separation selectivity was optimized by varying the composition of the mobile phase. Detection limits of about 1 μM were possible for the inorganic cations as well as for the small amines, while the amino acids could be quantified down to about 10 μM. The separation of 12 amino acids was achieved in the relatively short time of 10 min.     

Uniform Mesoporous Silica Hexagon and Its Two‐Dimensional Colloidal Crystal

Well‐ordered mesoporous silica nanoparticles with uniform hexagonal disk shapes are synthesized under dilute alkaline conditions with a two‐step process, separating the nucleation and growth process. The resulting monodisperse hexagons can be arranged in a 2‐dimensional (2D) ordered periodical super‐structure. The hexagonal symmetry is similar in both scales. A statistical mechanical cell model is applied to analyze consequences of the interesting packing structure, including osmotic bulk modulus, phase separation and defects.     

Synthesis of Alginate Oligosaccharides Containing L‐Guluronic Acids

Making ends meet : The L ‐gulopyranosyl trichloroacetimidate and 1,6‐anhydro‐2,3‐di‐O‐benzyl‐β‐L ‐gulopyranose, efficiently derived from L ‐ascorbic acid, were individually used as a starting unit and an elongation unit to prepare L ‐guluronic acid linked alginate oligosaccharides in excellent α selectivity and in good yields.

     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

Synthesis of methyl N-Boc-(2S,4R)-4-methylpipecolate

An efficient stereoselective synthesis of fully protected (2S,4R)-4-methylpipecolic acid has been developed. The synthesis was achieved by initial asymmetric α-alkylation of glycine with a chiral iodide, affording the linear precursor as a single stereoisomer. Subsequent aldehyde formation using OsO(4)/NaIO(4) followed by immediate intramolecular cyclization afforded an enamine that was then subjected to hydrogenation to give the final compound in 23% yield over 10 steps.     

Regioselective one-pot protection of D-glucosamine

A highly regioselective one-pot transformation of 2-azido-2-deoxy-1,3,4,6-tetra-O-trimethylsilyl-d-glucopyranose via sequential additions of various reagents was systematically studied, yielding the fully protected derivatives and the 1-, 3-, 4-, as well as 6-alcohols, respectively.     

Effects of temperature and physical state on heterogeneous oxidation of oleic acid droplets with ozone

The heterogeneous reactions of pure micrometer-sized oleic acid droplets with ozone were studied as a function of temperature and physical state. Oxidation reactions were monitored using attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FT-IR) and UV-vis spectrometry. Variations in droplet morphology due to the extent of oxidation were monitored using an optical microscope. Oleic acid droplets were maintained in either solid or liquid state at 9.0 °C. The physical state of the aerosol was determined from the IR absorbance spectra. Oxidation of solid state oleic acid with ozone at 9.0 °C was rapidly converted to the liquid state, which was most likely due to the presence of oxidation products on the surface of the droplets. The fast melting process that resulted from exposure of solid-phase droplets to ozone produced an oxidation rate similar to that for liquid-phase droplets exposed to ozone at the same temperature. Analysis of the carboxylic IR absorbance ratio for esters vs carboxylic acids indicates that the larger ester C═O-to-carboxylic acid C═O ratios at higher temperature appeared to correspond to the production of α-acyloxyalkyl hydroperoxide oligomers and polymers. The wide variation in product yields will result in vastly different physical properties of aerosol particles under different ambient environmental conditions.