A bijective proof of the hook-length formula

A well-known theorem of Frame, Robinson, and, Thrall states that if λ is a partition of n, then the number of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape λ, and the set of pairs (T, S), where T is a Standard Young Tableau of shape λ and S is a “Pointer” Tableau of shape λ.     

The carbon-13 NMR spectrum of dimethoxybilirubin dimethyl ester

The ¹³C NMR spectrum of dimethoxybilirubin dimethyl ester (1) is reported. Methoxyvinylneoxanthobilirubinic acid methyl ester, a compound containing one endo-vinylpyrromethene unit, was prepared and used in the assignment of the spectrum of 1.     

Further properties of discrete analytic functions

Analogs of the classical theorems of Liouville, Phragmén-Lindelöf, and Paley-Wiener are proved in the class of discrete analytic functions.     

Recent developments in positron research

Positrons are proving to be powerful analysis tool in the field of materials research and development. Due to positrons highly interactive nature and how they affect their surroundings when they become trapped in a defect, positron spectroscopy is a fast growing field. Positron spectroscopy may be utilized to detect and analyze defects on and near the surfaces of many materials used in engineering applications today. This spectroscopy technique is very useful in the silicon and semiconductor industry for locating and analyzing the low amount of defects present in those materials. The research in this field is growing every year, with more types of positron sources being developed and more complete systems being operated. This article attempts to provide a recent review of the research in the positron fields of materials analysis and spectroscopy research.     

Binding of Cd2+ to self-assembled bilayers bearing pyridine terminal groups: attenuated total reflection Fourier transform infrared spectroscopic studies

Various aspects of the adsorption of Cd2+ on self-assembled monolayers (SAMs) of 4-heptadecylpyridine (HDpy) and 7-tridecyl-4-methyl-1,10-bipyridine (TMbipy) supported on an octadecylsilane (ODS)-modified Ge prism have been examined both ex situ (dry) and in situ (in the presence of aqueous solutions) using attenuated total reflection Fourier transform infrared spectroscopy. In direct analogy with the behavior found for a variety of genuine pyridine (py) and bipyridine (bipy) metal ion complexes, Cd2+ binding to both SAMs led to shifts in the skeletal vibrational modes of the corresponding uncoordinated ligands in the region 1650-1400 cm(-1), toward higher energies. Analysis of spectra acquired ex situ for HDpy/ODS/Ge and TMbipy/ODS/Ge before and after exposure to 0.1 mM [Cd2+] yielded values for the fraction of uncoordinated py, theta(py(un)), and uncoordinated bipy, theta(bipy(un)), of about 0.5 and about 0.1, respectively. The features attributed to the py(un) groups for spectra collected for HDpy/ODS/Ge in situ and ex situ were found to be virtually identical, making it possible to isolate by graphical means the most prominent band of py(co) centered at about 1615 cm(-l) for HDpy/ODS/Ge collected in situ. The resulting bands for pure py(co) and pyun in situ were then used to deconvolute spectra recorded in situ for HPpy/ODS/Ge in contact with Cd2+containing solutions in the range 10 nM < [Cd2+] < 0.1 mM, from which information regarding the adsorption isotherm was obtained.     

The size of the set of <Emphasis Type="Italic">μ</Emphasis>-irregular points of a measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a compactly suppported positive measure on the real line. A point x∈supp?[μ] is said to be μ-regular, if, as n→∞,

$\sup_{\deg\, (P) \le n}\left(\frac{|P(x)|}{{\|P\|}_{L_{2}(d\mu)}}\right )^{1/n}\to1.$

Otherwise it is a μ-irregular point. We show that for any such measure, the set of μ-irregular points in {μ′>0} (with a suitable definition of this set) has Hausdorff \(m_{h_{\beta}}\) measure 0, for \(h_{\beta}(t) =\left(\log \frac{1}{t}\right)^{-\beta}\), any β>1.

     

Exceptional electrochemical performance of Si-nanowires in 1,3-dioxolane solutions: a surface chemical investigation

The effect of 1,3-dioxolane (DOL) based electrolyte solutions (DOL/LiTFSI and DOL/LiTFSI-LiNO(3)) on the electrochemical performance and surface chemistry of silicon nanowire (SiNW) anodes was systematically investigated. SiNWs exhibited an exceptional electrochemical performance in DOL solutions in contrast to standard alkyl carbonate solutions (EC-DMC/LiPF(6)). Reduced irreversible capacity losses, enhanced and stable reversible capacities over prolonged cycling, and lower impedance were identified with DOL solutions. After 1000 charge-discharge cycles (at 60 °C and a 6 C rate), SiNWs in DOL/LiTFSI-LiNO(3) solution exhibited a reversible capacity of 1275 mAh/g, whereas only 575 and 20 mAh/g were identified in DOL/LiTFSI and EC-DMC solutions, respectively. Transmission electron microscopy (TEM) studies demonstrated the complete and uniform lithiation of SiNWs in DOL-based electrolyte solutions and incomplete, nonuniform lithiation in EC-DMC solutions. In addition, the formation of compact and uniform surface films on SiNWs cycled in DOL-based electrolyte solutions was identified by scanning electron microscopic (SEM) imaging, while the surface films formed in EC-DMC based solutions were thick and nonuniform. X-ray photoelectron spectroscopy (XPS) and Fourier transform infrared (FTIR) spectroscopy were employed to analyze the surface chemistry of SiNWs cycled in EC-DMC and DOL based electrolyte solutions. The distinctive surface chemistry of SiNWs cycled in DOL based electrolyte solutions was found to be responsible for their enhanced electrochemical performances.