An improved approximation algorithm for maximum edge 2-coloring in simple graphs

We present a polynomial-time approximation algorithm for legally coloring as many edges of a given simple graph as possible using two colors. It achieves an approximation ratio of $\frac{468}{575}$. This improves on the previous best (trivial) ratio of $\frac{4}{5}$.     

Highly powerful and practical acylation of alcohols with acid anhydride catalyzed by Bi(OTf)(3).

Bi(OTf)(3)-catalyzed acylation of alcohols with acid anhydride was evaluated in comparison with other acylation methods. The Bi(OTf)(3)/acid anhydride protocol was so powerful that sterically demanding or tertiary alcohols could be acylated smoothly. Less reactive acylation reagents such as benzoic and pivalic anhydride are also activated by this catalysis. In these cases, a new technology was developed in order to overcome difficulty in separation of the acylated product from the remaining acylating reagent: methanolysis of the unreacted anhydride into easily separable methyl ester realized quite easy separation of the desired acylation product. The Bi(OTf)(3)/acid anhydride protocol was applicable to a wide spectrum of alcohols bearing various functionalities. Acid-labile THP- or TBS-protected alcohol, furfuryl alcohol, and geraniol could be acylated as well as base-labile alcohols. Even acylation of functionalized tertiary alcohols was effected at room temperature.     

Synthesis of pyrazine acetals by the biased reaction of symmetric 2,5-bis(chloromethyl) or 2,3,5,6-tetrakis(chloromethyl) substituents on pyrazine ring with sodium alkoxides

2,5-Bis(chloromethyl)pyrazine reacted with sodium alkoxide to give unexpected 2-dialkoxymethyl-5-methylpyrazine along with normal substitution product, 2,5-bis(alkoxymethyl)pyrazine. The reaction of 2,3,5,6-tetrakis(chloromethyl)pyrazine with sodium alkoxide afforded similar results to yield 2,6-bis(dialkoxymethyl)-3,5-dimethylpyrazine along with other alkoxymethylpyrazines. The ratio of products depended on the solvent and alkoxide used. A general discussion of the mechanism of such a pyrazine acetal synthesis in the basic conditions is given.     

Crystal growth and electric‐property change by rubidium or cesium doping on potassium‐sodium‐niobate

Alkali metals (Na, Rb or Cs) co‐doped with fiber‐ and bulk‐shaped KNbO₃ single crystals were grown using two original methods by means of doping together of small ionic Na and large ionic Rb or Cs into KNbO₃. Single‐phase crystals could be grown with an orthorhombic system at room temperature as well as pure KNbO₃. Piezoelectric and ferroelectric property changes by the co‐doping of Rb or Cs with Na were estimated using d₃₃ values and a polarization‐electric field hysteresis curve in fiber‐ and bulk‐shaped crystals. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrocatalytic Nitrite Reduction to Nitrogen Oxide by a Synthetic Analogue of the Active Site of Cu‐Containing Nitrite Reductase Incorporated in Nafion Film

The electrocatalytic reduction of nitrite to NO by [CuMe₂bpa(H₂O)(ClO₄)]⁺ ( 1 ), which is a model for the active site of copper‐containing nitrite reductase, incorporated in Nafion film was investigated. The Cu complex in the Nafion matrix exhibits an intense band at 267 nm and a broad band around 680 nm, assigned to d–d and ligand field transitions, respectively. The 77‐K EPR spectrum of 1 in the Nafion matrix reveals the typical axial signals (g_//=2.28, g =2.08, A_//=13.3 mT) of a tetragonal Cu²⁺ chromophore. The redox potential, which is related to the Cu⁺/Cu²⁺ couple, was ?146 mV (ΔE=72 mV) at pH 5.5. The redox reaction of 1 in Nafion was not dependent on pH and was a diffusion‐controlled process. The electronic structure and redox properties of 1 in the negatively charged polymer matrix were almost the same as those in aqueous solution. In the presence of nitrite, an increase in the cathodic current was observed in the cyclic voltammogram of 1 in the Nafion matrix. The current increase was dependent on the nitrite concentration and pH in solution. Upon reaching ?400 mV, a linear generation of NO was observed for the 1 /Nafion film coated electrode. The relationship between the rate of NO generation and the nitrite concentration in solution was analyzed with the Michaelis–Menten equation, where V_max=45.1 nM s^?1 and K_m=15.8 mM at pH 5.5. The Cu complex serves the function of both the catalyst and electron transport in the Nafion matrix. The sensitivity of the electrode was estimated to be 3.23 μA mM^?1 in the range of 0.1–0.4 mM nitrite.     

The second main theorem of hypersurfaces in the projective space

We deal with a holomorphic map from the complex plane ${\mathbb{C}}$ to the n-dimensional complex projective space ${\mathbb{P}^{n}(\mathbb{C})}$ and prove the Nevanlinna Second Main Theorem for some families of non-linear hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ . This Second Main Theorem implies the defect relation. If the degree of the hypersurfaces are sufficiently large, the defect of the map is smaller than one. This means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate. To prove the Second Main Theorem, we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces. A meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the Wronskian operator. By resolving the singularity and pulling back a meromorphic partial projective connection, we also prove the Second Main Theorem for singular hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ , and prove the Second Main Theorem for smooth hypersurfaces in ${\mathbb{P}^{2}(\mathbb{C})}$ which are not normal crossing.     

Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation

Based on the integrable discrete hungry Toda (dhToda) equation, the authors designed an algorithm for computing eigenvalues of a class of totally nonnegative matrices (Ann Mat Pura Appl, doi:10.1007/s10231-011-0231-0). This is named the dhToda algorithm, and can be regarded as an extension of the well-known qd algorithm. The shifted dhToda algorithm has been also designed by introducing the origin shift in order to accelerate the convergence. In this paper, we first propose the differential form of the shifted dhToda algorithm, by referring to that of the qds (dqds) algorithm. The number of subtractions is then reduced and the effect of cancellation in floating point arithmetic is minimized. Next, from the viewpoint of mixed error analysis, we investigate numerical stability of the proposed algorithm in floating point arithmetic. Based on this result, we give a relative perturbation bound for eigenvalues computed by the new algorithm. Thus it is verified that the eigenvalues computed by the proposed algorithm have high relative accuracy. Numerical examples agree with our error analysis for the algorithm.