On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

We consider the equation \(\rm f^{\prime\prime}+{A}(z){f}=0\) with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of \(\rm f^{\prime\prime}+{e}^P{f}=0\) satisfies λ(f) = ∞. Finally we consider the particular equation \(\rm f^{\prime\prime}+({e}^Z-K){f}=0\) where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.     

Mass spectra of halogenated esters. 7. Methyl esters of 2-chloro C2?C20 n-alkanoic acids

The mass spectral fragmentation of a homologous series of methyl esters of 2-chloro n-alkanoic acids ranging from acetic (C₂) to eicosanoic (C₂₀) acid on electron impact has been investigated. The fragmentation pathways were elucidated with the aid of the first field-free region metastable ions, the results being presented with one compound, i.e. with ionized methyl 2-chloro-octauoate. Owing to the Cl/H exchanges and to the formation of the non-chlorinated parent esters prior to the fragmentations the spectra show the peak pairs with and without the chlorine atom. The effects become more evident with increasing chain length; shown most visually by the abundance ratios of the McLafferty rearrangement ions atm/z108/110 and 74, and fragments at m/z121/123 and 87.     

Synthesis of C-C-bridged bis-isoflavones

Unique C-C-bridged bis-isoflavones 5, 8, and 9 were obtained by reaction of 2-bromomethyl-7,4'-dimethoxyisoflavone 4 with ethyl cyanoacetate anion or tetraethylammonium cyanide or by Pd-catalyzed ethoxycarbonylation, respectively. The phenolic carboxylic acid 7 is available from 5 in two steps.     

Corrigendum to a Paper by Charak and Laine

This is a corrigendum to our paper, "On a class of prime entire functions", published in Acta Math. Sin., Engl. Ser., 25, 1647-1652 (2009).     

Pore formation coupled to ion transport through lipid membranes as induced by transmembrane ionic charge imbalance: atomistic molecular dynamics study

We have employed atomic-scale molecular dynamics simulations to address ion transport through transient water pores in phospholipid membranes. The formation of a water pore is induced by a transmembrane ionic charge imbalance, which gives rise to a significant potential difference across the membrane. The subsequent transport of ions through the pore discharges the transmembrane potential and makes the water pore metastable, leading eventually to its sealing. The findings highlight the importance of ionic charge fluctuations in spontaneous pore formation and their role in ion leakage through protein-free lipid membranes.     

Aqua[bis(pyrimidin‐2‐yl‐κN)amine](carbonato‐κ2O,O′)copper(II) dihydrate

The title mononuclear complex, [Cu(CO₃)(C₈H₇N₅)(H₂O)]·2H₂O, was obtained by fixation of CO₂ by a mixture of copper(II) tetra­fluoro­borate and the ligand bis­(pyrimidin‐2‐yl)­amine in ethanol/water. The Cu^II ion of the complex has a distorted square‐pyramidal environment, with a basal plane formed by two N atoms of the ligand and two chelating O atoms of the carbonate group, while the apical position is occupied by the O atom of the coordinating water mol­ecule. In the solid state, hydrogen‐bonding interactions are dominant, the most unusual being the Watson–Crick‐type coplanar ligand pairing through two N—H?N bonds. Lattice water mol­ecules also participate in hydrogen bonding.