On uniform distribution of double sequences

In this paper a new notion of uniform distribution of double sequences is introduced. In the most elementary case a double sequence (x_mn)_{m, nN} is called uniformly distributed in this sense if all finite matrices of equal size appear with the same frequency as submatrices of the infinite matrix (x_mn). Some special double sequences with such a distribution property are studied and metric theorems are proved.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Path integral formulation of general diffusion processes

A method is given for the derivation of a path integral representation of the Green's function solutionP of equationsP/t=L P,L being some Liouville operator. The method is applied to general diffusion processes.Feynman's path integral representation of the Schrödinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.     

Diphosphine-phosphenium coordination complexes representing monocations with pendant donors and ligand tethered dications

Homoatomic P-P coordinate bonding is exploited to prepare the first examples of triphosphorus monocations and tetraphosphorus dications using dimethylphosphenium or diphenylphosphenium Lewis acceptors with diphosphinomethane, diphosphinoethane, diphosphinohexane, or diphosphinobenzene ligands. Solid-state structures and spectroscopic characterization data for complexes involving bis(diphenylphosphino)methane ligands show coordination of only one donor site of the diphosphine ligand in the monocations, and chelate complexation is not observed. Tetraphosphorus dications are observed with longer diphosphines, in which the ligand tethers two phosphenium acceptors. The structural preferences between monocations with pendant phosphines and tethered dications are dependent on intramolecular steric interactions and the flexibility of the tether.     

Structure of gas phase [C5H6]+˙ ions

Charge-stripping spectra have been used to differentiate ionized cyclopentadiene from its acyclic isomers. The minimum amounts of translational energy lost during the charge-stripping processes and the relative charge-stripping efficiencies, which are also structurally important parameters, have been measured for these ionic species. [C₅H₆]⁺˙ ions, formed by dissociative ionization of various precursors in the ion source are found, usually, to be a mixture of cyclic and acyclic ions. In contrast, [C₅H₆]⁺˙ ions, derived from the dissociation of metastable molecular ions from a series of organic compounds, have the cyclopentadienyl structure. This structure was confirmed by collision-induced dissociation of ions formed in the first field-free region of a triple sector mass spectrometer.     

Transition metal complexes of 2-Carbethoxypyridine (Ethyl picolinate)

Summary A variety of metal(II) complexes of 2-carbethoxypyridine (L) have been prepared and characterised. With metal(II) chlorides the bis complexes can be formulated [ML₂Cl₂]^o (M=Cu^II, Ni^II, Co^II, Fe^II or Mn^II). The complexes are six-coordinate with 2-carbethoxypyridine acting as a bidentate ligandvia the pyridine nitrogen and the carbonyl group of the ester. The chloro complexes are nonelectrolytes in nitroethane; magnetic susceptibility measurements, i.r. and d-d electronic spectra are reported. With metal(II) perchlorate salts the complexes can be formulated as six-coordinate [ML₂ (OH₂)₂] [ClO₄]₂ species containing ionic perchlorate. The ester exchanges of some of these complexes with a variety of primary alcohols have been investigated.     

Molecular weight distribution studies. I. Radiation-induced polymerization of liquid α-methylstyrene

The concentration of water in purified and BaO-dried α-methylstyrene was found to be 1.1 × 10^?4M. The radiation-induced bulk polymerization of the α-methylstyrene thus prepared was studied in the temperature range of ?20°C to 35°C. The polymerization rate varied as the 0.55 power of the dose rate. The theoretical molecular weights and molecular weight distribution were calculated from a proposed kinetic scheme and these values were then compared with those found experimentally. The agreement between these two was reasonably close, and therefore it was concluded that, from the molecular weight distribution point of view, the proposed kinetic scheme for the cationic polymerization of α-methylstyrene is an acceptable one. The rate constant for chain transfer to monomer k_f changed with temperature and was found to be responsible for the decrease in the molecular weight of the polymer with increase in temperature. k_f and k_p at 20°C were found to be 0.95 × 10⁴ l./mole-sec and 0.99 × 10⁶ l./mole-sec, respectively.     

Determination of the sulfur components of gasoline streams by capillary column gas chromatography with sulfur chemiluminescence detection

A method using a sulfur chemiluminescence detector (SCD) has been developed for the qualitative and quantitative determination of sulfur components in stabilized gasoline-range process streams, including blended gasolines containing between 1 and > 4000 ppm total sulfur. The detection limit per sulfur component was approximately 50 ppb. On-column injection was employed for optimum precision and accuracy. A new probe material was found which did not soften under FID operating conditions. A new method, using a hydrogen sulfide permeation tube, was developed for rapid alignment of the probe in the FID.     

Nucleosides. Part LX. Synthesis and Characterization of Monomeric Cordycepin-Vitamin and Cordycepin-Lipid Conjugates Model Substances for Biodegradable Ester and Carbonate Linkages in Conjugates and Potential Inhibitors of HIV-1 Replication

Monomeric 3′-deoxyadenosine (cordycepin) was modified at the 2′-O- ( 13–18 ) and 5′-O-position ( 25–29 ) by the vitamins E, D₂, and A and by the two lipids 1,2-di-O-palmitoylglycerol and 1,2-di-O-hexadecylglycerol via succinate or carbonate linkages. The base-labile conjugates afforded protection groups like the 2-(4-nitro-phenyl)ethoxycarbonyl (npeoc) and monomethoxytrityl group (MeOTr) that are cleavable without harming the ester and carbonate bonds, respectively. Monomeric conjugates of cordycepin and vitamin E, vitamin D₂, 1,2-di-O-palmitoylglycerol, and 1,2-di-O-hexadecylglycerol (see 13, 14, 17, 18, 25, 26, 28 , and 29 ) inhibited HIV-1-induced syncytia formation 1.7 to 6.2 fold compared to 1.5-fold for cordycepin (see Table); IC₅₀ values for 25 and 28 were 257 and 267 m?M , respectively. In addition, the monomeric cordycepin-vitamin and -lipid conjugates inhibited HIV-1 RT activity 28–49% which compares with a 13% inhibition of HIV-1 RT observed for cordycepin. The minimal inhibition of HIV-1-induced syncytia formation and HIV-1 RT activity did not proceed by the activation of RNase L. The monomeric conjugates tested ( 13, 14 ) increased PKR expression.     

The crystal structure of Mn(NO)3P(C6H5)3: The first x-ray analysis of a mononuclear metal trinitrosyl complex

The structure of Mn(NO)₃PPh₃ has been analyzed by single-crystal X-ray diffraction. It shows a tetrahedral geometry with essentially linear nitrosyl groups, and an eclipsed configuration around the MnP bond. Average distances and angles are: MnN 1.686(7) Å, MnP 2.315(2) Å, NO 1.165(10) Å, PC 1.815(4) Å, MnNO 177.2(7)°, PMnN 103.6(2)°, NMnN 114.7(4)°. Final R factor 7.3% for 2064 non-zero reflections. The structure of the five-coordinate nitrito complex Mn(NO)₂(ONO)(PEt₃)₂ is also mentioned briefly.