A simple sum rule for total radiative decay rates in diatomics

The quantity (ν³μ_e²), which occurs in expressions for the total radiative decay rate for an excited vibronic state of a diatomic, is evaluated as a simple expectation value for the initial state. In test calculations this expectation value agrees with the exact sum over states within ≈0.2%.     

Determination of moisture in cheese and cheese products

Variables related to oven-drying samples of cheese and cheese products to determine moisture content were examined to provide more efficient and reproducible methods. Over 6500 samples of cheese were analyzed in an effort to modify the current AOAC procedure. The gravity atmospheric oven was unsuitable for use in accurate moisture analysis because of wide temperature differentials within the oven cavity. Use of this for oven moisture determination resulted in higher variance, which corresponded to the high temperature variation within the oven. Cheese sample preparation using an Oster blender yielded consistently lower variance in final moisture content than did preparation of cheese samples with a hand grater, rotary grater, and plug and plunger. Sample size of 3 +/- 0.25 g maximized surface area-to-volume ratios and yielded a lower error in final moisture content because of better control of ambient weight loss rates. Use of combination of disposable 5.5 cm diameter aluminum sample pans with 5.5 cm diameter glass fiber filter pads for covers produced a smaller standard deviation for moisture analysis than did the AOAC pan and insert cover and filter paper covers. All pans must be pre-dried for at least 3 h at 100 degrees C, and the glass fiber covers should be pre-dried for 1 h under the same conditions. All dried pans and covers must be stored in a desiccator with active desiccant. Equipment upgrades from the existing AOAC standard methods provide safer more efficient methods of analysis.     

Thermal and autoxidation reactions of poly-3-methylenecyclobutene and poly-1-methyl-3-methylenecyclobutene

The thermal ring-opening reactions, autoxidation and hydrogenation of polymethy-lenecyclobutene (PMCB) and poly-1-methyl-3-methylenecyclobutene (PMMCB), were investigated. Both polymers were prepared by cationic polymerization and consisted almost entirely of 1,5-repeating units containing cyclobutene rings in the polymer backbone. Both polymers showed well behaved exothermic processes at elevated temperatures which apparently resulted in crosslinking. These processes were investigated by differential scanning calorimetry and interpreted to involve thermal ring-opening reactions. Autoxidation occurred very rapidly in PMCB but much more slowly in PMMCB as predicted by Bolland's rules. Attempts to hydrogenate the cyclobutene rings in both polymers resulted in the occurrence of hydrogenolysis in PMCB and little or no reaction with PMMCB for a Pd-catalyzed reaction and partial hydrogenation of the latter for a diimide reaction.     

A simplified proof of a theorem about group representations

It is a standard theorem of group representation theory that the dimension of an irreducible representation is a divisor of the order of the group. This paper gives a new, relatively simple proof, intended to make the theorem understandable to readers unfamiliar with algebraic integers.     

Extension of a theorem of P. Wolfe on singular integral equations

The integral operator which sends f (ζ) into $$\int_{|\xi |< 1} {\tfrac{{f(\xi )}}{{|x - \xi |^{p - 1} }}} d\xi = g(x)(|x|< 1)$$ x and ζ ∈ R^P, is shown to map the space of functions f (ζ) on |ζ|<1, square integrable with respect to the weight (1-|ξ|²)^1/2 one to one onto the space of functions g (x) on |x|<1 which possess gradients δg(x) in the sense of distribution theory, with both g(x) and δg(x) being square integrable with respect to the weight (1-|x|²)^1/2. This extends to p≥3 a result of P. Wolfe's in the case p=2.     

Existence,uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

We study positive solutions of the Dirichlet problem: u(x)+f(u(x))=0,xD ⁿ ,u(x)=0,xD ⁿ , whereD ⁿ is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.Research supported in part by NSF Contract Number MCS 80-02337     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

Anharmonicity in rotational and vibrational excitation of H2 by Li+ collisions

Integral cross sections for pure rotational and vibrational-rotational excitation of H₂(X¹Σ⁺_g) by Li⁺(¹S) impact are computed by close-coupling methods at 0.2, 0.6, and 1.2 eV in the c.m. system using vibrational functions that are numerical solutions of the one-dimensional radial Schrödinger equation for harmonic, Morse, and adiabatically corrected Kolos-Wolniewicz (KW) potential functions. Comparison of results employing KW and Morse functions shows excellent agreement for all transitions studied. Findings using harmonic oscillator functions, however, differ noticeably from KW and Morse values for vibrational (0 → 1) and very large rotational (Δj = 10) transitions, but are satisfactory for lower order (0 → 2, 4, 6, 8) rotational transitions.     

Studies in the heterocyclic series. XI. Reactions of 2-Amino-3-hydroxypyridine with 2-Chloro-3-nitropyridine

The reactions of 2-amino-3-hydroxypyridine and 2-chloro-3-nitropyridine in various solvents and in the presence of certain catalysts were studied. A dilute acid-catalysed procedure led to a 45% yield of 3-hydroxy-3-nitro-2,2-dipyridylamine which was also obtained by refluxing in alcoholic potassium hydroxide for several hours. The diarylamine was converted to 1,9-diazaphenoxazine by base-catalysed intramolecular condensation in refluxing DMSO. The same diazaphenoxazine was obtained directly by prolonged base-catalysed reaction of the starting pyridine compounds in DMSO. From a study of the infrared spectrum of the diarylamine precursor, it was found that strong NHO hydrogen bonding is responsible for the failure of the cyclization step in either aqueous or alcoholic base. This problem was overcome by the use of DMSO as the reaction solvent. Structural assignments were made by a study of their ultraviolet, infrared, nmr and mass spectra and the mechanistic pathways were also discussed.