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Opening a copy of The Mathematical Intelligenceryou may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

A practical synthesis of 2,3-dihydro-2-benzofurancarboxylic acid: A general route to 2,3-dihydrobenzofurans

A practical synthesis of 2,3-dihydro-2-benzofurancarboxylic acid is reported in five steps and approximately 40% overall yield. The methodology offers a useful new route to 2-substituted-2,3-dihydrobenzofurans.     

Random Minimum Length Spanning Trees in Regular Graphs

r -regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that . Received: Februray 9, 1998     

Formation of alkyl nitrates from the reaction of branched and cyclic alkyl peroxy radicals with NO

The yields of C₅ and C₆ alkyl nitrates from neopentane, 2-methylbutane, 2-methylpentane, 3-methylpentane, and cyclohexane have been measured in irradiated CH₃ONONO-alkane-air mixtures at 298 ± 2 K and 735-torr total pressure. Additionally, OH radical rate constants for neopentyl nitrate, 3-nitro-2-methylbutane, 2-nitro-2-methylpentane, 2-nitro-3-methylpentane, and cyclohexyl nitrate, relative to that for n-butane, have been determined at 298 ± 2 K. Using a rate constant for the reaction of OH radicals with n-butane of 2.58 × 10^?12 cm³ molecule^?1 s^?1, these OH radical rate constants are (in units of 10^?12 cm³ molecule^?1 s^?1): neopentyl nitrate, 0.87 ± 0.21; cyclohexyl nitrate, 3.35 ± 0.36; 3-nitro-2-methylbutane, 1.75 ± 0.06; 2-nitro-2-methylpentane, 1.75 ± 0.22; and 2-nitro-3-methylpentane, 3.07 ± 0.08. After accounting for consumption of the alkyl nitrates by OH radical reaction and for the yields of the individual alkyl peroxy radicals formed in the reaction of OH radicals with the alkanes studied, the alkyl nitrate yields (which reflect the fraction of the individual RO₂ radicals reacting with NO to form RONO₂) determined were: neopentyl nitrate, 0.0513 ± 0.0053; cyclohexyl nitrate, 0.160 ± 0.015; 3-nitro-2-methylbutane, 0.109 ± 0.003; 2-nitro-2methylbutane, 0.0533 ± 0.0022; 2-nitro-2-methylpentane, 0.0350 ± 0.0096; 3- + 4-nitro-2-methylpentane, 0.165 ± 0.016; and 2-nitro-3-methylpentane, 0.140 ± 0.014. These results are discussed and compared with previous literature values for the alkyl nitrates formed from primary and secondary alkyl peroxy radicals generated from a series of n-alkanes.     

Inversion of the exciton fine structure splitting in quantum dots

The fine structure splitting of exciton state was measured for a large number of single InAs quantum dots in GaAs. It is shown to decrease as the exciton confinement decreases, crucially passing through zero and changing sign. Degeneracy of the exciton spin states is an important step to producing entangled photons from the biexciton cascade. Thermal annealing reduces the exciton confinement and thereby increases the number of degenerate dots in a particular sample.     

Bulk modulus for some simple molecular fluids

The bulk modulus B of several molecular fluids composed of rigid molecules has been calculated from p-p-T data obtained with a high-pressure vibrating tube densitometer. The data of all the substances studied, including Ar, can be described by a single master curve when plotted versus the reduced density *, in agreement with the predictions of the Gubbins-Gray perturbation theory for fluids with the same reference system. Combination of p-p-T and C_v data with the virial theorem has allowed the calculation of the exponent characterizing the repulsive branch of the intermolecular potential n. The different values of n suggest that different reference systems should be used for each substance, in contradiction with the conclusions obtained from the B versus * curves. This indicates that p-p-T data are less sensitive to the details of the intermolecular potential than their combination with other thermal properties like C_v, internal energy or residual entropy.     

Nitroarene products from the gas-phase reactions of volatile polycyclic aromatic hydrocarbons with the OH radical and N2O5

The nitroarene products of the gas-phase reactions of acenaphthylene, acenaphthene, phenanthrene, and anthracene-d₁₀ with N₂O₅ and the OH radical (in the presence of NO_x) are reported. The calculated atmospheric lifetimes of these polycyclic aromatic hydrocarbons (PAH), as well as those of naphthalene, 1- and 2-methylnaphthalene, biphenyl, fluoranthene, pyrene, and acephenanthrylene, show that reaction with the OH radical is the dominant loss process for these PAH, with the exception of acenaphthylene, acenaphthene, and acephenanthrylene which contain an external cyclopenta-fused ring. For these latter PAH, reaction with the NO₃ radical, and for acenaphthylene and acephenanthrylene reaction with O₃, are also expected to be important atmospheric loss processes. The nitroarenes observed as products of the atmospherically-important gas-phase reactions of the PAH in environmental chamber studies are compared with the nitroarenes measured in ambient air samples collected in California. It is concluded that although nitroarenes are formed in low yields (?5%) from the OH radical-initiated reactions of the PAH, atmospheric formation of nitroarenes may contribute significantly to ambient nitroarene concentrations.     

Nuclear magnetic relaxation of α-13C nuclei of helical poly(γ-hexyl-L-glutamate) and poly(γ-benzyl-L-glutamate)

Spin-lattice relaxation times (T₁), spin-spin relaxation times (T₂), and nuclear Overhauser enhancements (NOE), at 75.5 MHz are reported for α-¹³C nuclei of poly (γ-benzyl-L -glutamate) in deuterated dimethylformamide at 60°C and of poly(γ-hexyl-L -glutamate) in cyclohexanone at 48 and 79°C. It is shown that for molecular weights above 10⁵, the polypeptides cannot be considered as essentially rigid helices with internal librational motions; additional backbone flexing motions contribute to the relaxation behavior.