Error estimates for generalized compound quadrature formulas

We define quadrature formulas for integrals with weight functionsby applying a given approximation method locally. This allowsthe generalisation of different quadrature formulas, e.g., thecompound Newton-Cotes formulas, Gauss summation formulas, orGregory's formulas, to the case of weighted integrals, as wellas to construct new quadrature formulas, and to derive errorestimates for all these quadrature formulas. The estimates consideredhere are mainly of the form |R[f]|c||f^(r)||, provided the underlyingapproximation method is exact for polynomials of degree <r(R[f] is the quadrature error). Explicit, asymptotically sharperror estimates are obtained for arbitrary integrable weightfunctions. Further, estimates are obtained for the case thatthe quadrature error is of higher order than the approximationerror.     

Further Revelations on L.S.D.

In a recent paper the author showed that the distribution ofleading significant digits, l.s.d., resulting from successivemultiplications is logarithmic. In this paper these resultsare extended by establishing, still without any assumptionsof invariance to scaling, that this distribution remains invariantunder all further arithmetic operations—both multiplicativeand additive. The fact that 30% of decimal numbers have l.s.d.1 persists.     

SPECTROSCOPIC INVESTIGATION OF DIHYDRONICOTINAMIDES-I: CONFORMATION, ABSORPTION, AND FLUORESCENCE

Abstract— The 1(N)-(2,6-dichlorobenzyl)-1,4-dihydronicotinamide (I), N-methyl- and N,N-dimethyl-1(N)-(2,6-dichlorobenzyl)-1,4-dihydronicotinamide (II and III), respectively), and 1(N)-(2,6-dichloro-benzyl)-2-aminomethyl-1,4-dihydronicotinic acid lactame (IV) were synthesized as model compounds for natural coenzymes, and systematically studied by 1H NMR, UV/V1S absorption and fluorescence spectroscopy. The absorption at ∼ 340 nm argues for an effective conjugation between dihydropyridine and carboxamide π-system, and rules out any severely twisted conformation. For the natural coenzymes NADH and NMNH, as well as for I and II (with no or only one N-amide substituent), 1H NMR definitively establishes a transoid conformation in solution, with the carbonyl O close to 2-H of the dihydropyridine ring. N,N-dimethyl substitution effectively inverts the carboxamide orientation into the cisoid form. The 1H NMR data (as well as molar extinctions) for the fused-ring derivatives IV and V, with a fixed cisoid and transoid structure, respectively, provide final proof for the conformational assignment.
Absorption maxima are shifted to lower energies with increasing solvent polarity. In solvents which can act as hydrogen bond acceptors to the carboxamide N-H, absorption shows a general blue-shift of ∼ 10 nm. H-bond donor solvents do not affect absorption maxima but enhance molar extinction. Fluorescence maxima show a similar dependence on solvent polarity but no specific hydrogen-bonding effect. Fluorescence quantum yields appear increased tenfold in solvents donating H-bonds to the carboxamide C=O group. These results are interpreted in terms of the vinylogous amide resonance between C=O function and ring-N lone pair being the electronic interaction dominating in the ground state of dihydronicotinamides.     

PHOTOPRODUCTS OF BROMOURACIL-LABELED DNA AND THE STRUCTURE OF 5-BROMODEOXYURIDYLYL-(3' → 5')-THYMIDINE PHOTOPRODUCT

Abstract— –An attempt was made to identify some of the ultraviolet (u.v.) photoproducts of 5-bromouracil-labeled DNA (BrU-DNA). Two synthetic dinucleotides, 5-bromodeoxyuridylyl-(3' →5 ')-thymidine (BrdUpT) and 5-bromodeoxyuridylyl-(3' → 5')-deoxycytidine (BrdUpdC), were prepared. Each gave a single u.v. photoproduct which in turn gave a single acid hydrolysis product. 2-¹⁴C-BrU-DNA. prepared from E. coli B3, was irradiated (275–280 nm), hydrolyzed, and paper chromatographed in four systems. Comparison with the two synthetic photoproducts showed that if present at all, BrdUpT and BrdUpdC photoproducts could account for no more than 10 and 3.5 per cent respectively of the total photoproducts. At 55 per cent conversion of BrU into photoproducts, the major ¹⁴C-photoproduct was uracil (78 per cent); the remaining 22 per cent was made up of at least six products, three of which were reversed by 232 nm irradiation.
The debrominated cyclobutane structure proposed by Haug for BrdUpT photoproduct has been shown to be incorrect. It was found to contain one atom of bromine per molecule. On the basis of nuclear magnetic resonance and u.v. spectra, two possible structures are proposed for the photoproduct, each containing an eight-membered ring.     

Complexation in dense gases: concentrations of argon monomers,dimers and trimers

A classical statistical mechanical cluster formalism is presented for computing the thermodynamic properties and equilibrium state of aggregation of a fluid composed of structureless monomeric units which are capable of forming weakly bound van der Waals complexes. The general procedure is not restricted to any particular definition of the clusters, provided that the definition used satisfies certain reasonable criteria. The formalism is illustrated by applying it to two different cluster definitions. Calculations based on these definitions have been performed for the concentrations of argon monomers, dimers and trimers. It is found that gas imperfections due to cluster-cluster interactions can significantly affect the calculated values of these concentrations.     

Intramolecular vibrational relaxation in aromatic molecules. 2: An experimental and computational study of pyrrole and triazine near the IVR threshold

The threshold region of vibrational energy redistribution (IVR) presents a great experimental and computational challenge for organic molecules with more than 10 degrees of freedom. The density of states ρ_tot is high and requires high resolution measurements over a wide range to cover all relevant timescales experimentally. Yet ρ_tot is sufficiently low that IVR quantities like the initial relaxation time τIVR or the number of participating states N_eff are very sensitive to the coupling structure. To highlight the competing effects of molecular symmetry and mode localization on the accessible density of states, this work complements a study of benzene (Callegari, A., Merker, U., Engels, P., Srivastava, H. K., Lehmann, K. K., and Scoles, G., 2000, J. chem. Phys., 113, 10583) by measuring the CH overtone spectra of pyrrole (C₄H₄NH) and 1,2,3-triazine (C₃N₃H₃) using eigenstate-resolved double-resonance spectroscopy. Large scale computations of IVR dynamics were undertaken, applying filter diagonalization to analytically fitted fourth-order ab initio force fields. With an overall adjustment to the anharmonicity of the potential, the modelled N_eff and τ_IVR agree with the experimental quantities within a factor of 2 to 3, which is reasonable for a rate theory in the threshold regime. The models also correctly predict the experimentally observed trends of τ_IVR and N_eff for the two molecules, and provide insight into the highly off-resonant coupling mechanism, which yields very sharp linewidths.