The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Synthetic Applications of the beta-Lithiation of beta-Aryl Secondary Amides: Diastereoselective and Enantioselective Substitutions

The sequence of beta-lithiation and electrophilic substitution of beta-aryl secondary amides is reported. The lithiations occur regioselectively at the beta-position, and the resulting lithiated intermediates can be reacted with a wide range of electrophiles to give substituted products. Reactions of beta-lithiated amides bearing an alpha-substituent provide substituted products with high diastereoselectivity. Electrophilic substitutions of beta-lithiated N-methylamides in the presence of the chiral diamine (-)-sparteine provide highly enantioenriched products. The methodology is used to synthesize enantioenriched beta-aryl beta-substituted amides, acids, and lactones.     

Nonspecific medium effects versus specific group positioning in the antibody and albumin catalysis of the base-promoted ring-opening reactions of benzisoxazoles

The mechanisms by which solvents, antibodies, and albumins influence the rates of base-catalyzed reactions of benzisoxazoles have been explored theoretically. New experimental data on substituent effects and rates of reactions in several solvents, in an antibody, and in an albumin are reported. Quantum mechanical calculations were carried out for the reactions in water and acetonitrile, and docking of the transition state into a homology model of antibody 34E4 and an X-ray structure of human serum albumin was accomplished. A microenvironment made up of catalytic polar groups (glutamate in antibody 34E4 and lysine in human serum albumin) surrounded by relatively nonpolar groups is present in both catalytic proteins.     

Irradiation of aromatic polymers: The effect of irradiation temperature on crosslinking and scission in polystyrene

Polystyrene of narrow molecular weight distribution, M _w/M _n ≈ 1.03, was subjected to γ irradiation at 100 and 150°C. The yields of scission and crosslinking, G(S) and G(X), where determined from the changes in molecular weight distribution using gel permeation chromatography, (GPC) with supporting evidence from osmometry and viscometry. The ratio G(S)/G(X) increased from 0.02 at 30°C, obtained previously, to 2.8 at 150°C. This was mainly due to a tenfold increase in G(S), whereas G(X) apparently decreased slightly. These results are compatible with increased disproportion of chain scission radicals relative to their combination, analogous to the temperature dependence of mutual termination in the free radical polymerization of styrene. There was no obvious discontinuity through the glass transition temperature, although there may be a change in sign of the temperature coefficient of G(X). This system provides an excellent example of the applicability of measurements of molecular weight distributions and averages to determinations of G(S) and G(X) under conditions when gel measurements are inappropriate, either because of the failure of the system to form gel, or, as in the present case, because of the very large radiation doses required for gel formation.     

Thermodynamics of mixtures of hexane and heptane isomers with normal and branched hexadecane

Molar excess mixing enthalpies h ^E , Gibbs free energies g ^E and hence entropies s ^E have been obtained using calorimetry and the vapor sorption method at 25°C for hexane isomers+2,2,4,4,6,8,8-heptamethylnonane, a highly branched C ₁₆ . The h ^E and g ^E are negative while Ts ^E are positive, but small. The values are explained by the Prigogine-Flory theory through negative free volume contributions to h ^E and Ts ^E , counterbalanced in the case of Ts ^E by the positive combinatiorial Ts ^E for mixing molecules of different size. No contribution is seen from the interaction between methyl and methylene groups. The excess quantities are also obtained for hexane and heptane isomers mixed with n-hexadecane. Values of h ^E and Ts ^E are now strongly positive, while those of g ^E are only slightly less negative. The interpretation requires two recently advanced contributions in addition to those of the Prigogine-Flory theory: 1) a decrease of order when correlations of orientations between n-C ₁₆ molecules in the pure liquid are replaced in the solution by weaker correlations whose strengths depend on the shapes of the lower alkane isomers. For lower alkane isomers of the same shape, but highly sterically hindered, h ^E and Ts ^E are small, manifesting, 2) a negative contribution, ascribed to a rotational ordering of n-C ₁₆ segments on the sterically-hindered molecule. Enthalpy-entropy compensation is observed for these new contributions, arising from their rapid fall-off with increase of temperature.     

Investigation of the enzymatic and nonenzymatic cope rearrangement of carbaprephenate to carbachorismate

The dimethyl esters of carbaprephenate and 4-epi-carbaprephenate were prepared by modification of published procedures. In methanol these compounds are converted quantitatively to isomeric 6-hydroxytricyclo[3.3.1.0(2,7)]non-3-en-1,3-dimethyl esters via a two-step sequence involving an initial Cope rearrangement, followed by intramolecular Diels-Alder reaction of the dimethyl carbachorismate or 4-epi-carbachorismate intermediates. Carbaprephenate and its epimer were obtained by alkaline hydrolysis of the corresponding dimethyl esters. These compounds, in contrast to their ester precursors, undergo spontaneous acid-catalyzed decarboxylation in aqueous solution. Only at high pH does the Cope rearrangement compete with decarboxylation. At pH 12 and 90 degrees C, carbaprephenate slowly rearranges to carbachorismate, which rapidly loses water to give 3-(2-carboxyallyl)benzoic acid as the major product. A small amount of the intramolecular Diels-Alder adduct derived from carbachorismate is also observed by NMR as a minor product. Carbaprephenate is not a substrate for the enzyme chorismate mutase from Bacillus subtilis (BsCM), nor does carbaprephenate inhibit the normal chorismate mutase activity of this enzyme, even when present in 200-fold excess over chorismate. Its low affinity for the enzyme-active site is presumably a consequence of placing a methylene group rather than an oxygen atom proximal to the essential cationic residue Arg90. Nevertheless, BsCM variants that lack this cation (R90G and R90A) do not accelerate the Cope rearrangement of carbaprephenate either, and a catalytic antibody 1F7, which exhibits modest chorismate mutase activity, is similarly inactive. Poor substrate binding and the relatively high barrier for the Cope compared to the Claisen rearrangement presumably account for the lack of detectable catalysis. Acceleration of this sigmatropic rearrangement apparently requires more than an active site that is complementary in shape to the reactive substrate conformer.     

Correlations of13C chemical shifts and geometry modifications

The effect of geometry modifications of¹³C chemical shifts has been investigated in a small subset of molecules using both LO-INDO and Gaussian 70 (4–31) calculations. The Gaussian calculations, while known to give poor absolute shifts, compare well to the reparameterized semi-empirical INDO determinations in calculated shift changes. In virtually all cases the signs of the shift changes were found to be opposite to that of the changes in the calculated electronic energy.     

Army ants algorithm for rare event sampling of delocalized nonadiabatic transitions by trajectory surface hopping and the estimation of sampling errors by the bootstrap method

The most widely used algorithm for Monte Carlo sampling of electronic transitions in trajectory surface hopping (TSH) calculations is the so-called anteater algorithm, which is inefficient for sampling low-probability nonadiabatic events. We present a new sampling scheme (called the army ants algorithm) for carrying out TSH calculations that is applicable to systems with any strength of coupling. The army ants algorithm is a form of rare event sampling whose efficiency is controlled by an input parameter. By choosing a suitable value of the input parameter the army ants algorithm can be reduced to the anteater algorithm (which is efficient for strongly coupled cases), and by optimizing the parameter the army ants algorithm may be efficiently applied to systems with low-probability events. To demonstrate the efficiency of the army ants algorithm, we performed atom-diatom scattering calculations on a model system involving weakly coupled electronic states. Fully converged quantum mechanical calculations were performed, and the probabilities for nonadiabatic reaction and nonreactive deexcitation (quenching) were found to be on the order of 10(-8). For such low-probability events the anteater sampling scheme requires a large number of trajectories ( approximately 10(10)) to obtain good statistics and converged semiclassical results. In contrast by using the new army ants algorithm converged results were obtained by running 10(5) trajectories. Furthermore, the results were found to be in excellent agreement with the quantum mechanical results. Sampling errors were estimated using the bootstrap method, which is validated for use with the army ants algorithm.     

An examination of intermolecular and intramolecular hydrogen bonding in biomolecules by AM1 and MNDO/M semiempirical molecular orbital studies

The recent NMDO/M modification and parameterization of the MNDO molecular orbital method has been used to analyze intermolecular hydrogen bonding between amino acids and water, and intramolecular hydrogen bonding in monosaccharides. The results have been compared to AM1 calculations on the same systems. The MNDO/M calculations gave values which were similar to ab initio calculations with respect to the intermolecular interactions, but yielded significantly poorer results for the intramolecular interactions. The AM1 procedure performed better on the intramolecular interactions than the MNDO/M procedure, but frequently provided unfavorable three-centered hydrogen bonding geometries for the intermolecular interactions.