Molecular integrals over Gaussian-type geminal basis functions

We present formulas for the evaluation of molecular integrals over basis functions with an explicit Gaussian dependence on interelectronic coordinates. These formulas use expansions in Hermite Gaussian functions and represent an extension to the work of McMurchie and Davidson to two-electron basis functions. Integrals that depend on the coordinates of up to four electrons are discussed explicitly. A key feature of this approach is that it allows full exploitation of the shell structure of the orbital part of the basis. Received: 24 February 1997 / Accepted: 4 March 1997     

Solubility of phenobarbital in aqueous cosolvent mixtures revisited: IKBI preferential solvation analysis

The preferential solvation parameters of phenobarbital in aqueous binary mixtures of 1,4-dioxane, t-butanol, n-propanol, ethanol, propylene glycol and glycerol were derived from solution thermodynamic properties by using the IKBI method. This drug is sensitive to preferential solvation effects in all these mixtures. The preferential solvation parameter by the cosolvent (δx_1,3) is negative in almost all the water-rich mixtures but positive in mixtures with similar proportions of solvents and cosolvent-rich mixtures, except in 1-propanol + water mixtures, where negative values are also found in mixtures with x₁ ≥ 0.70. Hydrophobic hydration around the non-polar ethyl and phenyl groups of this drug in water-rich mixtures could play a relevant role in drug solvation. Otherwise, in mixtures of similar solvent compositions and in cosolvent-rich mixtures the preferential solvation by cosolvent could be due to the acidic behaviour of the drug.     

Limits of elliptic hypergeometric integrals

In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level. The author was supported in part by NSF Grant No. DMS-0401387.     

Integral representation formula for generalized normal derivations

For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation formulas hold:

 

and

 

whenever is a Hilbert-Schmidt class operator and is a Lipschitz class function on Applying this formula, we extend the Fuglede-Putnam theorem concerning commutativity modulo Hilbert-Schmidt class, as well as trace inequalities for covariance matrices of Muir and Wong. Some new monotone matrix functions and norm inequalities are also derived.

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

A Radial Derivative with Boundary Values of the Spherical Poisson Integral

A formula of a radial derivative is obtained with the aid of derivatives with respect to and to of the functions closely connected with the spherical Poisson integral and the boundary values are determined for . The boundary values are also found for partial derivatives with respect to the Cartesian coordinates .     

Algebraic independence of the periods of abelian integrals

Abelian integrals of the first and the second kind are proved to have two algebraically independent periods. Some corollaries concerning the algebraic independence of the values of Euler's beta and gamma functions at rational points are derived.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 681–691, November, 1996.     

Functional integrals in the statistical theory of turbulence and the burgers model equation

The functional integrals appearing in master equations for turbulent flow of an incompressible fluid and for the Burgers model equation are treated. A possible way is described to define the integration properly and related problems are discussed. For the simple example of the Burgers model equation of turbulence some results are presented.