Fitting complex potential energy surfaces to simple model potentials: application of the simplex-annealing method

A stochastic method of optimization, which combines simulated annealing with simplex, is implemented to fit the parameters of a simple model potential. The main characteristic of the method is that it explores the whole space of the parameters of the model potential, and therefore it is very efficient in locating the global minimum of the cost function, in addition to being independent of the initial guess of the parameters. The method is employed to fit the complex intermolecular potential energy surface of the dimer of water, using as a reference the spectroscopic quality anisotropic site-site potential of Feller et al. The simple model potential chosen for its reparameterization is the MCY model potential of Clementi et al. The quality of the fit is assessed by comparing the geometry of the minimum, the harmonic frequencies, and the second virial coefficients of the parameterized potential with the reference one. Finally, to prove more rigorously the robustness of this method, it is compared with standard nonstochastic methods of optimization.     

Kinetic Spectrophotometric Determination of Ritodrine Hydrochloride in Dosage Forms

A simple and sensitive kinetic method was developed for the determination of ritodrine hydrochloride in pharmaceutical preparations. The method is based upon a kinetic investigation of the oxidation reaction of the drug with alkaline potassium permenganate at room temperature for a fixed 12.5 min; the absorbance of the colored permenganate ion was measured at 609 nm. The absorbance concentration plots were rectilinear over the range of 1.6–11.2 μg mL^?1 (r = 0.9992) with a minimum detectability of 0.096 μgmL^?1 (2.88 × 10^?7 M). The different experimental parameters affecting the development and stability of color were carefully studied and optimized. The determination of ritodrine hydrochloride by the fixed concentration and rate constant methods is also feasible with the calibration equations obtained, but the fixed time method has been found to be more applicable. The proposed method was applied successfully to the determination of ritodrine hydrochloride in tablets and ampoules with average recoveries of 100.37 ± 0.93% and 100.42 ± 0.87%, respectively. The results obtained were in good agreement with those obtained using a reference method for comparison. A proposal of the reaction pathway is also presented.     

Polarographic behaviour and determination of spironolactone

The polarographic behaviour of spironolactone in Britton-Robinson buffers containing 40% methanol as a solubilizer was studied. Over the pH range 3-12 a cathodic wave was produced. The wave was characterized as irreversible, diffusion-controlled and partially affected by adsorption phenomena. The number of electrons involved in the reduction was found by coulometric measurements at a controlled potential. A method was developed for the determination of spironolactone in tablets and the results obtained were in agreement with those obtained by the B.P. 1988 method. The mechanism of the electrode reaction is discussed.AvH scholar: 1989–1991     

Randomized algorithms for the separation of point sets and for solving quadratic programs

A randomized algorithm for finding a hyperplane separating two finite point sets in the Euclidean space ^d and a randomized algorithm for solving linearly constrained general convex quadratic problems are proposed. The expected running time of the separating algorithm isO(dd! (m + n)), wherem andn are cardinalities of sets to be separated. The expected running time of the algorithm for solving quadratic problems isO(dd! s) wheres is the number of inequality constraints. These algorithms are based on the ideas of Seidel's linear programming algorithm [6]. They are closely related to algorithms of [8], [2], and [9] and belong to an abstract class of algorithms investigated in [1]. The algorithm for solving quadratic problems has some features of the one proposed in [7].This research was done when the author was supported by the Alexander von Humboldt Foundation, Germany.On leave from the Institute of Mathematics and Mechanics, Ural Department of the Russian Academy of Sciences, 620219 Ekaterinburg, S. Kovalevskaya str. 16, Russia.     

An invariant trace formula for the universal covering group of SL(2,ℝ)

Automorphic forms of arbitrary real weight can be considered as functions on the universal covering group of SL(2, ). In this situation, we prove an invariant form of the Selberg trace formula for Hecke operators. For this purpose, the Fourier transforms of weightet orbital integrals, obtained by J. Arthur, R. Herb and P. Sally, jr., are explicitly calculated. Our formula does not follow from Arthur's invariant trace formula, since the group has infinite centre, and vector-valued automorphic forms with respect to non-congruence lattices are considered.     

Density computations for real quadratic units

In order to study the density of the set of positive integers for which the negative Pell equation is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in as the factorization of is a natural part of the input for the values of we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm .

     

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

Convergence of distorted Brownian motions and singular Hamiltonians

We prove a convergence theorem for sequences of Diffusion Processes corresponding to Dirichlet Forms of the kind .We obtain convergence in total variation norm of the corresponding probability measures on the path space C(₊;^d) under hypotheses which, for example, are satisfied in the case of H _loc ¹ ( ^d )-convergence of the 's, but we can allow more singular situations as regards the approximating sequences. We use then these results to give a criterion of convergence for generalized Schrödinger operators in which the potential function should not necessarily exists as a measurable function. We obtain convergence not only in strong resolvent sense, but we also obtain convergence in the uniform operator topology up to sets of arbitrarily small Lebesgue measure. Applications to the problem of the approximation of ordinary Schrödinger operators by generalized ones corresponding to zero-range interactions are given.     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

On quadratic forms of height two and a theorem of Wadsworth

Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .