Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation

Product integration rules generalizing the Fej?r, Clenshaw-Curtis,and Filippi quadrature rules respectively are derived for integralswith trigonometric and hyperbolic weight factors. The Chebyshevmoments of the weight functions are found to be given by well-conditionedexpressions, in terms of hypergeometric functions ₀F₁. An a priori error estimator is discussed which is shown bothto avoid wasteful invocation of the integration rule and toincrease significantly the robustness of the automatic quadratureprocedure. Then, specializing to extended Clenshaw-Curtis (ECC) rules,three types of a posteriori error estimates are considered andthe existence of a great risk of their failure is demonstratedby large scale validation tests. An empirical error estimator,superseding them for slowly varying integrands, is found toresult in a spectacular increase in the output reliability. Finally, enhancements in the control of the interval subdivisionstrategy aiming at increasing code robustness is discussed.Comparison with the code DQAWO of QUADPACK, with about a hundredthousand solved integrals, is illustrative of the increasedrobustness and error estimate reliability of our computer codeimplementation of the ECC rules.     

An Improved Large Scale Synthesis of 2-Amino-4-chloropyridine and Its Use for the Convenient Preparation of Various Polychlorinated 2-Aminopyridines

An efficient large scale synthesis of 2-amino-4-chloropyridine (3) has been achieved through a modification of existing literature procedures. Compound 3 was used to prepare the previously unreported 2-amino-4,5-dichloropyridine (4). The known 2-amino-3,4-dichloropyridine (5) and 2-amino-3,4,5-trichloropyridine (6) were prepared from 3 by new routes and in higher yields than previously reported.     

Thermodynamic Complexation of Dopamine with Zinc(II) in Media with Different Dielectric Constants

The complexation of zinc(II) with dopamine has been investigated by spectrophotometric measurements in mixed solvent system at an ionic strength of 0.2 mol•dm－3 sodium chloride, employed at at (15±0.1), (25±0.1), (35±0.1) ℃ at inin a pH range of ca. 6 to ca. 7 with a high ratio of ligand to metal. The effect of solvent systems on protonation and complexation are was was discussed. Linear relationships are werewere observed by plotting lg K versus 1/D, where K and D show stability and dielectric constants, respectively.     

Nonlinear interference in a mean-field quantum model

Using similar nonlinear stationary mean-field models for both a 2D axisymmetricalBose-Einstein Condensate and an electron pair in a parabolic trap, we propose to describethe original many-particle ground state as a one-particle mixed state (in contrast to apure state), i.e. as a statistical ensemble of several one-particle quantum states. Thesequantum states are the eigenfunctions of the corresponding stationary nonlinearSchrödinger equation (hence called “nonlinear eigenstates”). Due to their nonlinearity,they are not orthogonal. Therefore, taking the simple example of a two-level system, weshow that each of these two nonlinear eigenstates |i? and|j? occurs with a probability (or statistical weight) that isdefined by their non-orthogonality ?i|j? 0. We givethe corresponding density matrix. We search for physical grounds in the interpretation ofour two main results, namely, a quantum-classical nonlinear transition and theinterference between two “nonlinear eigenstates”.     

The far-infrared absorption of a periodic 2DEG in the transition regime between weak and strong modulation

We study the optical absorption of arrays of quantum dots and antidots in a perpendicular homogeneous magnetic field. The electronic system is described quantum mechanically using a Hartree approximation for the mutual Coulomb interaction of the electrons. The evolution of the absorption is traced from the homogeneous to the strongly modulated case identifying the ensuing collective modes, the magnetoplasmons, and their correlations with inherent length scales of the system.     

Harmonic morphisms from the Grassmannians and their non-compact duals

In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases. Mathematics Subject Classifications (2000): 58E20, 53C43, 53C12The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.