Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718     

BaSnF4 fast ion conductor: Variations versus the method of preparation and anomalous temperature variation of the quadrupole splitting

A new method of preparation of high performance fluoride ion conductor, BaSnF₄, by water leaching of newly discovered barium tin(II) chloride fluorides, has been designed, and the materials have been studied and compared to the solid prepared by the usual dry method. The unit-cell parameters and crystallite dimensions were found to vary with the method of preparation. In addition, the crystallite dimensions were found to be highly anisotropic for the samples obtained by the wet method. The Mössbauer spectrum is made of a large tin(II) quadrupole doublet, and a broad tin(IV) oxide peak due to surface oxidation. The tin(II) spectrum is in agreement with covalently bonded tin(II) having a strongly stereoactive lone pair. An unusually high dependence of the quadrupole splitting at low temperatures was observed (5.8 times larger than for α-SnF₂).     

Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y^′=f(t,y)$y^{\prime }=f(t,y)$ with initial conditions y(_t0)=_y0$y\left(t_0\right)=y_0$. Its formula uses y^′$y^{\prime }$, y^″$y^{″}$ and y^?$y^{?}$ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.     

Global solution curves for a class of elliptic systems

We study curves of positive solutions for a system of elliptic equations of Hamiltonian type on a unit ball. We give conditions for all positive solutions to lie on global solution curves, allowing us to use the analysis similar to the case of one equation, as developed in P. Korman, Y. Li and T. Ouyang [An exact multiplicity result for a class of semilinear equations, Commun. PDE 22 (1997), pp. 661–684.] (see also T. Ouyang and J. Shi [Exact multiplicity of positive solutions for a class of semilinear problems, II, J. Diff. Eqns. 158(1) (1999), pp. 94–151].). As an application, we obtain some non-degeneracy and uniqueness results. For the one-dimensional case we also prove the positivity for the linearized problem, resulting in more detailed results.     

Simulation from Wishart Distributions with Eigenvalue Constraints

Abstract

This article provides an efficient algorithm for generating a random matrix according to a Wishart distribution, but with eigenvalues constrained to be less than a given vector of positive values. The procedure of Odell and Feiveson provides a guide, but the modifications here ensure that the diagonal elements of a candidate matrix are less than the corresponding elements of the constraint vector, thus greatly improving the chances that the matrix will be acceptable. The Normal hierarchical model with vector outcomes and the multivariate random effects model provide motivating applications.     

An efficient and robust numerical algorithm for estimating parameters in Turing systems

We present a new algorithm for estimating parameters in reaction–diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer–Meinhardt reaction–diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.     

Proton transfers in aromatic and antiaromatic systems. How aromatic or antiaromatic is the transition state? An ab initio study

An ab initio study of six carbon-to-carbon identity proton transfers is reported. They refer to the benzenium ion/benzene (C6H7(+)/C6H6), the 2,4-cyclopentadiene/cyclopentadienyl anion (C5H6/C5H5(-)), and the cyclobutenyl cation/cyclobutadiene (C4H5(+)/C4H4) systems and their respective noncyclic reference systems, that is, [structure: see text], [structure: see text] and [structure: see text]. For the aromatic C6H7(+)/C6H6 and C5H6/C5H5(-) systems, geometric parameters and aromaticity indices indicate that the transition states are highly aromatic. The proton-transfer barriers in these systems are quite low, which is consistent with a disproportionately high degree of transition-state aromaticity. For the antiaromatic C4H5(+)/C4H4 system, the geometric parameters and aromaticity indices indicate a rather small degree of antiaromaticity of the transition state. However, the proton-transfer barrier is higher than expected for a transition state with a low antiaromaticity. This implies that another factor contributes to the barrier; it is suggested that this factor is angle and torsional strain in the transition state. The question whether charge delocalization at the transition state might correlate with the development of aromaticity was also examined. No such correlation was found, that is, charge delocalization lags behind proton transfer as is commonly observed in nonaromatic systems involving pi-acceptor groups.     

Frontispiece: Cationic Tetrylene-Iron(0) Complexes: Access Points for Cooperative,Reversible Bond Activation and Open-Shell Iron(−I) Ferrato-Tetrylenes

The open-shell cationic stannylene-iron(0) complex 4 ( 4 =[^PhiPDippSn⋅Fe⋅IPr]⁺; ^PhiPDipp={[Ph₂PCH₂Si(ⁱPr)₂](Dipp)N}; Dipp=2,6-ⁱPr₂C₆H₃; IPr=[(Dipp)NC(H)]₂C:) cooperatively and reversibly cleaves dihydrogen at the Sn−Fe interface under mild conditions (1.5 bar, 298 K), in forming bridging hydrido-complex 6 . The One-electron oreduction of the related Ge^II−Fe⁰ complex 3 leads to oxidative addition of one C−P linkage of the ^PhiPDipp ligand in an intermediary Fe^−I complex, leading to Fe^I phosphide species 7 . One-electron reduction reaction of 4 gives access to the iron(−I) ferrato-stannylene, 8 , giving evidence for the transient formation of such a species in the reduction of 3 . The covalently bound tin(II)-iron(−I) compound 8 has been characterised through EPR spectroscopy, SQUID magnetometry, and supporting computational analysis, which strongly indicate a high localization of electron spin density at Fe^−I in this unique d⁹-iron complex.     

Evaluation of binary QSAR models derived from LUDI and MOE scoring functions for structure based virtual screening

In today's world of high-throughput in silico screening, the development of virtual screening methodologies to prioritize small molecules as new chemical entities (NCEs) for synthesis is of current interest. Among several approaches to virtual screening, structure-based virtual screening has been considered the most effective. However the problems associated with the ranking of potential solutions in terms of scoring functions remains one of the major bottlenecks in structure-based virtual screening technology. It has been suggested that scoring functions may be used as filters for distinguishing binders from nonbinders instead of accurately predicting their binding free energies. Subsequently, several improvements have been made in this area, which include the use of multiple rather than single scoring functions and application of either consensus or multivariate statistical methods or both to improve the discrimination between binders and nonbinders. In view of it, the discriminative ability (distinguishing binders from nonbinders) of binary QSAR models derived using LUDI and MOE scoring functions has been compared with the models derived by Jacobbsson et al. on five data sets viz. estrogen receptor alphamimics (ERalpha_mimics), estrogen receptor alphatoxins (ERalpha_toxins), matrix metalloprotease 3 inhibitors (MMP-3), factor Xa inhibitors (fXa), and acetylcholine esterase inhibitors (AChE). The overall analyses reveal that binary QSAR is comparable to the PLS discriminant analysis, rule-based, and Bayesian classification methods used by Jacobsson et al. Further the scoring functions implemented in LUDI and MOE can score a wide range of protein-ligand interactions and are comparable to the scoring functions implemented in ICM and Cscore. Thus the binary QSAR models derived using LUDI and MOE scoring functions may be useful as a preliminary screening layer in a multilayered virtual screening paradigm.     

Synthesis and structure of iron(iii) diamine-bis(phenolate) complexes

The structures and properties of six new iron(iii) diamine-bis(phenolate) complexes are reported. Reaction of anhydrous FeX(3) salts (where X = Cl or Br) with the diprotonated tripodal tetradentate ligands 2-pyridylamino-N,N-bis(2-methylene-4-methyl-6-tert-butylphenol), H(2)[L(1)], and N,N-dimethyl-N',N'-bis(2-methylene-4-methyl-6-tert-butylphenol)ethylenediamine, H(2)[L(2)], produces the trigonal bipyramidal iron(iii) complexes, [L(1)]FeCl , [L(1)]FeBr , [L(2)]FeCl and [L(2)]FeBr . Reaction of FeX(3) with the related linear tetradentate ligand N,N'-bis(4,6-tert-butyl-2-methylphenol)-N,N'-bismethyl-1,2-diaminoethane, H(2)[L(3)], generates square pyramidal iron(iii) complexes, [L(3)]FeCl and [L(3)]FeBr . Complexes have been characterized using electronic absorption spectroscopy and magnetometry. Single crystal X-ray molecular structures have been determined for complexes 1, 3, 5 and 6.