On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

A neural networks study of quinone compounds with trypanocidal activity

This work investigates neural network models for predicting the trypanocidal activity of 28 quinone compounds. Artificial neural networks (ANN), such as multilayer perceptrons (MLP) and Kohonen models, were employed with the aim of modeling the nonlinear relationship between quantum and molecular descriptors and trypanocidal activity. The calculated descriptors and the principal components were used as input to train neural network models to verify the behavior of the nets. The best model for both network models (MLP and Kohonen) was obtained with four descriptors as input. The descriptors were T(5) (torsion angle), QTS1 (sum of absolute values of the atomic charges), VOLS2 (volume of the substituent at region B) and HOMO-1 (energy of the molecular orbital below HOMO). These descriptors provide information on the kind of interaction that occurs between the compounds and the biological receptor. Both neural network models used here can predict the trypanocidal activity of the quinone compounds with good agreement, with low errors in the testing set and a high correctness rate. Thanks to the nonlinear model obtained from the neural network models, we can conclude that electronic and structural properties are important factors in the interaction between quinone compounds that exhibit trypanocidal activity and their biological receptors. The final ANN models should be useful in the design of novel trypanocidal quinones having improved potency.

Towards to the trans-bromination of 2-styrylpyridine with a palladacycle intermediary and structure analysis for trans-1,2-dibromo-2-styrylpyridine

The simple halogenation of alkynes in conventional organic reactions gives a blend of cis and trans isomers. It is proposed then, a synthesis of stereospecific halogenation of alkynes in trans position, using palladacycle as intermediaries. The recrystallization of the compound obtained by bromination of 2-Styrylpyridine, with cyclepalladium intermediary results in a single crystal, which is subjected to X-ray diffraction. The crystal packing is established through weak interactions of three types. The first one is of the type pi x pi interactions, from symmetry operation, between the centroids. The second one is of the type C-X center dot center dot center dot pi interactions. And the last type is an anomalous intermolecular interaction between halogens, C-X center dot center dot center dot X-C, with bond distances smaller than the sum of the van der Waals radii. The conformation on the C=C bond is trans and the dihedral angle between the aromatic rings is (with esd approximate) 18.1(3)degrees. (C) 2010 Elsevier B.V. All rights reserved.