The <Emphasis Type=Italic>n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part II: Singularities

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singularities for as few as three bodies, whose orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces nonsingularity collisions, characterized by finite velocities and forces at the collision instant.     

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $PSL(2,\mathbb{Z})?\mathbb{H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma ?\mathbb{H}$ and odd and grotesque continued fractions, where $\Gamma ?{\mathbb{Z}}_{3}1{\mathbb{Z}}_3$ is the index two subgroup of $PSL(2,\mathbb{Z})$ generated by the order three elements $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill ?1\hfill & \hfill 1\hfill \end{array}\right)$, and having an ideal quadrilateral as fundamental domain.A similar connection between geodesics on $\Theta ?\mathbb{H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $PSL(2,\mathbb{Z})$ generated by $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.     

Fixed point theorems for Reich type contractions on metric spaces with a graph

Let (X,d)$(X,d)$ be a metric space endowed with a graph G$G$ such that the set V(G)$V\left(G\right)$ of vertices of G$G$ coincides with X$X$. We define the notion of G$G$-Reich type maps and obtain a fixed point theorem for such mappings. This extends and subsumes many recent results which were obtained for other contractive type mappings on ordered metric spaces and for cyclic operators.     

Positron annihilation spectroscopy studies of implanted polymer membranes

Two different polymer membranes, polyethylene and Nafion, have been sequentially implanted under vacuum at room temperature with 90 keV copper ions up to 10¹⁷ ions/cm²; after each implantation step, the targets were subjected to analysis with Positron annihilation lifetime spectroscopy and coincidence Doppler broadening spectroscopy. Both membranes revealed a similar behavior regarding the high fluence superficial implantation, but with different degree of magnitude. In order to endorse the Positrons annihilation spectroscopy results, scanning electron microscopy coupled with energy‐dispersive energy has been performed on the same targets, despite its lower sensitivity.     

The Vlasov–Poisson System for Stellar Dynamics in Spaces of Constant Curvature

We obtain a natural extension of the Vlasov–Poisson system for stellar dynamics to spaces of constant Gaussian curvature \({\kappa \ne 0}\): the unit sphere \({\mathbb S^2}\), for \({\kappa > 0}\), and the unit hyperbolic sphere \({\mathbb H^2}\), for \({\kappa < 0}\). These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov–Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.     

Some approximation properties of a Durrmeyer variant of q‐Bernstein–Schurer operators

In this paper, we will propose a Durrmeyer variant of q‐Bernstein–Schurer operators. A Bohman–Korovkin‐type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.     

Enzyme-catalyzed synthesis of (R)- and (S)-3-heteroaryl-3-hydroxy-propanoic acids and their derivatives

Starting from the racemic 3-benzofuranyl- and 3-benzo[b]thiophenyl-3-hydroxypropanoic acid ethyl esters as substrates, various multistep enzymatic procedures were developed for the efficient synthesis of the corresponding highly enantiomerically enriched (R)- and (S)-3-heteroaryl-3-hydroxypropanoic acids.     

How much can an intermediate state influence competing reactive pathways?

A molecule undergoing reaction may form a short-lived intermediate. Under certain conditions, the rate at which the reaction proceeds toward the product state via the intermediate may exceed that of a simple, direct path. The competition of two alternative reactive pathways is analyzed here in terms of a stochastic model. The approach allows one to diagnose this competition as a function of the energy of the intermediate relative to the barrier heights of the potential surface and values of the reactive vibrational modes. The result has applications to a variety of problems in chemical physics, ranging from the "lock-and-key" mechanism for the enzymatic activity to control of temporal evolution of complex systems by optimal laser fields.     

Electron spin resonance spectra of polymers during fluorination

ESR spectra characteristic of peroxy radicals appeared rapidly in all of eleven hydrogen-containing polymers examined when treated with dilute fluorine. These radicals presumably result from the reaction of hydrocarbon and fluorocarbon radicals, existing at undetectably low steady-state concentrations, with the oxygen impurity content of commercial fluorine. In poly(vinylidene fluoride) films of thickness 11 and 58 μm the radical contents were nearly proportional to surface area rather than volume, in agreement with earlier reports of a shallow depth of penetration. Some polymers exhibited also or exclusively a broad spectral component, varying in character with the polymer; examples are polystyrene, polyethylene, poly (vinyl chloride), poly(vinylidene chloride), polyoctafluoropentadiene, polyhexafluoropropene, and a fluorinated graphite. The broad spectral component did not react with ordinary radical scavengers such as propylene and oxygen, and is probably not due to a fluorocarbon radical but to unknown transition metal fluorides.     

SERS assessment of the cancer-specific methylation pattern of genomic DNA: towards the detection of acute myeloid leukemia in patients undergoing hematopoietic stem cell transplantation

In this label-free surface-enhanced Raman scattering (SERS) study of genomic DNA, we demonstrate that the cancer-specific DNA methylation pattern translates into specific spectral differences. Thus, DNA extracted from an acute myeloid leukemia (AML) cell line presented a decreased intensity of the 1005 cm⁻¹ band of 5-methylcytosine compared to normal DNA, in line with the well-described hypomethylation of cancer DNA. The unique methylation pattern of cancer DNA also influences the DNA adsorption geometry, resulting in higher adenine SERS intensities for cancer DNA. The possibility of detecting cancer DNA based on its SERS spectrum was validated on peripheral blood genomic DNA samples from n = 17 AML patients and n = 17 control samples, yielding an overall classification of 82% based on the 1005 cm⁻¹ band of 5-methylcytosine. By demonstrating the potential of SERS in assessing the methylation status in the case of real-life DNA samples, the study paves the way for novel methods of diagnosing cancer.

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