Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Detection of fentanyl in binary mixtures with cocaine by use of surface-enhanced Raman spectroscopy

Abstract

The ongoing epidemic pertaining to overdose deaths has been attributed to the synthetic opioid fentanyl due to its use as an adulterant in other, less potent drugs of abuse. Detection of low quantities of fentanyl would, therefore, be extremely useful in a forensic science laboratory. While Raman spectroscopy is particularly effective at distinguishing between classes of drugs, weak signatures can prove difficult when dealing with microscopic samples. Surface-enhanced Raman scattering spectroscopy provides the enhancement necessary to make Raman a viable approach for the detection of small amounts of fentanyl. This work explores the use of a paper-based substrate loaded with silver nanoparticles for the recovery of small quantities of fentanyl in cocaine, where it was identified at a lower limit of 500?ng (～65?ppm) in mixtures. Linear relationships were investigated between intensity and concentration for diagnostic peaks associated with fentanyl and cocaine, which in turn sheds light on the attenuation of the enhancement intensity as a result of competitive binding to silver nanoparticles. This work demonstrates a potentially simple and qualitative pathway for the forensic analysis of fentanyl as an adulterant in cocaine.     

Transmit-Only/Receive-Only Radiofrequency System for Hyperpolarized 13C MRS Cardiac Metabolism Studies in Pigs

Hyperpolarized ¹³C magnetic resonance spectroscopy in pig models enables metabolic activity mapping, providing a powerful tool for the study of the heart physiology, but requires the development of dedicated radiofrequency coils, capable of providing large field of view with high signal-to-noise ratio (SNR) data. This work describes the simulations and the tests of a transmit-only (TX) volume coil/receive-only (RX) surface coil both designed for hyperpolarized studies of pig heart with a clinical 3T scanner. The coil characterization is performed by developing an SNR model for coil performance in terms of coil resistance, sample-induced resistance and magnetic field pattern. In particular, coil resistances were calculated from Ohm’s law, while magnetic field patterns and sample-induced resistances were calculated using a numerical finite-difference time-domain algorithm. Experimental phantom chemical shift image, showed good agreement with the theoretical SNR-vs-depth profiles and highlighted the advantage of the novel configuration over the single transmit–receive coils throughout the volume of interest for cardiac imaging in pig. Finally, the TX-birdcage/RX-circular configuration was tested by acquiring metabolic maps with hyperpolarized [1-13C] pyruvate injected i.v. in a pig. The results of the phantom and pig experiments show the ability of the coil configuration to image well the metabolites distribution.     

The influence of hydroxyl groups on the ultraviolet spectra of substituted aromatic molecules adsorbed on silica surfaces

Infrared and uv spectroscopy have been used to study the interactions of a series of mono and di-substituted benzene molecules on both porous and nonporous high surface area silicas. It is confirmed that the strength of adsorption depends upon the presence and type of surface hydroxyl group but shown that the uv spectral shifts are not necessarily related to bond strength. Thus, when the surface OH groups reduce the effect of the electron donating side groups, stronger hydrogen bonds produce larger blue shifts in π-π* transitions. When n-π* transitions are involved, however, it is a dipole-dipole interaction which determines the magnitude of the red shift and not the strength of the hydrogen bond.     

“4,7-lactams”, intermediates for penems synthesis. I. skeletal conversion of penicillanic acid to (+)-2,2,5,5-tetramethyl-9-oxo-3-oxa-6-thia-azabicyclo[5,2,01,7]nonane

The synthesis of chiral “4,7-lactam“ $\text{5}$ has been accomplished starting from natural 6-aminopenicillanic acid.     

Analyse de complexité pour un théorème de Hall sur les fractions continues

We give a polynomial time controlled version of a theorem of M. Hall: every real number can be written as the sum of two irrational numbers whose developments into a continued fraction contain only 1, 2, 3 or 4. Mathematics Subject Classification: 03D15, 03F60, 11A55, 68Q25.     

Study of singlet—triplet coupling in glyoxal by level anticrossing spectroscopy. V. Nature of singlet—triplet interaction

We observe in glyoxal cooled in a supersonic free jet the fluorescence of individual rotational levels of the S₁ state excited by a cw laser. We use the technique of singlet—triplet magnetic resonance near an anticrossing to measure matrix elements V₃₁ as a function of rotational quantum numbers N_s, N_t, K_s, K_t. The experimental results are compared with theoretical models of singlet—triplet couplings and we show that the spin-vibronic interaction is the dominant singlet—triplet interaction in glyoxal.     

Electric field spectrum of pyridazine

An electric field induced spectrum has been obtained in the region of the 0⁰₀ band of pyridazine near 3752 Å. Since all intensity in this single-peaked spectrum is negative we interpret this to indicate field induced mixing with a very dense set of states.     

Error estimates on anisotropic elements for functions in weighted Sobolev spaces

In this paper we prove error estimates for a piecewise average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions.

Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses.

Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems.

Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements.

As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

     

Dreikörperkräfte und die Stabilität einfacher Kristallgitter

Observed stability of crystal structures for rare-gas atoms, alkali halides and for those II–VI and III–V compounds whose ions are isoelectronic with rare-gas atoms is correlated with atomic and ionic interactions in dense media. On the basis of central forces essential discrepancies between theory and experiment occur. Further analysis indicates that strong, simultaneous interactions of exchange type between three atoms or three ions play an important role in determining the stable crystal structure. An evaluation of such three-body interactions is carried out in first and second orders of perturbation theory, using effective-electron wave functions of Gaussian form. It is shown that the theory accounts for all observed stability relations on a quantitative basis. The stability of the face-centered cubic configuration for solids of heavy rare gases is explained. Also, the occurrence of the cesium chloride structure for some of the heavy alkali halides and the magnitude of observed transition pressures are reproduced by the theory. In addition, it is shown that three-ion interactions are responsable for the occurrence of the sphalerite (zincblende) and wurtzite configurations with II–VI and III–V compounds, without covalent bonding between the ions being involved. General laws are given which govern the relative stability of the sodium chloride, cesium chloride, sphalerite and wurtzite configurations for these types of compounds.