Classical Mechanics in Hilbert Space,Part 1

We consider the Hamilton formulation as well as the Hamiltonian flows on a symplectic (phase) space. These symplectic spaces are derivable from the Lie group of symmetries of the physical system considered. In Part 2 of this work, we then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces so obtained.     

Simple and efficient relaxation methods for interfaces separating compressible fluids,cavitating flows and shocks in multiphase mixtures

Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002–3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664–698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAM Journal of Multiscale and Modeling and Simulation (5) (2006) 84–115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425–467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248] is adapted to the present formulation.     

Raman spectroscopic study of the metatellurate mineral: xocomecatlite Cu3TeO4(OH)4

The mineral xocomecatlite is a hydroxy metatellurate mineral with Te⁶⁺ O₄ units. Tellurates may be subdivided according to their formula into three types of tellurate minerals: type (a) (AB)_m (TeO₄)_pZ_q, type (b) (AB)_m(TeO₆)^·xH₂O and (c) compound tellurates in which a second anion including the tellurite anion, is involved. The mineral xocomecatlite is an example of the first type. Raman bands for xocomecatlite at 710, 763 and 796 cm⁻¹, and 600 and 680 cm⁻¹ are attributed to the ν₁(TeO₄)²⁻ symmetric and ν₃ antisymmetric stretching mode. Raman bands observed at 2867 and 2926 cm⁻¹ are assigned to TeOH stretching vibrations and enable estimation of the hydrogen bond distances of 2.622 Å (2867 cm⁻¹), 2.634 Å (2926 cm⁻¹) involving these OH units. The hydrogen bond distances are very short implying that they are necessary for the stability of the mineral. Copyright © 2008 John Wiley & Sons, Ltd.     

Synthesis and Raman spectroscopic study of Mg/Al,Fe hydrotalcites with variable cationic ratios

Hydrotalcites of formula Mg₆(Al,Fe)₂(OH)₁₆(CO₃)·4H₂O formed by intercalation with the carbonate anion as a function of divalent/trivalent cationic ratio have been successfully synthesised. The XRD patterns show variation in the d‐spacing attributed to the size of the cation. Raman and infrared bands in the OH stretching region are assigned to (1) brucite layer OH stretching vibrations, (2) water stretching bands and (3) water strongly hydrogen bonded to the carbonate anion. Multiple (CO₃)²⁻ symmetric stretching bands suggest that different types of (CO₃)²⁻ exist in the hydrotalcite interlayer. Increasing the cation ratio (Mg/Al,Fe) resulted in an increase in the combined intensity of the two Raman bands at around 3600 cm⁻¹, attributed to Mg OH stretching modes, and a shift of the overall band profile to higher wavenumbers. These observations are believed to be a result of the increase in magnesium in the structure. Raman spectroscopy shows a reduction in the symmetry of the carbonate, leading to the conclusion that the anions are bonded to the brucite‐like hydroxyl surface and to the water in the interlayer. Water bending modes are identified in the infrared spectra at positions greater than 1630 cm⁻¹, indicating that water is strongly hydrogen bonded to both interlayer anions and the brucite‐like surface. Copyright © 2009 John Wiley & Sons, Ltd.     

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e^|k|/k_d{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e^{-C(k/k_d) ln(|k|/k_d)}{{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C _* = 1/ ln 2. The same behavior with a universal constant C _* is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.     

Littrow-type discretely tunable, Q-switched Nd:YAG laser around 1.3 μm

An all-solid-state, side diode array pulse pumped Nd:YAG laser tunable for six wavelengths ranging from 1318.8 nm to 1356.0 nm is developed. The tunability is obtained by using a grating in Littrow mode that also serves as an output coupler. The configuration ensures a line width as low as 0.04 nm. Thermal effects limit the maximum average power to 250 mW for an average absorbed pump power of 8.0 W in the free-running condition. An acousto-optic Q-switching of the laser provides pulses of width 251 ns with peak power of 733 W for an average pump power of 11.5 W. The laser may find application in microsurgery and dermatology. PACS 42.55.Xi; 42.60.-v; 42.60.Fc; 42.60.Gd; 42.62.Be     

Measurements of the Decay KL-->e+ e- mu+ mu-

The KTeV experiment at Fermilab has isolated a total of 132 events from the rare decay K(L)-->e+ e- mu+ mu-, with an estimated background of 0.8 events. The branching ratio of this mode is determined to be [2.69+/-0.24(stat)+/-0.12(syst)]x10(-9), with a radiative cutoff of M(2)(ee mu mu)/M(2)(K)>0.95. The first measurement using this mode of the parameter alpha from the D'Ambrosio-Isidori-Portolès (DIP) model of the K(L)gamma*gamma* vertex yields a result of -1.59+/-0.37, consistent with values obtained from other decay modes. Because of the limited statistics, no sensitivity is found to the DIP parameter beta. We use this decay mode to set limits on CP and lepton violation.     

Strong Bragg gratings photoinduced by 633-nm illumination in evaporated As2Se3 thin films

Bragg gratings are used in several photonic devices to reflect, and thus to isolate, specific wavelengths of light. Gratings can be photoinduced in chalcogenide glasses by illumination of bandgap light in an interference pattern. We used holographic interferometry to create Bragg gratings in amorphous As2Se3 thin films with a period of 0.56 microm by illumination with 633-nm light. The quality of the gratings was tested in real time, and refractive-index modulations as high as 0.037 were measured. These gratings were found to be stable over a period of several months if they were kept in the dark.     

Louder sounds can produce less forward masking: effects of component phase in complex tones

The influence of the degree of envelope modulation and periodicity on the loudness and effectiveness of sounds as forward maskers was investigated. In the first experiment, listeners matched the loudness of complex tones and noise. The tones had a fundamental frequency (F0) of 62.5 or 250 Hz and were filtered into a frequency range from the 10th harmonic to 5000 Hz. The Gaussian noise was filtered in the same way. The components of the complex tones were added either in cosine phase (CPH), giving a large crest factor, or in random phase (RPH), giving a smaller crest factor. For each F0, subjects matched the loudness between all possible stimulus pairs. Six different levels of the fixed stimulus were used, ranging from about 30 dB SPL to about 80 dB SPL in 10-dB steps. Results showed that, at a given overall level, the CPH and the RPH tones were louder than the noise, and that the CPH tone was louder than the RPH tone. The difference in loudness was larger at medium than at low levels and was only slightly reduced by the addition of a noise intended to mask combination tones. The differences in loudness were slightly smaller for the higher than for the lower F0. In the second experiment, the stimuli with the lower F0s were used as forward maskers of a 20-ms sinusoid, presented at various frequencies within the spectral range of the maskers. Results showed that the CPH tone was the least effective forward masker, even though it was the loudest. The differences in effectiveness as forward maskers depended on masker level and signal frequency; in order to produce equal masking, the level of the CPH tone had to be up to 35 dB above that of the RPH tone and the noise. The implications of these results for models of loudness are discussed and a model is presented based on neural activity patterns in the auditory nerve; this predicts the general pattern of loudness matches. It is suggested that the effects observed in the experiments may have been influenced by two factors: cochlear compression and suppression.