Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics

We consider a fifth-order partial differential equation (PDE) that is a generalization of the integrable Camassa–Holm equation. This fifth-order PDE has exact solutions in terms of an arbitrary number of superposed pulsons with a geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N==2. Numerical simulations show that the pulsons are stable, dominate the initial value problem, and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. But after demonstrating the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure and obtaining negative results from Painlevé analysis and the Wahlquist–Estabrook method, we assert that this fifth-order PDE is not integrable.     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.     

Calculation of Systematic Errors in Determining Light‐Scattering Matrix Elements

We consider systematic errors in determining lightscattering matrix elements by a device in which the polarization properties of probing radiation change depending on different angular positionings of the principal directions of the first phase plate, and we perform polarization analysis of scattered radiation using the Fourier expansion of the signal from a photodetector that records the intensity of radiation passed through the rotating second phase plate and a stationary analyzer. Formulas estimating the errors arising in separate deviation of the axes of the anisotropic optical elements in the illuminating and lightreceiving channels of the measuring device were derived. Using the method of mathematical simulation, we consider the combined influence of various inaccuracies in positioning anisotropic elements.     

Amplification Coefficient of the Anti‐Stokes Component of Raman Scattering

The theory of intracavity parametric interaction of the components of Raman scattering is developed in the prescribedintensity approximation, which takes into account the reverse reaction of the excited waves to the pumpingwave phase. It is shown that one can substantially improve the conversion efficiency or the amplification coefficient of the antiStokes component by selection of the intracavity geometry, the optimum phase relation between the interacting waves, the pumping level, and the phase mismatch.     

Phonon Band Filling and Photon Emission by Phonons

Phonon generation by electrons is supplied in n-type Si crystals in electric fields E100 kV/cm at the lattice temperature of 80 K employing the ensemble Monte Carlo technique. Electron transfer between equivalent energy valleys is accounted for the g-type- and f-type phonon absorption and emission. Acoustic phonons are accounted for the quasi-elastic scattering of electrons within the energy valleys. Excess phonon number is determined using numerical data on phonon generation rate and experimental values of phonon lifetimes. The feasibility of stimulated emission of infrared-range photons due to direct optical transitions between the phonon bands is discussed.     

The Ground State Energy of a Dilute Two-Dimensional Bose Gas

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E₀/N=(2²/m)|ln(a²)|^–1, to leading order, with a relative error at most O(|ln(a²)|^–1/5). Here N is the number of particles, =N/V is the particle density and a is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, E₀ is not simply N(N–1)/2 times the energy of two particles in a large box of volume (area, really) V. It is much larger.     

NMR phase noise in bitter magnets

We have studied the temporal instability of a high field resistive Bitter magnet through nuclear magnetic resonance (NMR). This instability leads to transverse spin decoherence in repeated and accumulated NMR experiments as is normally performed during signal averaging. We demonstrate this effect via Hahn echo and Carr--Purcell--Meiboom--Gill (CPMG) transverse relaxation experiments in a 23-T resistive magnet. Quantitative analysis was found to be consistent with separate measurements of the magnetic field frequency fluctuation spectrum, as well as with independent NMR experiments performed in a magnetic field with a controlled instability. Finally, the CPMG sequence with short pulse delays is shown to be successful in recovering the intrinsic spin--spin relaxation even in the presence of magnetic field temporal instability.     

Effect of Addition of MTES on the Structure of Siloxane-PPG Nanocomposites

Hybrid transparent and flexible siloxane-polypropyleneglycol (PPG) materials with covalent bonds between the inorganic (siloxane) and organic (polymeric) phases were prepared by sol-gel process. In order to improve the quality of the mechanical properties of these materials, different amounts of methyltriethoxysilane (MTES) were added to the initial sol. The effect of MTES addition on the structure of the composites was studied by Small-Angle X-Ray Scattering (SAXS) and ²⁹Si Nuclear Magnetic Resonance (²⁹Si NMR). In absence of MTES, SAXS spectra exhibit a peak that is assigned to spatial correlation due to short range order between the siloxane clusters embedded in the polymeric phase. The experimental results indicate that, for low MTES concentrations ([MTES]/[O] 0.8, O: ether-type oxygen of PPG), the silicon species resulting from hydrolysis and condensation of MTES fill the open spaces between polymeric chains, interacting with the ether-type oxygens. For larger MTES content ([MTES]/[O] 0.8), the number of free ether-type oxygen sites avalaible for reaction with such silicon species is not large enough. Consequently, a fraction of silicon species resulting from MTES addition graft to siloxane clusters formed by hydrolysis and condensation of the hybrid precursor. For all MTES concentrations the condensation degree of the siloxane phase, determined from ²⁹Si NMR spectroscopy, is high (>69%), as expected under neutral pH synthesis conditions.     

On the Determination of Free Evolution in the Lax--Phillips Scattering Scheme for Second-Order Operator-Differential Equations

The paper presents some necessary and sufficient conditions on abstract positive self-adjoint operators L under which the operator-differential equation determines a free evolution in the Lax--Phillips scattering scheme.     

Scattering matrices for the quantum body problem

Let be a generalized body Schrödinger operator with very short range potentials. Using Melrose's scattering calculus, it is shown that the free channel `geometric' scattering matrix, defined via asymptotic expansions of generalized eigenfunctions of , coincides (up to normalization) with the free channel `analytic' scattering matrix defined via wave operators. Along the way, it is shown that the free channel generalized eigenfunctions of Herbst-Skibsted and Jensen-Kitada coincide with the plane waves constructed by Hassell and Vasy and if the potentials are very short range.