Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Protein and solvent dynamics: how strongly are they coupled?

Analysis of Raman and neutron scattering spectra of lysozyme demonstrates that the protein dynamics follow the dynamics of the solvents glycerol and trehalose over the entire temperature range measured 100-350 K. The protein's fast conformational fluctuations and low-frequency vibrations and their temperature variations are very sensitive to behavior of the solvents. Our results give insight into previous counterintuitive observations that protein relaxation is stronger in solid trehalose than in liquid glycerol. They also provide insight into the effectiveness of glycerol as a biological cryopreservant.     

Design and construction of multicrystal analyser detectors using Rowland circles: application to MAD26 at ALBA

A concept is given for describing multicrystal analyser detectors (MADs), as they are in use for synchrotron powder diffraction, on the basis of the Rowland circle construction. The Rowland circle is typically used to describe focusing geometries and can be adapted for the case of MADs working at a single energy as well as in a limited energy range. With this construction it is also possible to quantify and optimize the walk of the beam along non‐central crystals which is inevitable in certain detector designs. The results of this geometrical inspection are correlated with a real detector design that is implemented at the ALBA synchrotron facility in Spain. An error budget is given to estimate the influence and amount of tolerance of the manufacturing process.     

Some remarks on quasilinear parabolic problems with singular potential and a reaction term

In this paper we deal with the following quasilinear parabolic problem $$\left\{\begin{array}{l@{\quad}l} (u^\theta)_t - \Delta_p {u} = \lambda \frac{u^{p - 1}}{|x|^{p}} + u^q + f,\,\, u \geq 0 \quad {\rm in} \;\;\Omega \times (0, T),\\ u(x, t) = 0 \quad\qquad\qquad\qquad\qquad\qquad\qquad {\rm on}\; \partial \Omega \times(0, T),\\ u(x, 0) = u_0(x), \,\,\, \qquad\qquad\qquad\qquad\qquad x \in\; \Omega,\end{array}\right.$$ where θ is either 1 or (p ? 1), \({N \geq 3, \,\Omega \subset \mathcal{IR}^N}\) is either a bounded regular domain containing the origin or \({\Omega \equiv \mathcal{IR}^N}\) , 1 < p < N, q > 0 and u ₀ ≥  0, f ≥  0 with suitable hypotheses. The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, we analyze blow-up phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle. We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power.     

Nonlinear elliptic problems with a singular weight on the boundary

We study existence of solutions to $$-\Delta u = \frac{u^p}{|x|^2}\quad u\, >\,0 \,{\rm in }\,\Omega$$ with u?=?0 on ???, where ?? is a smooth bounded domain in ${\mathbb {R}^N}$ , N??? 3 with ${0\,\in\,\partial \Omega}$ and ${1< p < \frac{N+2}{N-2}}$ . The existence of solutions depends on the geometry of the domain. On one hand, if the domain is starshaped with respect to the origin there are no energy solutions. On the other hand, in dumbbell domains via a perturbation argument, the equation has solutions.     

Fluorocompounds as antisolvents in hydrocarbons separation

The behaviour of perfluorotributylamine (FC43) with respect to hydrocarbons is investigated, and a comparison is made with perfluorocyclic oxide. The results obtained suggest the use of these fluorocompounds, especially of FC75, as antisolvents, in conjunction with a polar solvent, in the liquid-liquid extraction of hydrocarbons, particularly for the separation of aroma$\text{t}$ics from cycloparaffins.     

The trace of totally positive algebraic integers

For all totally positive algebraic numbers except a finite number of explicit exceptions, the following inequality holds:

\max(1.780022,1.66+\alpha_1), \end{displaymath}">

where is the degree of and its conjugates. This improves previous results of Smyth, Flammang and Rhin.

     

Microstructure evolution and grain size distribution in nanocrystalline FeNbBCu from synchrotron XRD and TEM analysis

The microstructure evolution during nanocrystallization of an Fe₇₇Nb₇B₁₅Cu₁ amorphous alloy is investigated using in situ synchrotron X-ray diffraction (XRD) and transmission electron microscopy (TEM). The microstructure of the nanocrystallized alloy consists in dispersion of bcc-Fe nanocrystals of 4–6 nm of diameter embedded in a stabilized amorphous remaining matrix. The grain size distribution of the nanocrystalline Fe₇₇Nb₇B₁₅Cu₁ alloy was obtained using three different methodologies: statistical analysis of TEM images, the Warren–Averbach and Langford methods to analyse the XRD patterns and modelling of the diffraction pattern from the Debye equation. A lognormal distribution function has been assumed in all three methods in order to obtain comparable results. A good agreement is found in the calculated average radius and dispersion although some deviations are found with the Langford approach. The microstructure evolution during crystallization was obtained from the XRD patterns during heating (5.0 · 10^?3 K s^?1) at temperatures between 700 and 900 K. A decrease and prompt saturation of the growth rate is obtained, indicative of the diffusion barrier caused to the overlap between the concentration gradients at the interface of growing grains (soft impingement). A simple model assuming nucleation and initial fast growth of the crystalline grains followed by reduced growth capable of predicting microstructural evolution is presented. The modelling results agree with the experimental observations.