Pattern formation from consistent dynamical closures of uniaxial nematic liquid crystals

Pattern formation in uniaxial polymeric liquid crystals is studied for different dynamic closure approximations. Using the principles of mesoscopic non-equilibrium thermodynamics in a mean-field approach, we derive a Fokker-Planck equation for the single-particle non-homogeneous distribution function of particle orientations and the evolution equations for the second and fourth order orientational tensor parameters. Afterwards, two dynamic closure approximations are discussed, one of them considering the relaxation of the fourth order orientational parameter and leading to a novel expression for the free-energy like function in terms of the scalar order parameter. Considering the evolution equation of the density of the system and values of the interaction parameter for which isotropic and nematic phases coexist, our analysis predicts that patterns and traveling waves can be produced in lyotropic uniaxial nematics even in the absence of external driving.     

Determination of Cu, Mn, Ni and Sn in gasoline by electrothermal vaporization inductively coupled plasma mass spectrometry, and emulsion sample introduction

Trace metals in fuels, except in the case of additives, are usually undesirable and normally they occur in very low concentrations in gasoline, requiring sensitive techniques for their determination. Coupling of electrothermal vaporization with inductively coupled plasma mass spectrometry minimizes the problems related to the introduction of organic solvents into the plasma. Furthermore, sample preparation as oil-in-water emulsions reduces problems related to gasoline analysis. In this work, a method for determination of Cu, Mn, Ni and Sn in gasoline is proposed. Samples were prepared by forming a 10-fold diluted emulsion with a surfactant (Triton X-100), after treatment with concentrated HNO₃. The sample emulsion was pre-concentrated in the graphite tube by repeated pipetting and drying. External calibration was used with aqueous standards in a purified gasoline emulsion. Six samples from different gas stations were analyzed, and the analyte concentrations were found to be in the μg l⁻¹ range or below. The limits of detection were 0.22, 0.02, 0.38 and 0.03 μg l⁻¹ for Cu, Mn, Ni and Sn, respectively. The accuracy of the method was estimated using a recovery test.     

Multilayered artifacts in the pre-Columbian metallurgy from the North of Peru

Three types of alloys were recognized when analyzing pre-Columbian artifacts from the North of Peru: gold, silver, and copper alloys; gilded copper and silver; silvered copper; tumbaga, i.e., copper or silver enriched on gold at the surface by depletion gilding. In this paper, a method is described to differentiate gold alloys from gilded copper and from copper–gold tumbaga, and silver alloys from silvered copper and copper–silver tumbaga. This method is based on the use of energy-dispersive X-ray fluorescence, i.e., on a sophisticated analysis of XRF-spectra carrying out an accurate determination of Cu(K _α /K _β ), Ag(K _α /K _β ), Au(L _α /L _β ), and Au-L _α /Cu-K _α or Ag-K _α /Cu-K _α ratios. That implies a dedicated software for the quantitative determination of the area of X-ray peaks. This method was first checked by a relevant number of standard samples and then it was applied to pre-Columbian alloys from the North of Peru.     

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

General necessary conditions for infinite horizon fractional variational problems

We consider infinite horizon fractional variational problems, where the fractional derivative is defined in the sense of Caputo. Necessary optimality conditions for higher-order variational problems and optimal control problems are obtained. Transversality conditions are obtained in the case state functions are free at the initial time.     

The Root Operator on Finite Zero Based Invariant Subspaces of the Weighted Bergman Space

For a nonnegative integer α, we study and compute the root functions ${R_{\alpha}^{I}(z, w) = (1-\overline{w}z)^{2+\alpha}K_{\alpha}^{I}(z, w)}$ of finite zero based invariant subspaces I of the weighted Bergman space ${A_{\alpha}^{2}}$ , where ${K_{\alpha}^{I}}$ is the reproducing kernel of I. Furthermore, we estimate ranks of the corresponding root operators.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(\mathbb{R},w)}$ , where ${p\in(1,\infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R}\cup\{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.