Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

Conjugate interpolation problems in spaces of entire functions

A generalization of the Taylor expansions of entire functions with variable center is used to illustrate the notion of conjugate interpolation problem. The radii of existence, uniqueness, and convergence for such problems are found. A relationship between the interpolation functional systems of two mutually conjugate problems is revealed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 616–628, April, 2000.     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.     

A New Differential Lattice Boltzmann Equation and Its Application to Simulate Incompressible Flows on Non-Uniform Grids

A new differential lattice Boltzmann equation (LBE) is presented in this work, which is derived from the standard LBE by using Taylor series expansion only in spatial direction with truncation to the second-order derivatives. The obtained differential equation is not a wave-like equation. When a uniform grid is used, the new differential LBE can be exactly reduced to the standard LBE. The new differential LBE can be applied to solve irregular problems with the help of coordinate transformation. The present scheme inherits the merits of the standard LBE. The 2-D driven cavity flow is chosen as a test case to validate the present method. Favorable results are obtained and indicate that the present scheme has good prospects in practical applications.     

Premixed flame response to oscillatory pressure waves

We numerically investigate the interaction of sinusoidal pressure waves and slightly corrugated premixed flames. Work up to now has demonstrated the well-known Rayleigh–Taylor instability by imposing a single pressure ramp function onto a corrugated premixed flame. This paper considers sinusoidal oscillations of pressure. Such inputs are important since observations of large-scale experiments suggest that the presence of acoustic waves might be expected to have a significant influence on the propagation of the flame. The numerical experiments reported in this paper show that oscillatory pressure waves of the order of 800 Hz can have a magnifying effect on the wrinkling of the flame, due to the effect of Rayleigh–Taylor instabilities, hence increasing the overall mass burning rate, and that a sinusoidal pressure wave interacting with a premixed flame causes the flame to increase its wrinkling with every cycle in pressure. The result of great interest is that this growing process reaches a maximum after a few cycles, creating a dynamic equilibrium in which the final time-averaged mass burning flux is larger compared to that of the initial flame. It is also shown that the final mass burning flux increases with amplitude of the pressure wave, and increases with increasing frequency over the range explored.     

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations (gℓm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new turbulence closure equations for compressible fluids in the EP variational framework.     

A moving-boundary technique for the measurement of diffusion in liquids. triton X-100 in water

A modified taylor dispersion technique is used to measure liquid-phase mutual diffusion coefficients D. Rather than inject a narrow band of solution of solute concentration C+C into a carrier stream of composition C, the carrier stream is switched from a solution of composition C-(C/2) to a solution of composition C+(C/2), forming an initially-sharp moving boundary at the tube inlet. D is calculated from the refractive index profile across the broadened boundary at the tube outlet. Since the mean of concentration of the diffusing solute (C) is constant throughout the run, the calculated value of D accurately represents the differential value at C, even if relatively large concentration differences are used or if D is sensitive to composition. The advantages of the technique are illustrated by measuring the diffusion of aqueous Triton X-100, a nonionic surfactant. D is found to drop sharply as the concentration is raised through the critical micelle concentration near 0.15 g-L^–1.     

A class of perturbations which do not affect local controllability

Given a finite family of analytic vector fieldsF in a neighborhood of the origin of ⁿ , we are interested in the following property: the origin is an interior point of the reachable set from the origin itself at each positive instant. We describe a class of small perturbations, acting on the terms of the Taylor expansion of the vector fields ofF from some order on, which do not destroy the property above.     

Taylor dispersion measurements at low electrolyte concentrations. I. Tetraalkylammonium perchlorate aqueous solutions

An equipment for the determination of mutual diffusion coefficients using the Taylor's dispersion technique is described. The radius of the capillary was determined with the help of various calibration methods. Diffusion coefficients of aqueous tetraalkylammonium perchlorates, Me₄NClO₄, and Et₄NClO₄, were measured at 25°C in the concentration range 10^–3 to 5×10^–2 mol-dm^–3, and the slightly soluble Pr₄NClO₄ up to 1×10^–2 mol-dm^–3. The slope of linear plots ofD vs. is in agreement with theory, in contrast to the limiting valuesD ₀, which all deviate by about –5% from the Nernst-Hartley values.     

有机物在超临界二氧化碳中的无限稀释扩散系数的测定

利用经改造的超临界流体色谱仪测定不同温度和压力下苯、丙酮在超临界二氧化碳中的无限稀释扩散系数,开发出一个针对本科高年级学生的开放性实验,旨在加深学生对超临界流体及其性质的理解。在熟悉超临界流体色谱仪的工作原理、结构及其应用并掌握其操作方法的基础上,拓展学生视野和思维的深度及广度,提升学生的实验操作技能,培养高素质化学人才。