Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U(y) and with stationary streamlines y=y₀ (such that U^′(y₀)=0), a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Nonlinear Exact Solutions of the 2-Dimensional Rotational Euler Equations for the Incompressible Fluid

In this paper, the Clarkson–Kruskal direct approach is employed to investigate the exact solutions of the2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2)and sech(kt/2) due to the rotational parameter k = 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.     

Galvanomagnetic and thermoelectric properties of multicomponent n-type solid solutions based on Bi and Sb chalcogenides

The galvanomagnetic and thermoelectric properties of n-Bi_2-x Sb_xTe_3-y-z Se_yS_z multicomponent solid solutions with atomic substitutions (Sb → Bi; Se, S → Te) are studied. The principal components of the effective mass tensor (m ₁, m ₂, m ₃) for the isotropic mechanism of charge carrier scattering are determined within a many-valley model of the energy spectrum for compositions 0.08 ≤ x ≤ 0.4 and 0.06 ≤ y = z ≤ 0.15. The effect of a variation in the parameters of the constant-energy surface on the thermoelectric efficiency is analyzed for different compositions and carrier concentrations in solid solutions.     

3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis

We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube Ω=[0,L]³ can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding regular solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. The new system is amenable to being integrated numerically using similar methods as in Euler equations. We propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite-time singularity problem of Euler equations, based on the conservation of quantities directly related to energy and circulation. The method of mapping to a regular system can be extended to any fluid equation that admits a Beale-Kato-Majda type of theorem, e.g. 3D Navier-Stokes, 2D and 3D magnetohydrodynamics, and 1D inviscid Burgers. We discuss briefly the case of 2D ideal magnetohydrodynamics. In order to illustrate the usefulness of the mapping, we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1D inviscid Burgers equation.     

Stability improvement and regenerative chatter suppression in nonlinear milling process via tunable vibration absorber

In this paper, a tunable vibration absorber set (TVAs) is designed to suppress regenerative chatter in milling process (as a semi-active controller). An extended dynamic model of the peripheral milling with closed form expressions for the nonlinear cutting forces is presented. The extension part of the cutting tool is modeled as an Euler–Bernoulli beam with in plane lateral vibrations (x–y directions). Tunable vibration absorbers in x–y directions are composed of mass, spring and dashpot elements. In the presence of regenerative chatter, coupled dynamics of the system (including the beam and x–y absorbers) is described through nonlinear delay differential equations. Using an optimal algorithm, optimum values of the absorbers' position and their springs' stiffness in both x–y directions are determined such that the cutting tool vibration is minimized. Results are compared for both linear and nonlinear models. According to the results obtained, absorber set acts effectively in chatter suppression over a wide range of chatter frequencies. Stability limits are obtained and compared with two different approaches: a trial and error based algorithm and semi-discretization method. It is shown that in the case of self-excited vibrations, the optimum absorber improves the process stability. Therefore, larger values of depth of cut and consequently more material removal rate (MRR) can be achieved without moving to unstable conditions.     

Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations

We study the convergence rate of the solutions of the incompressible Euler-α, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter α approaches zero. First we show the convergence in H ^s , s>n/2+1, in the whole space, and that the smooth Euler-α solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.     

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation

Many applications lead to large systems of linear equations with dense matrices. Direct matrix-vector products become prohibitive, since the computational cost increases quadratically with the size of the problem. By exploiting specific kernel properties fast algorithms can be constructed.A directional multilevel algorithm for translation-invariant oscillatory kernels of the type K(x, y) = G(x ? y)e^?k∣x?y∣, with G(x ? y) being any smooth kernel, will be presented. We will first present a general approach to build fast multipole methods (FMMs) based on Chebyshev interpolation and the adaptive cross approximation (ACA) for smooth kernels. The Chebyshev interpolation is used to transfer information up and down the levels of the FMM. The scheme is further accelerated by compressing the information stored at Chebyshev interpolation points using ACA and QR decompositions. This leads to a nearly optimal computational cost with a small pre-processing time due to the low computational cost of ACA. This approach is in particular faster than performing singular value decompositions.This does not address the difficulties associated with the oscillatory nature of K. For that purpose, we consider the following modification of the kernel K^u = K(x, y)e^??ku·(x?y), where u is a unit vector (see Brandt [1]). We proved that the kernel K^u can be interpolated efficiently when x ? y lies in a cone of direction u. This result is used to construct an FMM for the kernel K.Theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method. The paper ends with the presentation of 2D and 3D numerical convergence studies, and computational cost benchmarks.     

Thermoelectric properties of n-Bi2Te3 − x − y Se x S y solid solutions under high pressure

The thermoelectric properties of n-Bi₂Te_{3 ? x ? y} Se _x S _y solid solutions with atomic substitutions in the tellurium sublattice (x = 0.27, 0.3, y = 0, and x = y = 0.09) have been studied under a pressure to 8 GPa. It has been found that the Seebeck coefficient and the resistance decrease with increasing P, and power factor χ increases in all compositions and becomes maximal at pressures of 2–4 GPa. It has been shown that the power factor χ, which is proportional to the product of the effective mass of the density of states m/m m/m ₀ and the charge carrier mobility μ₀ in the form (m/m ₀)^3/2μ₀, increases with increasing pressure mainly due to the increase in the mobility and also depends on the solid solution composition. In the composition with substitution Te → Se + S (x = y = 0.09), the peculiarity of the dependence of m/m ₀ on P in the pressure range corresponding to maximal values of the power factor can be explained by the existence of an electronic topological transition. The increase in the power factor under pressure in n-type Bi₂Te_{3 ? x ? y} Se _x S _y solid solutions combined with similar data for p-type Bi_{2 ? x} Sb _x Te₃ solid solutions obtained earlier, including the estimations of possible changes in the thermal conductivity with increasing pressure, give grounds to design thermoelements with improved value of the thermoelectric figure-of-merit, which can be 50–70% at pressures of 2–4 GPa.     

Symmetry Reduction and Exact Solutions of the Euler–Lagrange–Born–Infeld,Multidimensional Monge–Ampere and Eikonal Equations

Abstract

Using the subgroup structure of the generalized Poincaré group P (1, 4), ansatzes which reduce the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which reduce the considered equations to linear ordinary differential equations. The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, some classes of exact solutions of the investigated equation have been presented.     

Interferometry for ellipso-height-topometry: Part 2: Measurement of the ellipsometric data, material identification, and correction of the measured height

Ellipso-height-topometry, EHT, as introduced by Leonhardt, et al. [Interferometry for ellipso-height-topometry, part 1, Optik 113 (2003) 513-519; Topometry for locally changing materials, Opt. Lett. 23 (1998) 1772-1774]; [Ellipso-height topometry, Optik 112 (2001) 413-420] is an extended topometry where topographies of the surface height H(x,y), the ellipsometric parameters Ψ(x,y) and Δ(x,y) (and optionally the intensity distribution I(x,y)) of the surface are measured on the same pixel raster and with high resolution. Thus, we can dispose over a set of (mutually coherent) topographies, and further topographies of quantities of interest can be calculated from this set: the local change in the complex refractive index N(x,y)=n(x,y)−k(x,y)i of bulk surfaces, the thickness distribution t(x,y) of locally changing (discontinuous) films and overlayers, and a correction of the optically measured local height H(x,y). The height error ΔH(x,y) can be calculated from the ellipsometric data and the true height h(x,y)=H(x,y)−ΔH(x,y) is thus obtained. ΔH(x,y) can assume large values when overlayers of oxidations or residues from lubrication oil or from processing are present. Much more information about the surface is gained with this concept.In part 1 of this work a z-scanning interferometric scheme with oblique incidence over the entire object field has been introduced, with the advantages of white-light interferometry, but with the additional capability of measuring material information to supply a complete set of topographies for extended topometry. In addition, very useful coherence properties for interferometry with oblique incidence were presented and discussed which allow to shape the interferogram in a simple way without any additional devices. In this part we derive the theory of the ellipsometric measurement and present first results by complete sets of two different samples.     

Synthesis,structure, and properties of BIFENBVOX

Solid solutions Bi₄V_2-x Fe _x/2Nb _x/2O_11-δ (х?=?0.05–1.0) and Bi₄V_2-х-y Fe _x Nb _y O_11–δ (with fixed x or y?=?0.2 and variable х or y?=?0.2–0.5 with step 0.1) were synthesized by the standard ceramic technology in the temperature range 773–1113 K and by mechanochemical activation method using Bi₂O₃, V₂O₅ Fe₂O₃, and Nb₂O₅ oxides as initial compounds. The formation of solid solutions was studied. Ranges of stability and temperature values of phase transitions for different polymorphous modifications were defined using dylatometric and thermo gravimetric studies. The morphology and the local chemical composition of the ceramic samples were studied. Samples with concentration of dopants x?>?0.3 contain two phases; both major and impurity phases are solid solutions of the BIFENBVOX type although the dopants atoms distribution between them is random. The thermal expansion coefficients (TEC) were measured. The electrical conductivity of ceramic samples was investigated in a wide range of temperatures. The highest conductivity values among the studied solid solutions are observed for the sample with a small amount of dopants x?=?0.25.     

Global regularity of the 4D restricted Euler equations

We are concerned with the critical threshold phenomena in the restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for the 3D and 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that the 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.     

Numerical optimization of In-mole fractions and layer thicknesses in AlxGa1−xN/AlN/GaN high electron mobility transistors with InGaN back barriers

The effects of the In-mole fraction (x) of an In_xGa_1−xN back barrier layer and the thicknesses of different layers in pseudomorphic Al_yGa_1−yN/AlN/GaN/In_xGa_1−xN/GaN heterostructures on band structures and carrier densities were investigated with the help of one-dimensional self-consistent solutions of non-linear Schrödinger-Poisson equations. Strain relaxation limits were also calculated for the investigated Al_yGa_1−yN barrier layer and In_xGa_1−xN back barriers. From an experimental point of view, two different optimized structures are suggested, and the possible effects on carrier density and mobility are discussed.     

Optical and vibrational properties of CdxHg1-x-y ZnyTe solid solutions

The lattice dynamics of Cd_xHg_1-x-y Zn_yTe solid solutions is studied theoretically and experimentally. The frequencies of the basic optical phonons of Cd_xHg_1-x-y Zn_yTe are calculated in terms of a modified random-element isodisplacement model. As a result, all basic vibrations of the crystal lattice that substantially affect the optical properties of this material in the spectral region corresponding to one-phonon resonance are identified. The optical properties of epitaxial Cd_xHg_1-x-y Zn_yTe layers grown by liquid-phase epitaxy on Cd_1-x Zn_xTe substrates are studied. The calculated and experimental spectral dependences of the dielectric function of Cd_xHg_1-x-y Zn_yTe solid solutions of various compositions are compared at 295 and 78 K, and good agreement between them is reached. The additional lattice vibrations whose frequencies in the phonon density of states are lower than that of the HgTe mode are shown to be caused by the lattice defects of the Cd_xHg_1-x-y Zn_yTe solid solutions.     

EPR study of clusters of rare-earth ions in mixed fluoride crystals

The EPR spectra of the (BaF₂)_{1 ? x} (CeF₃) _x system are studied for the concentrations x = 0, 0.001, 0.002, 0.005, 0.01, and 0.02. The appearance of new tetragonal centers is detected beginning from x = 0.002, the intensity of these centers being maximal at x = 0.01. The (CaF₂)_{1 ? x ? y} (CeF₃) _x (YF₃) _y double solutions with x = 0.001 and y from 0 to 0.02 are also studied. In addition to the ordinary tetragonal center, beginning from y = 0.001, a new tetragonal center appears with the same structure as in the previously studied mixed crystals based on BaF₂—namely, the Ce³⁺-□-R³⁺ chain elongated along the fourfold axis substitutes the Ca²⁺-Ca²⁺-Ca²⁺ and Ba²⁺-Ba²⁺-Ba²⁺ chains in regular CaF₂ and BaF₂ crystals (□ is the cation vacancy, and R³⁺ is the Ce³⁺, La³⁺, or Y³⁺ trivalent ion).     

Features of the absorption spectra of thin films of M 2Ag1 − x CuxI3 solid solutions (M = Rb, Cs)

The absorption spectra of thin films of (MI)_{1 ? y} (Ag_{1 ? x} Cu_xI)_y solid solutions (M = Rb, Cs) with the initial molar concentration y = 0.33 have been investigated. It is established that, at low concentrations x, a local exciton band due to Cu⁺ ions is split off from the main long-wavelength exciton bands. In Rb₂Ag_{1 ? x} Cu_xI₃ solutions, the concentration shift of exciton bands indicates the formation of a persistent-type exciton spectrum. However, in Rb₂Ag_{1 ? x} Cu_xI₃ with x ≥ 0.5 and in Cs₂Ag_{1 ? x} Cu_xI₃ with x > 0.2, exciton spectra of amalgamation type are observed, which are related to the formation of more stable M ₃Ag_{2 ? 2x} Cu_2x I₅ solid solutions. The formation of these solutions leads to broadening of the exciton bands and to the concentration transition from persistent-to amalgamation-type exciton spectra.     

Interferometry for Ellipso-Height-Topometry part 3: Correction of the aspect errors and reduction of the dispersion of the ellipsometric parameters

Ellipso-Height-Topometry, EHT, is an extended optical topometry, where both the topographies of the surface height H(x,y) and the ellipsometric local parameters Ψ(x,y) and Δ(x,y) of surfaces with locally changing materials are measured on the same pixel raster with high resolution and using the same data sets. Further quantities can be calculated from these measurements on the base of locally confined surface models: the local refractive index, the thickness t(x,y) of overlayers or films, or other parameters of layered systems.     

Stationary solitons of the fifth order KdV-type. Equations and their stabilization

Exact stationary soliton solutions of the fifth order KdV type equation, u_t + αu^pu_x + βu_3x + γu_5x = 0, are obtained for any p (> 0) in case αβ > 0, Dβ > 0, βγ < 0 (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case p ≥ 5. Various properties of these solutions are discussed. In particular, it is shown that for any p these solitons are lower and narrower than the corresponding γ = 0 solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when D, α, β, γ are all > 0 and discuss its various properties.