Spectral Equation Research Articles

Enhancing upconversion emission of Er3+ in single β-NaYF4 microrod through constructing different inert and active shells with doping Yb3+ ions

The construction of core-shell structure helps in enhancing upconversion luminescence in lanthanide-doped micro/nanomaterials. This study aimed to synthesize a series of NaYF4 and NaErF4 core-shell microrods based on epitaxial growth technology so as to enhance the upconversion emission. The upconversion luminescence properties of single NaYF4:Yb3+/Er3+ and NaErF4 microrods with different core-shell structures were carefully studied using a confocal microscope spectroscopy test system under 980-nm near-infrared excitation. The upconversion emission intensity and red-to-green emission intensity ratio of Er3+ ions were remarkably enhanced in different single microrods by building different inert and active core-shell structures. The higher red-to-green ratio was mainly due to the cross-relaxation between Er3+ ions and the energy back-transfer process from Er3+ to Yb3+ ions, which were explained on the basis of spectral characteristics and rate equations. These tunable NaYF4 and NaErF4 core-shell microrods had the potential to be applied to displays, micro-lasers, and anti-counterfeiting.

On basicity of the system of eigenfunctions of one discontinuous spectral problemfor second order differential equation for grand-Lebesgue space

On basicity of the system of eigenfunctions of one discontinuous spectral problemfor second order differential equation for grand-Lebesgue space

Uncertainty analysis of higher-order sandwich beam using a hybrid stochastic time-domain spectral element method

A stochastic time-domain spectral element method (STSEM) is applied to sandwich beam structures. STSEM couples the computational efficiencies of the spectral stochastic finite element method and the time-domain spectral element method. A higher-order sandwich panel theory incorporating the core flexibility is considered for the analysis. The material properties are modeled as random fields and represented by the Karhunen-Loéve expansion. The stochastic nodal displacements are expressed using the polynomial chaos expansion, and the expansion coefficients are computed by solving the spectral stochastic finite element equations. The computational efficiency of the proposed method is compared with Monte-Carlo simulation.

A nonlinear spectral rate-dependent constitutive equation for electro-viscoelastic solids

In this communication a spectral constitutive equation for nonlinear viscoelastic-electroactive bodies with short-term memory response is developed, using the total stress formulation and the electric field as the electric independent variable. Spectral invariants, each one with a clear physical meaning and hence attractive for use in experiment, are used in the constitutive equation. A specific form for constitutive equation containing single-variable functions is presented, which are easy to analyze compared to multivariable functions. The effects of viscosity and an electric field are studied via the results of boundary value problems for cases considering homogeneous distributions for the strains and the electric field, and some these results are compared with experimental data.

Spectral Features of Systems With Chaotic Dynamics

Using the Lorentz model and Hamiltonian systems without dissipation as an example, spectral methods for analyzing the dynamics of systems with chaotic behavior are considered. The insufficiency of the traditional approach to the study of perturbation dynamics based on an analysis of the roots of the classical spectral equation is discussed. It is proposed to study nonlinear systems using the method of constructing spectral equations with different eigenvalues, which allows one to take into account the randomness and multiplicity of states. The spectral features of instability and chaos for systems without dissipation are shown by the example of short-wave perturbations of a flow of a weakly ionized plasma gas.

Fast Exponentially Convergent Solution of Electromagnetic Scattering From Multilayer Concentric Magnetodielectric Cylinders by the Spectral Integral Method

The multilayer surface integral equations (SIEs) for electromagnetic scattering by infinitely long magnetodielectric cylinders with an arbitrary number of layers are derived and solved by the spectral integral method (SIM). Singularity subtraction for the 2-D Green's function is used to enhance the computation accuracy and achieve exponential convergence. The final matrix equation after discretization is formed in the Fourier spectral domain rather than the spatial domain, which greatly expedites the SIM solution by accelerating the convolution via the fast fourier transform (FFT) algorithm. A recursive method is proposed to solve the spectral integral equations instead of using an iterative method to lower the computation complexity. Numerical examples for ordinary multilayer cylinders and invisibility cloak cylinders are presented to validate the SIM results by comparing the total fields, scattered fields, and radar cross section (RCS) to analytical solutions or finite-element simulations. They verify that the recursive solution has a complexity of O(MN log N) for an M-layer cylinder with N discrete points on each interface. Meanwhile, the SIM outperforms the analytical method because only the 0th-order and 1st-order special functions (Bessel functions and Hankel functions) are used in the SIM but higher-order functions are necessary for the analytical method to maintain the accuracy.

Model reduction of nonlinear cyclic structures based on their cyclic symmetric properties

Displacement control of bladed-disks is of primary stake for turbo-engineers. Such structures present cyclic symmetric properties that allow some specific reduction techniques. For linear problems, the equation of motion projected on spectral components, also called nodal diameters, gives a system of equations in which the unknowns are uncoupled. Each nodal diameter can therefore be considered independently. However in real applications, the presence of nonlinear terms couples the different nodal diameters and makes the spectral equation of motion more complex to handle.This paper deals with this difficulty and presents two main results. It first gives an analytical derivation to determine which nodal diameters get coupled by friction nonlinearities. Such procedure reduces the size of the model but also the number of unknowns in the system by considering only the interacting nodal diameters. This method is general and allows to tackle a wide range of industrial problems. However this may not be sufficient for an efficient resolution of the nonlinear system since the nonlinear forces must first be evaluated in the physical domain, and, a priori, for all the sectors of the cyclic system before being computed in the spectral domain. The second main originality of the paper is the development of different strategies on this matter. One is analytical, valid for any excitation and is straightforward to implement, while the others are based on specific assumptions on the deformed shape but offer further reduction.The new methodologies are validated on a simplified bladed-disk with different excitation forces and different friction’s laws. They show very good accuracy and a substantial computation time reduction.

Modelling statistical wave interferences over shear currents

Wave forecasting in ocean and coastal waters commonly relies on spectral models based on the spectral action balance equation. These models assume that different wave components are statistically independent and as a consequence cannot resolve wave interference due to statistical correlation between crossing waves, as may be found in, for instance, a focal zone. This study proposes a statistical model for the evolution of wave fields over non-uniform currents and bathymetry that retains the information on the correlation between different wave components. To this end, the quasi-coherent model (Smit & Janssen, J. Phys. Oceanogr., vol. 43, 2013, pp. 1741-1758) is extended to allow for wave-current interactions. The outcome is a generalized action balance model that predicts the evolution of the wave statistics over variable media, while preserving the effect of wave interferences. Two classical examples of wave-current interaction are considered to demonstrate the statistical contribution of wave interferences: (1) swell field propagation over a jet-like current and (2) the interaction of swell waves with a vortex ring. In both examples cross-correlation terms lead to development of prominent interference structures, which significantly change the wave statistics. Comparison with results of the SWAN model demonstrates that retention of cross-correlation terms is essential for accurate prediction of wave statistics in shear-current-induced focal zones.

A consistent isotropic spectral constitutive equation: The infinitesimal strain depends nonlinearly on the stress

Recently, there have been several developments in nonlinear infinitesimal-strain stress constitutive relations to describe the response of elastic bodies. However, some of the nonlinear constitutive equations presented in the literature are unable to model both compressible and incompressible bodies. In this paper, we develop a spectral constitutive equation, where the infinitesimal strain depends nonlinearly on the stresses, and the incompressible behaviour is simply described as a special case obtained from the compressible elastic model, by just letting the value of the Poisson’s ratio equal to a half. The constitutive equation satisfies the Baker-Ericksen inequality and a specific expression for the constitutive equation is proposed to fit experimental data for the compression of a sample of rock. Several boundary value problems with homogeneous deformations and stresses are analysed. Spectral components for the fourth order tensor that is required for the analysis of the propagation of P- and S-waves are given.

Spinor vertices and bubbles in the old conformal bootstrap in AdS/CFT and the fermion mass hierarchy problem: The Yukawa model

Attempts to resolve the longstanding Fermion mass hierarchy (FMH) problem in frames of the AdS/CFT correspondence demand the knowledge of bulk fermion masses. The approach of the "old" conformal bootstrap in AdS context permitted to calculate bulk masses of scalar fields [1], [2]. In the present paper this approach is extended to physically more interesting spin 1/2 bulk fields. Calculation of spinor-scalar vertices is performed in physical AdS space, and unexpectedly simple expressions for spinor-scalar bubbles (2-point one-loop self-energy Witten diagrams) are obtained. The "double-trace from UV to IR flow" subtraction of UV divergences used earlier in calculations of the UV-finite bulk tadpoles is applied to bubbles. This permitted to write down in frame of the "old" conformal bootstrap approach the nontrivial spectral equations for the bulk fermion masses; SU(N) Yukawa model of spinor fields interacting with conformal invariant scalar field in case of four boundary dimensions is considered as a specific example.

Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Here we study theoretically and compare experimentally with the methods developed in [1,2] an efficient method for solving systems of algebraic equations A˜αu˜h=f˜h, 0<α<1, where A˜ is an N×N matrix coming from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in Rd, d=1,2,3. The proposed methods are based on the best uniform rational approximation (BURA) rα,k(t) of tα on [0,1]. Here rα,k is a rational function of t involving numerator and denominator polynomials of degree at most k.The approximation of u˜h=A˜−αf˜h is then w˜h=λ1−αrα,k(λ1A˜−1)f˜h, where λ1 is the smallest eigenvalue of A˜. We show that the proposed method is exponentially convergent with respect to k and has some attractive properties. First, it reduces the solution of the nonlocal system to solution of k systems with matrix (A˜+cjI˜) and cj>0, j=1,2,…,k. Thus, good computational complexity can be achieved if fast solvers are available for such systems. Second, the original problem and its rational approximation in the finite difference case are positivity preserving. In the finite element case, this valid for schemes obtained by mass lumping under certain mild conditions on the mesh. Further, we prove that the lumped mass schemes still have the expected rate of convergence, at times assuming additional regularity on the right hand side. Finally, we present comprehensive numerical experiments on a number of model problems for various α in one and two spatial dimensions. These illustrate the computational behavior of the proposed method and compare its accuracy and efficiency with that of other methods developed by Harizanov et al. [1] and Bonito and Pasciak [2].

On Drazin Spectral Equation for the Operator Products

Let X be a Banach space and A, B, C, D be bounded linear operators on X. In this paper we show that the Drazin (resp. generalized Drazin) spectrum of AC coincides with the Drazin (resp. generalized Drazin) spectrum of BD under certain assumptions. Moreover, several interesting Banach space properties, such as SVEP and polaroidness, are considered.

An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the $$d+1$$ dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.

UV-finite \u201cold\u201d conformal bootstrap on AdS: scalar case

The double-trace from UV to IR flow subtraction of infinities used earlier for the UV-convergent calculations of the Witten tadpole diagrams being applied to the bubble self-energy diagrams gives for them the amazingly simple expressions in case of the four- dimensional boundary space. For every N = 1 … 4 in the O(N) symmetric scalar fields model with the conformal Hubbard-Stratonovich field there are three roots of the “old” conformal bootstrap spectral equations that obey unitarity bound demand.

INVESTIGATION OF THE AMPLITUDE STABILITY RELATION OF A MATHEMATICAL MODEL OF A CENTRIFUGAL PUMP OF A PUMPING UNIT AS AN OBJECT OF CONTROL WITH A DEGREE OF GAS PRESSURE

Using the experimental data obtained intheprocesso fchange inthetime of the degree of increase in gas pressure, as well as the performance of the centrifugal supercharger of the gas pumping unit, . Digitization of graphs of change of degree of increase of pressure of gas, and also productivity of the centrifugal supercharger of gas-pumping unitis made with the help of software Graph2Digit. The digitization wasper formed with a sampling period of 0.3 s. The number of values forth edegree of increase of gaspressure, as well as the performance of the centrifugal pump gas pumping unitis 581 each.&#x0D; The experimental data are divide dinto ranges (1 - 811 sec - 818 sec; 2 - 818 sec - 825 sec; 3 - 811 sec - 831 sec; 4 - 825 sec - 831 sec; 5 - 827 pp. - 832 pp .; 6 - 829 pp. - 831 pp.). In the course of the study, the mathematical expectation for each of the intervals, as well as the spectral density equation for the degree of gas pressure increase and the spectral density equation for the performance of the centrifugal supercharger of the gaspumping unit, were determined at appropriate intervals.&#x0D; Using there lation between the spectral densitiesand the fact function of the Matlab software product, which factorizes fractional-rational expression, the gear function saredetermine date achinter valand the stability stocks are determine dusing the Nyquist criterion.&#x0D; The plot of the stability reserve of the centrifugal supercharger of the gas pumping unit as an object of control from the mathematical expectation, which is obtained on the basis of the spectral density ratios, is made. This dependency is approximated by the CurveExpert Professional software to select the optimalma them at icalmodel.&#x0D; From the set of results, the one with the high estcorrelation coefficient and the small eststandard error isselected.

Third-order sparse grid generalized spectral elements on hexagonal cells for uniform-speed advection in a plane

This paper investigates sparse grids on a hexagonal cell structure using a Local-Galerkin method (LGM) or generalized spectral element method (SEM). Such methods allow sparse grids to be used, known as serendipity grids in square cells. This means that not all points of the full grid are used. Using a high-order polynomial, some points of each cell are eliminated in the discretization, and thus saving Central Processing Unit (CPU) time. Here a sparse SEM scheme is proposed for hexagonal cells. It uses a representation of fields by second-order polynomials and achieves third-order accuracy. As SEM, LGM is strictly local for explicit time integration. This makes LGM more suitable for multiprocessing computers compared with classical Galerkin methods. The computer time depends on the possible timestep and program implementation. Assuming that these do not change when going to a sparse grid, the potential saving of computer time due to sparseness is 1:2. The projected CPU saving in 3-D from sparseness is by a factor of 3:8. A new spectral procedure is used in this paper, called the implied spectral equation (ISE). This procedure allows for some collocation points to use any finite difference scheme of high order and the time derivatives of other spectral coefficients are implied.

Spectral synthesis and partial differential equations

We offer a new approach to spectral synthesis on affine groups. The idea is to consider solution spaces of systems of linear partial differential equations corresponding to invariant differential operators with respect to some closed subgroup of the general linear group on $$\mathbb{R}^n$$. We show that a certain type of spherical spectral analysis and spectral synthesis holds for such solution spaces.

Fully dynamic earthquake sequence simulation of a fault in a viscoelastic medium using a spectral boundary integral equation method: does interseismic stress relaxation promote aseismic transients?

Along subduction interfaces or some major faults, a seismogenic layer in the upper crust is underlain by a zone of slow-slip events (SSEs) and tremors, and seismicity disappears at greater depths. The transition between seismogenic and aseismic behavior may be caused by changes in the frictional properties of the fault or changes in viscoelastic properties of the surrounding medium. Although aseismic transients have been numerically generated in previous studies by changing the frictional properties and compared to SSEs, the effect of viscoelasticity on the transition remains to be studied. In this study, we implemented interseismic viscoelastic stress relaxation in a simulation code for two-dimensional antiplane fully dynamic earthquake sequences in a uniform elastic material based on a spectral boundary integral equation method. In the implementation, we developed a suitable algorithm in which the viscoelastic relaxation is calculated by evolution of an “effective slip,” which gives viscoelastic stress change on the fault if convolved with a static Green’s function. We conducted parametric studies for a fault with a rate-weakening patch on two parameters: viscoelastic relaxation time t_{c} and characteristic length of the state evolution in the rate- and state-dependent friction law L. The behavior of the simulated fault can be classified into four classes, earthquakes (EQ), aseismic transients (AT), stable sliding (SS), and stuck (ST), in which the central part of the rate-weakening patch has a diminishingly small slip rate and is permanently locked. A phase diagram of the fault behavior shows that there are two different types of seismogenic–aseismic transition. As L increases, an EQ patch changes to an AT patch before becoming an SS patch, as has been reported in previous studies in an elastic limit. The boundary between AT and SS can be explained via linear stability analysis of a system composed of a spring, a dashpot, and sliders. As t_{c} decreases, the recurrence interval of the earthquakes diverges, and an EQ patch changes to an ST patch unless L is within a narrow range. Therefore, the transition associated with SSEs and tremors is dominated by changes in frictional properties rather than changes in viscoelastic properties.

RAP-method (random perturbation method) for finding 𝑆-minimax control vectors and parameter estimates for some linear systems with random coefficients

Abstract The spectral equations for the minimax estimates of the parameters of some systems are obtained.

Enhanced solar radiation absorption by carbon dioxide in thermodynamic nonequilibrium: A computational study

The conversion of carbon dioxide (CO2) into higher value products using solar energy can help fulfill the increasing need for fuels and chemicals while mitigating environmental emissions. Solar thermochemical conversion approaches rely on the absorption of radiative energy by a solid catalytic medium. The direct absorption of solar energy by CO2 before reaching the catalyst could lead to more efficient conversion. Greater solar photon absorption can be achieved through the use of an electrical discharge to drive CO2 to a low-temperature plasma state in which free electrons and heavy-species are in a state of Non-Local Thermodynamic-Equilibrium (NLTE). The absorption of solar energy by a plasma-enhanced solar reactor for atmospheric pressure CO2 conversion is analyzed via zero- and one-dimensional models. The models are comprised by the spectral radiative transfer equation (RTE) and energy conservation equations for electrons and heavy-species. Radiative properties are computed by a line-by-line radiative code and transport properties using kinetic theory. Simulation results for a representative 10 cm long solar-plasma reactor show up to 48% and 90% net-absorption of solar radiation by CO2 in NLTE at electron temperatures of 1 and 2 eV, respectively, and heavy-species temperatures of ~2000 K, common in solar thermochemical processes. In contrast, less than 0.2% of solar radiation is absorbed by CO2 in LTE under similar conditions. The results suggest that solar-plasma direct-receiver reactors, due to increased direct solar energy absorption, have the potential for greater efficiency CO2 decomposition processes.