Spectral Equation Research Articles

Estimation of thermal emission from mixture of CO2 and H2O gases and fly-ash particles

Abstract In the present work the thermal emission from gases and particle mixture is studied comprehensively. The mixture consist of CO2 and H2Ogases along with fly-ash particles and their spectral radiative properties are obtained from HITEMP database and Mie theory, respectively. The Line-by-line method, known for high accuracy is used to estimate radiative heat transfer from gases and particle mixture. Three distinct cases, i.e, pure absorption/emission, pure scattering, and their combined effects are investigated. The spectral radiative transfer equation is solved by finite angle method (FAM) and the correctness of the results is ensured by principle of energy conservation. The anisotropic scattering is modeled using the Henyey-Greenstein method. The influence of radiative properties of the participating medium, along with wall and cavity temperatures, on heat transfer and emission spectrum is evaluated. The radiative heat transfer enhances with introduction of the particles in the gaseous mixture. For pure scattering medium, the emission spectrum on each wall differs and also varies from the emission spectrum of the emitting wall. The particle influence grows notably with higher volume fractions in gaseous mixture. While there is a notable contribution of anisotropic scattering in a purely scattering medium, its significance diminishes in an absorbing, emitting, and scattering medium.

Phaseless inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers in the soft x-ray regime

Soft x-ray wafer-metrology experiments are characterized by low signal-to-noise ratios and lack phase information, which both cause difficulties with the accurate three-dimensional profiling of small geometrical features of structures on a wafer. To this end, we extend an existing phase-based inverse-scattering method to demonstrate a sub-nanometer and noise-robust reconstruction of the targets by synthetic soft x-ray scatterometry experiments. The targets are modeled as three-dimensional finite dielectric scatterers embedded in a planarly layered medium, where a scatterer’s geometry and spatial permittivity distribution are described by a uniform polygonal cross section along its height. Each cross section is continuously parametrized by its vertices and homogeneous permittivity. The combination of this parametrization of the scatterers and the employed Gabor frames ensures that the underlying linear system of the spatial spectral Maxwell solver is continuously differentiable with respect to the parameters for phaseless inverse-scattering problems. In synthetic demonstrations, we demonstrate the accurate and noise-robust reconstruction of the parameters without any regularization term. Most of the vertex parameters are retrieved with an error of less than λ/13 with λ=13.5nm, when the ideal sensor model with shot noise detects at least five photons per sensor pixel. This corresponds to a signal-to-noise ratio of 3.5 dB. These vertex parameters are retrieved with an accuracy of λ/90 when the signal-to-noise ratio is increased to 10 dB, or approximately 100 photons per pixel. The material parameters are retrieved with errors ranging from 0.05% to 5% for signal-to-noise ratios between 10 dB and 3.5 dB.

Investigation of the Sigma Approximation Technique for the Solution of the Time Spectral Equation System

Abstract The time spectral method is commonly utilized to analyze periodic unsteady flows within turbomachines. However, the time spectral solutions in regions with large temporal and spatial gradients, such as wakes, shocks, and boundary layers, can be plagued with unphysical oscillations, known as the Gibbs phenomenon. This paper presents an investigation into the sigma approximation technique, which is capable of significantly attenuating the Gibbs phenomenon by dampening high-frequency nonlinear components in the time spectral solutions. However, the optimal combination of the sigma factors remains to be ascertained considering the trade-off between the accuracy of solutions and the reduction of unphysical oscillations. In this study, a two-row compressor configuration is first selected, and numerical simulations are performed to evaluate the efficacy of this technique under near peak-efficiency and near-stall operating conditions. It is found that Gibbs-type unphysical oscillations can be effectively mitigated without notably compromising solution accuracy through an optimal combination of sigma factors. The NASA Stage 35 case study further verifies the effectiveness of the proposed optimal sigma factor combination in preserving the accuracy of time spectral solutions based on the data from test rig. Moreover, the von Neumann analysis reveals that the sigma approximation technique can enhance the numerical stability of the time spectral equation system, thereby allowing the use of aggressive numerical parameters to accelerate convergence. The stability analysis results are verified by a two-dimension bump under two different flow conditions and further assessed using the two-row compressor.

Optical soliton solutions and their nonlinear dynamics described by a novel time-varying spectral Hirota equation with variable coefficients

Optical soliton solutions and their nonlinear dynamics described by a novel time-varying spectral Hirota equation with variable coefficients

Scalar spectral functions from the spectral functional renormalization group

We compute spectral functions in a scalar ϕ4-theory in three spacetime dimensions via the spectral functional renormalization group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalization. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwinger equation. Published by the American Physical Society 2024

A generalized strain model for spectral rate-dependent constitutive equation of transversely isotropic electro-viscoelastic solids

We model the constitutive equation for nonlinear electro-viscoelastic transversely isotropic solids with short term memory via a generalized strain method, where the method is a change with respect to the methods that have been done in the last decades regarding mechanics of nonlinear solids. Our generalized strain model uses spectral invariants with a clear physical interpretation and hence they are attractive for use in experiments. The constitutive equation contains single-variable functions, which are easy to deal with when compared to multivariable functions. The effects of viscosity and electric fields are analyzed via the boundary value problem results. The efficacy the proposed prototype is scrutinized by comparing our theory with experimental data.

Non-local gravity wave turbulence in presence of condensate

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

Gravitational waves from binary neutron star mergers with a spectral equation of state

Gravitational waves from binary neutron star mergers with a spectral equation of state

Bound states from the spectral Bethe-Salpeter equation

We compute the bound state properties of three-dimensional scalar ϕ4 theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations. Published by the American Physical Society 2024

Constrained Wave-telescope Technique

Constrained Wave-telescope Technique

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

Transfer of near infrared calibration for gasoline octane number based on screening consistent wavelengths combined with direct standardization algorithm

In order to share multivariate calibration models of gasoline research octane number between different near infrared spectrometers, a novel calibration transfer method, namely combination of screening consistent wavelengths and direct standardization (SWCSS-DS) was proposed. Firstly, screening wavelengths with consistent and stable signals (SWCSS) between instruments was used to select the wavelengths with best stability, and then direct standardization (DS) further corrected the systematic errors that still exist after the SWCSS was implemented. The spectra of 120 standard gasoline samples collected on two near infrared spectrometers of the same type were investigated in detail to verify the validity of the new algorithm. Compared results of other transfer methods such as SWCSS, Slope/Bias (S/B), direct standardisation (DS), and piecewise direct standardization (PDS), the root mean squared error for prediction (RMSEP) of SWCSS-DS algorithm for target samples was decreased from 5.75 to 0.295, and the Akaike information criterion value decreased from 1516 to 640, which were better than those of the SWCSS, S/B, DS and PDS algorithms. Therefore, the joint algorithm of SWCSS-DS has not only improved the universality of the master model, but also reduced the dimension of the spectral matrix and calibration equation, that would provide a more efficient model transfer strategy for the practical applications.

On Equation Manifolds, the Vinogradov Spectral Sequence, and Related Diffeological Structures

We consider basic diffeological structures that can be highlighted naturally within the theory of the Vinogradov spectral sequence and equation manifolds. These interrelated features are presented in a rigorous and accurate way, that complements some heuristic formulations appearing in very recent literature. We also propose a refined definition of the Vinogradov spectral sequence using diffeologies.

On a characterization of the Rellich–Kondrachov theorem on groups and the Bloch spectral cell equation

On a characterization of the Rellich–Kondrachov theorem on groups and the Bloch spectral cell equation

Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products

<abstract><p>We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) $ \sharp $ and semi-tensor products $ \ltimes $. Indeed, for each real number $ t $ and two positive definite matrices $ A $ and $ B $ of arbitrary sizes, the $ t $-weighted SGM $ A \, \diamondsuit_t \, B $ of $ A $ and $ B $ is a unique positive solution $ X $ of the equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A^{-1}\,\sharp\, X \; = \; (A^{-1}\,\sharp\, B)^t. $\end{document} </tex-math></disp-formula></p> <p>We then established fundamental properties of the weighted SGMs based on MGMs. In addition, $ (A \, \diamondsuit_{1/2} \, B)^2 $ is positively similar to $ A \ltimes B $ and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.</p></abstract>

Dual ODE: Spatial–Spectral Neural Ordinary Differential Equations for Hyperspectral Image Super-Resolution

Dual ODE: Spatial–Spectral Neural Ordinary Differential Equations for Hyperspectral Image Super-Resolution

Three-Dimensional Forward Modeling of Transient Electromagnetic Method Considering Induced Polarization Effect Based on Spectral Element Method

The transient electromagnetic method (TEM) is widely used in the exploration of mineral, petroleum, and geothermal resources due to its sensitivity to low-resistivity bodies, limited site constraints, and strong resistance to interference. In practical applications, the TEM often uses a long wire source instead of an idealized horizontal electric dipole (HED) source as the excitation source. This is due to the complex external conditions and the relatively large distance between the receiving zone and the transmitter source. Compared to the HED, the long wire source can provide a larger excitation current, generating stronger signals to meet the requirements of a higher signal-to-noise ratio or deeper exploration. It also produces longer-duration signals, thereby providing better resolution. Additionally, for the interpretation of TEM data, three-dimensional forward modeling plays a crucial role. However, the mature traditional TEM forward method is based on a simple, sometimes inappropriate model, as it is well established that the induced polarization (IP) effect is widely present in the deep earth, especially in oil and gas reservoirs. The presence of the IP effect results in negative responses in field data that do not conform to the traditional theoretical decay law of TEM, which can significantly impact data processing and inversion results. To address this issue, a TEM forward modeling method considering the IP effect based on the spectral element method (SEM) has been developed in this study. Firstly, starting from the Helmholtz equation satisfied by the time domain electric field, we introduce the Debye model with polarization information into the forward modeling by utilizing the differential form of Ohm’s law. As a result, we derive the boundary value problem for the time domain electric field that considers the induced polarization effect. Using Gauss–Lobatto–Legendre (GLL) polynomials as the basis functions, the SEM is employed to discretize the governing equations at each time step and obtain spectral element discretization equations. Then, temporal discretization equations are derived using the second-order backward Euler formula, and the linear system of equations is solved using the Pardiso direct solver. Finally, the electromagnetic responses at any time channel are obtained via SEM interpolation and numerical integration, thereby achieving three-dimensional TEM forward modeling considering the IP effect. The results indicate that this method can effectively reflect the spatial distribution of polarizable subsurface media. It provides valuable references for studying the polarization parameters of subsurface media and performing a three-dimensional inversion of TEM data considering the induced polarization effect.

Instability of energy spectrum disturbance for ship turbulent wakes: SAR imaging simulation and analysis

We present a ship turbulent wake synthetic aperture radar (SAR) simulator (STW-SARSim) producing two-dimensional true-scale turbulent wakes in SAR images that extend for kilometers. In this paper, we innovatively overcome the sea surface time-varying problem in solving the spectral energy balance equation by transforming the unsteady problem into a steady problem and considering the dark trailing centerline region of the turbulent wake in SAR images as a texture modulation effect of the Bragg wave for sea surface scattering. We also present SAR images simulated in three different bands: (i) Sentinel-1A C-band, (ii) TerraSAR X-band, and (iii) ALOS PALSAR L-band to validate the reliability of our proposed method and analyze the effects of wind speed, wind direction, and dissipation rates on the turbulent wake simulation. This paper provides new insights into the simulation of turbulent wakes in SAR images of large scenes inspired by recent research on detection algorithms. It sets the stage for future dataset generation for detection.

A three-term projection method based on spectral secant equation for nonlinear monotone equations

A three-term projection method based on spectral secant equation for nonlinear monotone equations

On the Propagation Model of Two-Component Nonlinear Optical Waves

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions.