Spectral Decomposition Research Articles

Edge-illumination spectral phase-contrast tomography

Following the rapid, but independent, diffusion of x-ray spectral and phase-contrast systems, this work demonstrates the first combination of spectral and phase-contrast computed tomography (CT) obtained by using the edge-illumination technique and a CdTe small-pixel (62 μm) spectral detector. A theoretical model is introduced, starting from a standard attenuation-based spectral decomposition and leading to spectral phase-contrast material decomposition. Each step of the model is followed by quantification of accuracy and sensitivity on experimental data of a test phantom containing different solutions with known concentrations. An example of a micro CT application (20 μm voxel size) on an iodine-perfused ex vivo murine model is reported. The work demonstrates that spectral-phase contrast combines the advantages of spectral imaging, i.e. high-Z material discrimination capability, and phase-contrast imaging, i.e. soft tissue sensitivity, yielding simultaneously mass density maps of water, calcium, and iodine with an accuracy of 1.1%, 3.5%, and 1.9% (root mean square errors), respectively. Results also show a 9-fold increase in the signal-to-noise ratio of the water channel when compared to standard spectral decomposition. The application to the murine model revealed the potential of the technique in the simultaneous 3D visualization of soft tissue, bone, and vasculature. While being implemented by using a broad spectrum (pink beam) at a synchrotron radiation facility (Elettra, Trieste, Italy), the proposed experimental setup can be readily translated to compact laboratory systems including conventional x-ray tubes.

Resolvent model for aeroacoustics of trailing edge noise

This study presents a physics-based, low-order model for the trailing edge (TE) noise generated by an airfoil at low angle of attack. The approach employs incompressible resolvent analysis of the mean flow to extract relevant spanwise-coherent structures in the transitional boundary layer and near wake. These structures are integrated into Curle’s solution to Lighthill’s acoustic analogy to obtain the scattered acoustic field. The model has the advantage of predicting surface pressure fluctuations from first principles, avoiding reliance on empirical models, but with a free amplitude set by simulation data. The model is evaluated for the transitional flow (Re=5e4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Re} = 5e4$$\\end{document}) around a NACA0012 airfoil at 3 deg angle of attack, which features TE noise with multiple tones. The mean flow is obtained from a compressible large eddy simulation, and spectral proper orthogonal decomposition (SPOD) is employed to extract the main hydrodynamic and acoustic features of the flow. Comparisons between resolvent and SPOD demonstrate that the physics-based model accurately captures the leading coherent structures at the main tones’ frequencies, resulting in a good agreement of the reconstructed acoustic power with that of the SPOD (within 4 dB). Discrepancies are observed at high frequencies, likely linked to nonlinearities that are not considered in the resolvent analysis. The model’s directivity aligns well with the data at low Helmholtz numbers, but it fails at high frequencies where the back-scattered pressure plays a significant role in directivity. This modeling approach opens the way for efficient optimization of airfoil shapes in combination with low-fidelity mean flow solvers to reduce TE noise.Graphical abstract

A maturational frequency discrimination deficit may explain developmental language disorder.

Auditory perceptual deficits are widely observed among children with developmental language disorder (DLD). Yet, the nature of these deficits and the extent to which they explain speech and language problems remain controversial. In this study, we hypothesize that disruption to the maturation of the basilar membrane may impede the optimization of the auditory pathway from brainstem to cortex, curtailing high-resolution frequency sensitivity and the efficient spectral decomposition and encoding of natural speech. A series of computational simulations involving deep convolutional neural networks that were trained to encode, recognize, and retrieve naturalistic speech are presented to demonstrate the strength of this account. These neural networks were built on top of biologically truthful inner ear models developed to model human cochlea function, which-in the key innovation of the present study-were scheduled to mature at different rates over time. Delaying cochlea maturation qualitatively replicated the linguistic behavior and neurophysiology of individuals with language learning difficulties in a number of ways, resulting in (a) delayed language acquisition profiles, (b) lower spoken word recognition accuracy, (c) word finding and retrieval difficulties, (d) "fuzzy" and intersecting speech encodings and signatures of immature neural optimization, and (e) emergent working memory and attentional deficits. These simulations illustrate many negative cascading effects that a primary maturational frequency discrimination deficit may have on early language development and generate precise and testable hypotheses for future research into the nature and cost of auditory processing deficits in children with language learning difficulties. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Numerical study of multiple hydraulic fractures propagation in poroelastic media based on energy decomposition phase field methods

This paper proposes a novel phase-filed model for simulating hydraulic fracturing in poroelastic media under complex stress conditions. The main theoretical contribution lies in the fact that a generalized strain energy density decomposition is incorporated to hydraulic fracturing phase field model, which unifies hydrostatic-deviatoric and spectral decompositions and predicts material failure through classical Mohr-Coulomb failure criterion. Based on Biot’s poroelastic theory, an indicator function is used to distinguish between fluid flow in fractures and in pores. The fully coupled equations are solved by combining staggered schemes with Newton-Raphson methods. The proposed model is validated by comparing laboratory experimental results with numerical results. In order to demonstrate the influence of introducing hydrostatic-deviatoric and spectral energy decompositions, we simulated propagation of multiple hydraulic fractures and coalescence of hydraulic fractures with natural fractures. Finally, the effects of stress ratios (σx/σy) on multiple hydraulic fractures are analysed. It is found that when the stress ratio is 2, a more complex hydraulic fracture network is formed in the porous media.

Identifying dominant flow structures in a bubbling gas-particle fluidized bed using the spectral proper orthogonal decomposition

This study applies the spectral proper orthogonal decomposition (SPOD) to analyze the spatio-temporal characteristics associated with the flow fields of a bubbling fluidized bed. The results suggest there is no singular dominant frequency linked to the highest energy levels; rather, there is a spectrum of frequencies. These frequencies are consistent with the natural frequency from published correlations. SPOD analysis allows for the capture of quasi-periodicity associated with the flow that arises from the semi-periodic evolution of bubbles in the bed and reveals the presence of multiple co-existing spatio-temporal patterns in the particle volume fraction and velocity fields. The dominant SPOD modes can be used for prediction, control, design, optimization, and reduced-order system representation. The capability of a few dominant modes to yield a low-rank reconstruction of the flow fields is demonstrated. The intermittent behavior of the bed is examined through frequency-time analysis of the gas pressure field.

Three-dimensional time-dependent convection model for asteroseismology

Context. Due to an ill-depicting model of the convective layers below the photosphere in 1D stellar models (structural contribution) and/or a misrepresentation of the coupling between convection and oscillations (modal contribution), a well-known deviation appears between observed and theoretical frequencies, which grows towards high frequencies; the so-called surface effects. While satisfying solutions have been found regarding the structural contribution, the accurate modeling of the modal effect still represents a challenge. Alongside the frequency, the interaction between convection and oscillations also impacts the damping rate of the modes and forms an important part of the driving mechanism behind the stellar oscillations of low-mass stars. With increasing observational capabilities at our disposal with Kepler and TESS, shortcomings in modeling constitute the main limitation to accurate seismic probing of solar-like and red giant stars. Aims. We present the formalism of an approach that changes the current paradigm by addressing three-dimensional space. This new formalism consists in an original nonadiabatic 3D time-dependent convection model for asteroseismology. Methods. We aim to keep the entire 3D structure of the astrophysical flow in these superficial layers in order to fully account for the nature of turbulence in our model via the use of advanced hydrodynamic simulation. We use the perturbative approach and introduce a spectral decomposition approach that results in an entirely new formalism describing standing waves in 3D. This formalism is set to solve the quasi-radial global nonadiabatic oscillation equations in a full 3D framework. Results. Based on physical assumptions, we establish an eigenvalue problem describing the 3D quasi-radial global nonadiabatic stellar oscillation. We also provide a prescription for its numerical resolution alongside a proposed iteration method for our formalism. Finally, we derive the peculiar 3D work integral and establish the expression of the damping rate. We show how our formalism offers the possibility to probe the complex structure of stars and is able to precisely locate regions of the driving and damping of the modes as well as their physical origin.

Embedded shear layers in turbulent boundary layers of a NACA0012 airfoil at high angles of attack

An investigation of turbulent boundary layers (TBLs) is presented for a NACA0012 airfoil at angles of attack 9 and 12 deg. Wall-resolved large eddy simulations (LES) are conducted for a freestream Mach number M=0.2 and chord-based Reynolds number Re=4×105, where the boundary layers are tripped near the airfoil leading edge on the suction side. For the angles of attack analyzed, mild, moderate and strong adverse pressure gradients (APGs) develop over the airfoil. Despite the strong APGs, the mean flow remains attached along the entire airfoil suction side. Similarly to other APG-TBLs investigated in the literature, a secondary peak appears in the Reynolds stress and turbulence production profiles. This secondary peak arises in the outer layer and, for strong APGs, it may overcome the first peak typically observed in the inner layer. The analysis of the turbulence production shows that other components of the production tensor become important in the outer layer besides the shear term. For moderate and strong APGs, the mean velocity profiles depict three inflexion points, the third being unstable under inviscid stability criteria. In this context, an embedded shear layer develops along the outer region of the TBL leading to the formation of two-dimensional rollers typical of a Kelvin–Helmholtz instability which are captured by a spectral proper orthogonal decomposition (SPOD) analysis. The most energetic SPOD spatial modes of the tangential velocity show that streaks form along the airfoil suction side and, as the APG becomes stronger, they grow along the spanwise and wall-normal directions, having a spatial support along the entire boundary layer.

The expected sample allele frequencies from populations of changing size via orthogonal polynomials

In this article, discrete and stochastic changes in (effective) population size are incorporated into the spectral representation of a biallelic diffusion process for drift and small mutation rates. A forward algorithm inspired by Hidden-Markov-Model (HMM) literature is used to compute exact sample allele frequency spectra for three demographic scenarios: single changes in (effective) population size, boom-bust dynamics, and stochastic fluctuations in (effective) population size. An approach for fully agnostic demographic inference from these sample allele spectra is explored, and sufficient statistics for stepwise changes in population size are found. Further, convergence behaviours of the polymorphic sample spectra for population size changes on different time scales are examined and discussed within the context of inference of the effective population size. Joint visual assessment of the sample spectra and the temporal coefficients of the spectral decomposition of the forward diffusion process is found to be important in determining departure from equilibrium. Stochastic changes in (effective) population size are shown to shape sample spectra particularly strongly.

Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers

An improved resolvent analysis is proposed in the regime of compressible turbulent boundary layers. To better model nonlinear processes in the input, the resolvent framework is augmented by adding eddy viscosity. To this end, we propose two eddy-viscosity models: a modified Cess eddy-viscosity model coupling the compressibility transformation and outer-layer correction, and a new eddy-viscosity model based on an empirical relationship and mixing-length theory. Both are incorporated into the resolvent operator to examine the performance of the eddy-viscosity-improved resolvent-based reduced-order modelling. Results of the augmented resolvent analysis are compared qualitatively and quantitatively with the first leading mode of spectral proper orthogonal decomposition, by checking the profiles and cross-spectral densities of velocities, density and temperature in two hypersonic turbulent boundary layers under different wall conditions. Higher accuracy of the turbulence prediction is achieved by adding the proposed eddy-viscosity models, particularly for the energetic cycle in the outer-layer region where strong nonlinear energy transfer exists.

Seismic attributes and instantaneous spectral analysis-based temperature-porosity simulations: An example of imaging shale gas-bearing basin floor fans of early-cCretaceous delta sequences, Southeast Asian Basin

Basin floor fans (BFF) of deltaic depositional systems are important stratigraphic traps, which are sourced by the regionally developed meandering channel belts and associated deposits of point bar facies. The vertical and lateral changes in the tectonics and sea level adversely affect the temperature (θ) and porosity constraints of petroleum plays within BFF of the lowstand system tracts. These fluctuations impact the sedimentary controls on thickness, lithology-impedance contrast, phase, and frequency of BFF. As a result of the reliance on full-spectrum seismics, quantifying thin-bedded stratigraphic wet gas-bearing intervals (WGI) and light oil-bearing intervals of petroleum systems becomes extremely difficult. This study utilizes seismic attributes and instantaneous spectral analysis-based temperature-porosity reservoir simulations (TPRS) to quantify the BFF of the Early-Cretaceous delta of the Miano gas field, Central Pakistan. The 47-Hz peak-tuned frequency component of spectral decomposition [SpecDecomp] images a 154 km2 NEE-SWW traversed and regionally developed stratigraphic prospect. Linear regression of R2 > 0.80 between spectral decomposition-based tuning frequency of 47-Hz and gross reservoir thickness of BFF validates the presence of gas-bearing shales. The 47-Hz wavelet-based TPRS has resolved the phase reversals and seismic-doublet signatures for thin-bedded shale PETROS at θ of 170°F to 200°F and porosities of 22 % to 32 % of WGI within BFF. TPRS at 47-Hz SpecDecomp profile between the frequency bandwidth spectrums of 12–62-Hz has reappraised the exact morphology of the continental shelf, shelf-slope, and BFF. A 21 m thick aggradational parasequence of BFF was deposited at an angle of 2° during the onset of subsidence and rapid fall, followed by a subsequent standstill of sea-level at θ of 170 to 200°F, ∼26 to 35 % porosities, and laterally fractured subsidized regimes of WGI.. A 20 m thick retrogradational fluvial-dominated facies was deposited at an angle of 35° during a rising followed by a standstill of sea level at a θ of 140 to 165°F, ∼08 % to 18 % porosity, and poorly fractured uplifted compressional regimes of light oil-bearing intervals. The aggradational WGI of Lowstand System Tracts was completely embedded by retrogradational shales of transgressed system tracts from the top of the aggradational and laterally. The uncertainty analysis has prevailed with a moderate-to-good R2 > 0.75 for BFF and a very poor R2 < 0.1 for fluvial-dominated point bar traps. Our workflow has implications in terms of exploring the tectonics and stratigraphic frameworks within the petroleum systems for imaging the most productive zones in a single snapshot of 47-Hz tuning frequency.

Optimal Diophantine exponents for SL(n)

The Diophantine exponent of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the SLn(Z[1/p])-action on the generalized upper half-space SLn(R)/SOn(R), lies in [1,1+O(1/n)], substantially improving upon Ghosh–Gorodnik–Nevo's method which gives the above range to be [1,n−1]. We also show that the exponent is optimal, i.e. equals one, under the assumption of Sarnak's density hypothesis. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the temperedness of the underlying representations, the crucial assumption in Ghosh–Gorodnik–Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local L2-norms of the Eisenstein series.

Experimental study on acoustic resonance of subsonic and slightly underexpanded impinging jets

The aeroacoustic feedback loops in high-speed circular jets that impinge on a large flat plate are investigated via acoustic measurements and schlieren visualizations. In the present experiments, the nozzle pressure ratio ranges from 1.39 to 2.20, the corresponding ideally expanded jet Mach number $M_j$ is from 0.70 to 1.12 and the nozzle-to-plate distance ( $H$ ) is from 4.0 $D$ to 6.0 $D$ , where $D$ is the nozzle exit diameter. The results of acoustic measurements show that the strongest tones are generated in a limited frequency band. The empirical dispersion relations obtained from the fluctuating greyscales along the jet centreline of time-resolved schlieren images have good agreement with the dispersion relations from the vortex-sheet model. The coherent flow structures at tonal frequencies are extracted by spectral proper orthogonal decomposition and are analysed in detail. For the $M_j&lt;0.82$ jets, the upstream-propagating guided jet mode is progressively confined to the potential core of jets with increasing tonal frequency, which provides the first direct experimental support for theoretical results. The evolution in the structures of acoustic resonance loops is studied along a single frequency stage of axisymmetric impinging tones. When the acoustic resonance between the upstream- and downstream-propagating guided jet modes is formed at tonal frequencies, the impinging tones are intenser. Slightly underexpanded impinging jets can simultaneously produce impingement tones and screech tones. Shock-cell structures have modulatory effects on the downstream-propagating Kelvin–Helmholtz wavepacket and the upstream- and downstream-propagating guided jet modes. Due to the interaction between the flow structures at the frequencies of impinging and screech tones, tones of axisymmetric modes can be produced outside the frequency ranges in which the axisymmetric upstream-propagating guided jet modes are supported by jets.

Improving model choice in classification: an approach based on clustering of covariance matrices

This work introduces a refinement of the Parsimonious Model for fitting a Gaussian Mixture. The improvement is based on the consideration of clusters of the involved covariance matrices according to a criterion, such as sharing Principal Directions. This and other similarity criteria that arise from the spectral decomposition of a matrix are the bases of the Parsimonious Model. We show that such groupings of covariance matrices can be achieved through simple modifications of the CEM (Classification Expectation Maximization) algorithm. Our approach leads to propose Gaussian Mixture Models for model-based clustering and discriminant analysis, in which covariance matrices are clustered according to a parsimonious criterion, creating intermediate steps between the fourteen widely known parsimonious models. The added versatility not only allows us to obtain models with fewer parameters for fitting the data, but also provides greater interpretability. We show its usefulness for model-based clustering and discriminant analysis, providing algorithms to find approximate solutions verifying suitable size, shape and orientation constraints, and applying them to both simulation and real data examples.

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

This paper presents a method for parameterizing new Lorentz spacetime coordinates based on coupled parameters. The role of symmetry in rapidity in special relativity is explored, and invariance is obtained for new spacetime intervals with respect to the Lorentz transformation. Using the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz spacetime coordinates are obtained. The descriptions of particle motion using the old and new Lorentz spacetime coordinates as applied to plane laser pulses are compared in terms of the particle kinetic energy. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and that of a plane laser pulse, rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions. The new and old Lorentz spacetime coordinates are compared with the Fermi spacetime coordinates. The proper Lorentz groups SO(1,3) with coupled parameters using the old and new Lorentz spacetime coordinates are also compared. As a special case, the application of Lorentz spacetime coordinates to a relativistic hydrodynamic system with coupled parameters in 1+1 dimensions is demonstrated.

Machine Learning Application of Generalized Gaussian Radial Basis Function and Its Reproducing Kernel Theory

Gaussian Radial Basis Function Kernels are the most-often-employed kernel function in artificial intelligence for providing the optimal results in contrast to their respective counterparts. However, our understanding surrounding the utilization of the Generalized Gaussian Radial Basis Function across different machine learning algorithms, such as kernel regression, support vector machines, and pattern recognition via neural networks is incomplete. The results delivered by the Generalized Gaussian Radial Basis Function Kernel in the previously mentioned applications remarkably outperforms those of the Gaussian Radial Basis Function Kernel, the Sigmoid function, and the ReLU function in terms of accuracy and misclassification. This article provides a concrete illustration of the utilization of the Generalized Gaussian Radial Basis Function Kernel as mentioned earlier. We also provide an explicit description of the reproducing kernel Hilbert space by embedding the Generalized Gaussian Radial Basis Function as an L2−measure, which is utilized in implementing the analysis support vector machine. Finally, we provide the conclusion that we draw from the empirical experiments considered in the manuscript along with the possible future directions in terms of spectral decomposition of the Generalized Gaussian Radial Basis Function.

Quasinormal modes of Kerr black holes using a spectral decomposition of the metric perturbations

Quasinormal modes of Kerr black holes using a spectral decomposition of the metric perturbations

Probing solution conformations of l-DOPA: Integration of experiment and simulation via vibrational optical activity

l-DOPA plays a critical role as a precursor to dopamine and is a standard treatment for Parkinson's disease. Recent research has highlighted the potential therapeutic advantages of deuterated l-DOPA analogs having a longer biological half-life. For their spectroscopic characterization, the in-detail characterization of l-DOPA itself is necessary. This article presents a thorough examination of the vibrational spectra of l-DOPA, with a particular emphasis on chirally sensitive VOA techniques. We successfully obtained high-quality Raman and ROA spectra of l-DOPA in its cationic form, under low pH conditions, and at a high concentration of 100 mg/ml. These spectra cover a broad spectral range, allowing for precise comparisons with theoretical simulations. We also obtained IR and VCD spectra, but they faced limitations due to the narrow accessible spectral region. Exploration of l-DOPA's conformational landscape revealed its intrinsic flexibility, with multiple coexisting conformations. To characterize these conformations, we employed two methods: one involved potential energy surface scans with implicit solvation, and the other utilized molecular dynamics simulations with explicit solvation. Comparing ROA spectra from different conformer groups and applying spectral decomposition proved crucial in determining the correct conformer ratios. The use of explicit solvation significantly improved the quality of the final simulated spectral profiles. The accurate determination of conformer ratios, rather than solely relying on the number of averaged spectra, played a crucial role in simulation accuracy. In conclusion, our study offers valuable insights into the structure and conformational behavior of l-DOPA and represents a valuable resource for subsequent spectroscopic studies of its deuterated analogs.

The state of microcirculation in patients with periodontal diseases and disorders of the occlusal relationships of the dentition

Background. The study of microcirculation disorders, as one of the key factors in the pathogenesis of periodontal diseases, is relevant in clinical dentistry. Premature contacts, acting as provoking factors for occlusal disorders, cause overload of individual teeth and aggravate the severity of inflammatory periodontal pathology. Objective registration of capillary blood flow and associated changes in the microvessels of the arteriolar and venular units in patients with periodontopathies and occlusal dis-orders will allow us to assess the degree and nature of tissue blood flow disorders in the gum tissue.Goal. Increasing the efficiency of diagnosing microcirculatory disorders in periodontal tissues in patients with chronic generalized mild periodontitis and disorders of the occlusal relationships of the dentition.Materials and methods. Using laser Doppler flowmetry (LDF), the state of mi-crocirculation in the periodontium was studied in 78 people aged 21–35 years with intact dentition and physiological occlusion, of which 33 people (group 1) had clinically healthy periodontium and the absence of occlusal disorders (group 1), with generalized chronic pulmonary periodontitis degree and premature occlusal contacts – 45 people (group 2). The microcirculation index (M), standard deviation (σ), and coefficient of variation (КV) were assessed. To obtain objective diagnostic data, amplitude-frequency analysis of the harmonic rhythms of LDF-grams was used with spectral decomposition into harmonic components of physiological oscillations of tissue blood flow (wavelet analysis).Results. In patients of group 2, in comparison with patients of group 1, a decrease in the level of periodontal blood perfusion, a reduction in the fluctuation of erythro-cyte flow and vasomotor activity of microvessels is determined, while a decrease in “active” and an increase in «passive» modulation of tissue blood flow contributes to an increase in the volume of circulating blood at arteriovenular anastomoses when the intensity of blood flow through the vessels of the capillary bed decreases.Conclusions. Patients of group 1 were diagnosed with a normoemic type of LDF-gram, which provides a relatively high degree of intensity of tissue blood flow, balance of regulatory mechanisms, and optimal trophism of periodontal tissues. In patients of group 2 with a spastic type of LDF-gram, characterized by a reduced level of perfusion, flux, vasomotor activity of microvessels, indicators of tissue blood flow and trophic function are significantly reduced.

On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

Let A and E be self-adjoint matrices or operators on ℓ2(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2({{\\mathbb {N}}})$$\\end{document}, where A is fixed and E is a small perturbation. We study how the eigenvalues of A+E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+E$$\\end{document} depend on E, with the aim of obtaining second order formulas that are explicitly computable in terms of the spectral decomposition of A and a certain block decomposition of E. In particular we extend the classical Rayleigh-Schrödinger formulas for the one-parameter perturbation A+tE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+tE$$\\end{document} where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in {{\\mathbb {R}}}$$\\end{document} varies and E is held fixed, by dropping t and considering E as the variable.

Spectral feature extraction of rocket exhaust plume using spectral proper orthogonal decomposition

The spectral characterization of flow-field parameters provides a new perspective for understanding the spatiotemporal evolution of unsteady supersonic exhaust plumes and for extracting typical structures. In this study, a large-eddy simulation is performed to calculate the three-dimensional unsteady supersonic plume flow field of rocket engines, and a spectral proper orthogonal decomposition (SPOD) method with a spatiotemporal separation is established. This approach is used to extract the coherent structural features of the unsteady exhaust plume flow field and analyze the mode space structure at different frequencies. The three-dimensional reconstruction and denoising of the exhaust plume flow-field parameters can be achieved via the frequency- and time-domain reconstructions of the SPOD algorithm and oblique projection method, respectively. The ground rocket exhaust plume of ballistic evaluation motor-II is analyzed. The results indicate that the SPOD method can effectively extract the single-frequency mode structure of the reactive supersonic flow field, and that low-order behavior appears in the m = 0 and m = 1 azimuth modes. The potential core exhibits a high-frequency wave-packet structure that is affected by shock waves and shear layers. Time-domain reconstruction based on the oblique projection method facilitates the capture of the dynamic characteristics of the flow field. For the first-order SPOD mode, the frequency- and time-domain reconstruction errors are 3.3% and 1.5%, respectively. The frequency-domain reconstruction method exhibits a 4% improvement in denoising ability compared to low-pass filters. This study provides a novel method for the spectral characterization and spatiotemporal feature extraction of supersonic exhaust plume flow fields.