Eigenvalue Decomposition Research Articles

Spectrum of inner product type integral transformers on Hilbert spaces

In this paper, we have determined the spectrum of inner product type integral transformers on the Hilbert spaces. This has been achieved using the abelian properties of sets and the concept of commutators as in the von Neumann algebras. We have also shown that the spectrum of the inner product type integral transformers is contained in the closure of its numerical range using the classical spectral inclusion theorem.

The spectral representation method: A framework for simulation of stochastic processes, fields, and waves

The Spectral Representation Method (SRM) was developed in the 1970s to simulate Gaussian stochastic processes and fields from a Fourier series expansion according to the Spectral Representation Theorem. Since those early developments, the SRM has continuously evolved into a comprehensive framework for the simulation of stochastic processes, fields, and waves with a rigorous theoretical foundation. Its major advantages are conceptual simplicity and computational efficiency. In the 1990s, much of the theory for simulation of Gaussian stochastic processes, fields, and waves was firmly established and early methods for simulation of non-Gaussian processes, fields, and waves were introduced. In the 2000s and 2010s, methods that coupled the SRM with Translation Process Theory were improved to enable efficient and accurate simulations of stochastic processes, fields, and waves with strongly non-Gaussian marginal probability distributions. More recently, the SRM was extended for higher-order non-Gaussian processes, fields, and waves by extending the Fourier stochastic expansion to include non-linear wave interactions derived from higher-order spectra. This paper reviews the key theoretical developments related with the SRM and provides the relevant algorithms necessary for its practical implementation for the simulation of stochastic processes, fields, and waves that can be either stationary or non-stationary, homogeneous or non-homogeneous, one-dimensional or multi-dimensional, scalar or multi-variate, Gaussian or non-Gaussian, or any combination thereof. The paper concludes with some brief remarks addressing the open research challenges in SRM-based theory and simulations.

Coupled critical plane-pseudo excitation method for multiaxial fatigue analysis of structures under random vibration

This study proposes a coupled critical plane–pseudo excitation method for assessing multiaxial fatigue damage in mechanical structures subjected to random loads. Within the modal coordinate framework, we derive an expression for the spectral moment of the structural stress response using the pseudo excitation method, which facilitates the decoupling of spatial characteristics from frequency domain properties. For the input power spectral density, represented as an integer power function, we employ eigenvalue decomposition and the residue theorem to derive an analytical expression for the spectral moment of the modal displacement response. By constructing equivalent stress modes based on the maximum shear stress and normal stress criteria, we obtain an expression for the equivalent stress variance. Intelligent optimization algorithms are utilized to determine the critical plane position and the corresponding equivalent stress spectral moments, followed by the application of the Dirlik method for fatigue assessment. A numerical example involving random vibration multiaxial fatigue analysis of an L-shaped thin-walled plate demonstrates the effectiveness of the proposed coupled critical plane–pseudo excitation method in comparison to the traditional critical plane method that relies on the stress covariance matrix for calculating equivalent variance. The results confirm the accuracy and efficiency of our approach, and we further explore the impact of different forms of the excitation power spectral density and the damping ratios on the computational outcomes.

Ringing Out General Relativity: Quasinormal Mode Frequencies for Black Holes of Any Spin in Modified Gravity.

After black holes collide, the remnant settles to a stationary state by emitting gravitational waves. Once nonlinearities subside, these ringdown waves are dominated by exponentially damped sinusoids, or quasinormal modes. We develop a general method using perturbative spectral expansions to calculate the quasinormal-mode frequencies and damping times in a wide class of modified gravity theories for black holes with any subextremal spin. We apply this method to scalar-Gauss-Bonnet gravity to show its accuracy, thus enabling robust ringdown tests with gravitational wave data.

Evaluating the Impact of Interferogram Networks on the Performance of Phase Linking Methods

In recent years, phase linking (PL) methods in radar time-series interferometry (TSI) have proven to be powerful tools in geodesy and remote sensing, enabling the precise monitoring of surface displacement and deformation. While these methods are typically designed to operate on a complete network of interferograms, generating such networks is often challenging in practice. For instance, in non-urban or vegetated regions, decorrelation effects lead to significant noise in long-term interferograms, which can degrade the time-series results if included. Additionally, practical issues such as gaps in satellite data, poor acquisitions, or systematic errors during interferogram generation can result in incomplete networks. Furthermore, pre-existing interferogram networks, such as those provided by systems like COMET-LiCSAR, often prioritize short temporal baselines due to the vast volume of data generated by satellites like Sentinel-1. As a result, complete interferogram networks may not always be available. Given these challenges, it is critical to understand the applicability of PL methods on these incomplete networks. This study evaluated the performance of two PL methods, eigenvalue decomposition (EVD) and eigendecomposition-based maximum-likelihood estimator of interferometric phase (EMI), under various network configurations including short temporal baselines, randomly sparsified networks, and networks where low-coherence interferograms have been removed. Using two sets of simulated data, the impact of different network structures on the accuracy and quality of the results was assessed. These patterns were then applied to real data for further comparison and analysis. The findings demonstrate that while both methods can be effectively used on short temporal baselines, their performance is highly sensitive to network sparsity and the noise introduced by low-coherence interferograms, requiring careful parameter tuning to achieve optimal results across different study areas.

Analytic computation for stationary distributions of D-BMAP/D -MSP(a,b)/1 queueing system

ABSTRACT This paper analyses single server correlated batch arrival and correlated batch service queue with innumerable waiting space in discrete-time. The arrival and service processes are governed by discrete batch Markovian arrival process and discrete Markovian service process with general bulk service rule, respectively. Observing two consecutive random epochs, we construct general structure transition probability matrix and it is then reblocked to the desired M/G/1 structure. The UL-type RG-factorization approach combined with spectral expansion method is adopted to unravel the queue length distribution at random epoch. The relationships are undertaken to obtain the queue length distributions at pre-arrival, outside observer's, intermediate and post-departure epochs. Our most challenging effort is to derive the probability mass functions of waiting time and service batch size for arbitrary customer in an arriving batch. We address the warehouse management process for finished products from dealer to retailer as a potential practical use of our batch queue with correlation in both arrival and service, and provide the effect of our queueing behaviour in this application based on our analytical and computational investigation. Based on the implementation of parametrized numeric experiments with numerous categories, we impart our computational results to validate the analytical findings reported in this investigation.

Quantum algorithm for dynamic mode decomposition integrated with a quantum differential equation solver

We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O(polylogN) for an N-dimensional system. This is an exponential speedup over known classical algorithms with at least O(N) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Spectral expansion methods for prediction uncertainty quantification in systems biology

Uncertainty is ubiquitous in biological systems. For example, since gene expression is intrinsically governed by noise, nature shows a fascinating degree of variability. If we want to use a model to predict the behaviour of such an intrinsically stochastic system, we have to cope with the fact that the model parameters are never exactly known, but vary according to some distribution. A key question is then to determine how the uncertainties in the parameters affect the model outcome. Knowing the latter uncertainties is crucial when a model is used for, e.g., experimental design, optimisation, or decision-making. To establish how parameter and model prediction uncertainties are related, Monte Carlo approaches could be used. Then, the model is evaluated for a huge number of parameters sets, drawn from the multivariate parameter distribution. However, when model solutions are computationally expensive this approach is intractable. To overcome this problem, so-called spectral expansion (SE) methods have been developed to quantify prediction uncertainty within a probabilistic framework. Such SE methods have a basis in, e.g., computational mathematics, engineering, physics, and fluid dynamics, and, to a lesser extent, systems biology. The computational costs of SE schemes mainly stem from the calculation of the expansion coefficients. Furthermore, SE effectively leads to a surrogate model which captures the dependence of the model on the uncertainty parameters, but is much simpler to execute compared to the original model. In this paper, we present an innovative scheme for the calculation of the expansion coefficients. It guarantees that the model has to be evaluated only a restricted number of times. Especially for models of high complexity this may be a huge computational advantage. By applying the scheme to a variety of examples we show its power, especially in challenging situations where solutions slowly converge due to high computational costs, bifurcations, and discontinuities.

Spectral behavior and expansion dynamics of Gd plasma generated by dual-pulse laser irradiation

Laser-produced gadolinium plasma (Gd-LPP) emerges as a promising candidate for next-generation nanolithography light sources. In this study, a dual laser pulse scheme was implemented to achieve a narrow spectral peak. By varying the pre-main pulse delay and pre-pulse laser energy, optimal conditions of 40 ns delay and 50 mJ energy were identified to improve spectral purity. Radiation hydrodynamics simulations revealed that the improved spectral purity stems from a flatter density gradient at the ablation front and a lower average electron density in the EUV emission region. Additionally, reheating the pre-formed plasma with a short main pulse mitigated plasma squeezing, resulting in an even lower electron density and thus improved spectral purity. Our findings suggest that spectral narrowing in the dual-pulse scheme, essential for better matching with multilayer reflection bandwidths, can be optimized through precise control of pre-pulse energy, pre-main delay, and main-pulse duration.

Algorithm for Designing Waveforms Similar to Linear Frequency Modulation Using Polyphase-Coded Frequency Modulation

Linear frequency modulation (LFM) waveforms have been widely adopted due to their excellent performance characteristics, such as good Doppler tolerance and ease of physical implementation. However, LFM waveforms suffer from high autocorrelation sidelobes (ACSLs) and limited design flexibility. Phase-coded frequency modulation (PCFM) waveforms can be used to design waveforms similar to LFM, offering greater design flexibility to optimize ACSLs. However, it has been found that the initial PCFM waveform experiences spectral expansion during the ACSL optimization process, which reduces its similarity to LFM. Therefore, this article jointly optimizes the ACSLs and spectrum of the initial PCFM waveform, establishes an optimized mathematical model, and then solves it using the heavy-ball gradient descent algorithm. Numerical experiments indicate that the proposed method effectively addresses the problem of waveform similarity degradation caused by spectral expansion while reducing waveform ACSLs. At the same time, a balance between reducing waveform ACSLs and preserving waveform similarity can be achieved by adjusting the parameters.

Efficient Discrete Clustering With Anchor Graph.

Spectral clustering (SC) has been applied to analyze varieties of data structures over the past few decades owing to its outstanding breakthrough in graph learning. However, the time-consuming eigenvalue decomposition (EVD) and information loss during relaxation and discretization impact the efficiency and accuracy especially for large-scale data. To address above issues, this brief proposes a simple and fast method named efficient discrete clustering with anchor graph (EDCAG) to circumvent postprocessing by binary label optimization. First of all, sparse anchors are adopted to accelerate graph construction and obtain a parameter-free anchor similarity matrix. Subsequently, inspired by intraclass similarity maximization in SC, we design an intraclass similarity maximization model between anchor-sample layer to cope with anchor graph cut problem and exploit more explicit data structures. Meanwhile, a fast coordinate rising (CR) algorithm is employed to alternatively optimize discrete labels of samples and anchors in designed model. Experimental results show excellent rapidity and competitive clustering effect of EDCAG.

Evaluating genotype by environment interaction for growth traits in Limousine cattle

Environmental conditions affect the growth and health of animals, making it crucial to quantify heat stress and the genetic basis of heat tolerance in animal breeding. The main objective of this study was to evaluate heat stress on beef cattle growth and investigate the genetic background of tolerance to harsh environmental conditions in the Italian Limousine beef cattle. Three growth traits were analysed: average daily gain (ADG), weaning weight (WW), and yearling weight (YW). Data were collected from animals raised between 1991 and 2022 and combined with 14 environmental covariates. Records for ADG, WW, and YW encompassed 108 205, 100 058, and 24 939 individuals, respectively, with 4 617, 4 670, and 2 048 genotyped individuals. Climatic variables were compared for inclusion in a linear mixed model using the Deviance Information Criterion. Multiple-trait models and genomic information incorporated environmental conditions with the largest impact on the studied traits Genotype by environment interaction (G × E) was detected in all the studied traits, showing substantial heterogeneity of the variance components across the different environments (Env). Heritability for WW remains constant among Env; instead, for ADG and YW decreased under uncomfortable environmental conditions. YW showed the lowest genetic correlation (0.28) between divergent conditions (Env 2 and Env 5,) for ADG and WW correlations dropped below 0.50 among Env. The values of genetic correlations indicate that growth traits are moderately to strongly affected by G × E. Eigenvalue decomposition of the additive genetic (co)variance matrix for ADG, WW, and YW indicated that three components accounted for over 0.80 of the proportion of the variance explained, suggesting different animal performances across Env. Spearman rank correlations showed potential re-ranking of genotyped sires, because ADG, WW, and YW showed correlations between Env below 0.80. The accuracy of single-step genomic EBV was higher compared to EBV for al traits. Overall, the result confirms the existence of G × E for growth traits in the Italian Limousine population. Including G × E in the model allows for more environment-aware predictions, helping breeders understand how different genetic bases respond to varying conditions. Genomic predictions incorporating G × E could accelerate genetic gains and improve response to selection for heat tolerance.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

Nonlinear dynamics of the semi-infinite ferromagnetic samples with an easy-plane anisotropy

The modification of the inverse scattering problem is proposed to investigate solitons and dispersive waves in the framework of the Landau–Lifshitz model for semi-infinite ferromagnet with an ¡¡easy-plane¿¿ anisotropy under the mixed boundary conditions, corresponding to different degrees of spin pinning on the edge of the sample. New types of solitons are obtained and their elastic reflection from the edge of the sample is analyzed. Spectral expansions of the integrals of motion for solitons and waves are found. Additional integrals of motion are established that guarantee true boundary conditions for solitons, when they interact with the edge of the sample.

LS-based multipath channel measurement method using software defined radio platform

Multipath channel measurement is the foundation of multipath channel modeling and analysis. Recently, the software defined radio (SDR) has provided new ideas for the design of multipath channel measurement systems due to universality and reconfigurability. Therefore, this paper introduces a multipath channel measurement method applicable to various SDR platforms. At first, to enhance the resolution of multipath measurements, a signal processing approach combining the least squares (LS) estimation algorithm within the SDR framework is proposed. Furthermore, to mitigate LS estimation errors stemming from high correlation among baseband signals, a scrambled measurement signal is introduced and its effectiveness is demonstrated through eigenvalue decomposition. Next, the systematic errors of In-phase and Quadrature(IQ) imbalance and bandwidth limitation are further analyzed and addressed within the SDR framework. Finally, the proposed measurement method is implemented based on a practical universal software radio peripheral (USRP) device and validated under simulated and real channels. The proposed measurement method mainly focuses on the baseband signal processing within the software component of SDR, making it applicable to most general USRP devices. The experiment results demonstrate that multipath channel measurement results with high accuracy can be obtained.

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

Form-finding of tensegrity structures based on graph neural networks

Tensegrity structures, characterized by enhanced stiffness, slender struts, and superior buckling resistance, have found wide-ranging applications in fields such as engineering, architecture, art, biology, and robotics, attracting extensive attention from researchers. The form-finding process, a critical step in the design of tensegrity structures, aims to discover the self-equilibrated configuration that satisfies specific design requirements. Traditional form-finding methods based on force density often require repeated steps of eigenvalue decomposition and singular value decomposition, making the process complex. In contrast, this paper introduces a new intelligent form-finding algorithm that uses the force density method and combines the Coati optimization algorithm with Graph Neural Networks. This algorithm avoids the complex steps of eigenvalue and singular value decomposition and integrates the physical knowledge of the structure, making the form-finding process faster and more accurate. In this algorithm, various force densities are initially randomized and input into a trained Graph Neural Networks to predict a fitness function’s value. Through optimizing the constrained fitness function, the algorithm determines the appropriate structural force density and coordinates, thereby completing the form-finding process of the structure. The paper presents seven typical tensegrity structure examples and compares various form-finding methods. The results of numerical examples show that the method proposed in this paper can find solutions that align with the super-stable line more quickly and accurately, demonstrating its potential value in practical applications.

Spectral Domain Optical Theorem and Radiative Transfer in Random Media With Large-Scale Fluctuations

Spectral Domain Optical Theorem and Radiative Transfer in Random Media With Large-Scale Fluctuations