Eigenvalue Distribution Research Articles

Computing the Tracy-Widom Distribution for Arbitrary $\beta>0$

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

Spectrum Situation Awareness for Space-Air-Ground Integrated Networks Based on Tensor Computing.

The spectrum situation awareness problem in space-air-ground integrated networks (SAGINs) is studied from a tensor-computing perspective. Tensor and tensor computing, including tensor decomposition, tensor completion and tensor eigenvalues, can satisfy the application requirements of SAGINs. Tensors can effectively handle multidimensional heterogeneous big data generated by SAGINs. Tensor computing is used to process the big data, with tensor decomposition being used for dimensionality reduction to reduce storage space, and tensor completion utilized for numeric supplementation to overcome the missing data problem. Notably, tensor eigenvalues are used to indicate the intrinsic correlations within the big data. A tensor data model is designed for space-air-ground integrated networks from multiple dimensions. Based on the multidimensional tensor data model, a novel tensor-computing-based spectrum situation awareness scheme is proposed. Two tensor eigenvalue calculation algorithms are studied to generate tensor eigenvalues. The distribution characteristics of tensor eigenvalues are used to design spectrum sensing schemes with hypothesis tests. The main advantage of this algorithm based on tensor eigenvalue distributions is that the statistics of spectrum situation awareness can be completely characterized by tensor eigenvalues. The feasibility of spectrum situation awareness based on tensor eigenvalues is evaluated by simulation results. The new application paradigm of tensor eigenvalue provides a novel direction for practical applications of tensor theory.

ANALYSIS OF SPATIAL CORRELATION PROPERTIES AND RECEIVED SIGNAL CHARACTERISTICS OF LARGE DIMENSIONAL RIS-ASSISTED COMMUNICATION IN NEXT GENERATION RADIO NETWORKS

The reconfigurable intelligent surface (RIS) is envisioned to be a key serving technology in 6G wireless systems and it is also known as smart repeater or holographic radio. RIS is fully capable of controlling the EM waves in terms of reflections, refractions, scattering, amplitudes, phases, and polarization. Its response is adaptive over time as well as network conditions. It enhances the performance of a radio system in terms of capacity, energy efficiency, security, power consumption, and coverage. In this paper, a physically realizable RIS-assisted channel model is presented which uses the spatial correlation properties of the channel. The main objective of this work is to derive the fading distribution for RIS-assisted channels, characterize their spatial channel correlation, and minimize the channel training overhead. The proposed channel estimation model minimizes the number of required pilot signals for channel estimation. The channel properties are analyzed in terms of spatial correlation matrices rank, RIS physical geometry, and channel hardening. Monte Carlo simulations have verified the analytical results. The results proved that eigenvalue distribution and rank of spatial correlated matrix are favorable for lower dimensions of RIS reflecting elements.

Neumann Problem for Second-Order Differential Equation with Fractional Derivative in the Analysis and Modeling of Structures Made of Viscoelastic Elements

This article addresses a second-order differential equation containing a Gerasimov-Kaputo fractional differentiation operator of order less than two. The Neumann problem is formulated for this equation. A system of eigenfunctions and eigenvalues for the considered homogeneous boundary problem of the second kind is found. A conjugate boundary problem for the Gerasimov-Kaputo fractional derivative is introduced. A biorthogonal system is obtained that is orthogonal to the found system of eigenfunctions. Visualizations of the eigenfunction system, biorthogonal system, and an example of eigenvalue distribution on the real axis are provided.

A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

<abstract><p>Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.</p></abstract>

Dynamics of a delayed reaction–diffusion predator–prey model with nonlocal competition and double Allee effect in prey

In this paper, we study the effects of the nonlocal competition and double Allee effect in prey on a diffusive predator–prey model. We investigate the local stability of coexistence equilibrium in the predator–prey model by analyzing the eigenvalue spectrum. We study the consequence of double Allee effect on the prey population. Also, we discuss the existence of Hopf bifurcation under different parameters by using the gestation time delay of predators as a bifurcation parameter and analyzing the distribution of eigenvalues. By utilizing the normal form method and center manifold theorem, we have given some conditions that could determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Through our research, we obtain that the double Allee effect can affect the coexistence of prey and predator and induce periodic oscillations of the densities of prey and predator. In addition, the nonlocal competition can also affect the stability and homogeneity of the solutions, but may receive the consequences of the Allee effect as well as time delay. In the numerical simulation part, we further demonstrate the correctness of this conclusion by comparison.

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1].To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.

On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.

Distribution of eigenvalues of a perturbed differentiation operator on the interval

In this paper, we construct a characteristic determinant of the spectral problem of a first-order differential equation on an interval with an integral perturbation in the boundary value condition, which is an entire analytic function of the spectral parameter. Based on the formula for the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the perturbed spectral problem depending on the modulus of continuity of the subinteral function.

Stability Analysis of Autonomous Microgrid Operated at Constant-Frequency with the Consideration of Reactive Power Balance

Large-scale autonomous microgrids have potential application values as they can increase renewable energy penetration level without compromising the stability of the existing large power systems. Before their widespread implementation, critical issues like stability analysis etc need to be solved. This paper analyses the stability in an autonomous microgrid operated at constant frequency with the consideration of reactive power balance. Difference equations of reactive power for the grid-forming generator are constructed separately from those for the grid-supporting and grid-feeding generators while the difference equations of the real power for all the generators are the same. For the voltage source inverter with its current controlled by proportional resonant controller, at the fundamental frequency, its output current is disentangled from its terminal voltage and is controlled to trace its reference accurately, namely . Therefore, each inverter can be modelled as an equivalent current source and the equivalent circuits for d-component and q-component can be separated from each other. Then, the nodal equations in matrix form for the microgrid system can be established readily. With these, the system level state-space equations are built to study the distribution of eigenvalues. By choosing proper coefficients for real power and reactive power reference generations and controller’s parameters, one can make all the eigenvalues falling in the left-hand-side of the complex plane. Therefore, the system is stable. Such a research paves the way for systematically searching good sets of coefficients and controller’s parameters which make system operate safely away from unstable region with necessary margin.

A Helmholtz-type decomposition for the space of symmetric matrices

In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better understanding of the strain constraint space, which is important to the Navier–Stokes regularity problem. In particular, we give a full characterization the orthogonal complement of the strain constraint space and investigate the geometry of the eigenvalue distribution of matrices in the strain constraint space.

On Connected Graphs with Distance Eigenvalue −1 of Multiplicity at Least n−4

Let [Formula: see text] denote the set of all connected graphs on [Formula: see text] vertices having distance eigenvalue [Formula: see text] of multiplicity [Formula: see text]. By using the distribution of the third largest distance eigenvalue and the second least distance eigenvalue of a connected graph, in this paper we completely characterize the graphs in [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

Rotated block diagonal preconditioners for Navier-Stokes control problems

Fluid flow control problems play a crucial role in various industrial applications. The optimal control of the Navier-Stokes equations poses a significant challenge. By employing Oseen's approximation to linearize the problem, we encounter a series of large sparse structured linear systems. These coefficient matrices exhibit a two-by-two block structure with square blocks. Leveraging this block structure, we exploit matrix splitting preconditioners. Theoretical analysis indicates that the real parts of all eigenvalues of preconditioned matrix are equal to 1/2. A large number of eigenvalues are clustered in (1±i)/2. The imaginary part of the eigenvalues is bounded explicitly. The eigenvalue distribution predicts the fast convergence of Krylov subspace methods. To avoid solving the saddle point subsystems in the preconditioning procedure, we propose a practical variant of the preconditioner. The theoretical analysis demonstrates that if the approximation is sufficiently close to Schur complement, the eigenvalues of the modified preconditioned matrix will lie within a circle centered at the original eigenvalues with a radius less than 1. This implies that the modified preconditioner exhibits excellent performance in terms of preconditioning. The numerical experiments demonstrate that the generalized minimal residual method, when combined with the proposed preconditioners, proves to be an efficient and effective solution for Oseen's control optimization with a variable viscosity coefficient. The number of iterations required by the preconditioned methods remains unaffected by the mesh size used in finite element discretization and only marginally depends on the regularization parameter.

On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue α in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to μαn and σα2n respectively. It is also shown that μα and σα2 are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue 0, the constants μ0 and σ02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

Cooperative Spectrum Sensing Algorithm Based on Eigenvalue Fusion

A novel algorithm is introduced to improve collaborative spectrum sensing under low cognitive capabilities and insufficient signal-to-noise ratio. The algorithm is based on the difference of random matrix eigenvalues and uses the theory of random eigenvalues and the extreme distribution of the minimum eigenvalue. It makes use of the average, both arithmetic and geometric, as well as the minimum and maximum values of eigenvalues as the detection metric. It calculates the fusion power parameter through local energy spectrum sensing. Simulation results demonstrate that the algorithm outperforms the DMM algorithm and the NMME algorithm under users with low cognitive capabilities and Insufficient signal-to-noise ratio, making it more suitable for low signal-to-noise ratio environments.

On the limits of distinguishing seabed types via ambient acoustic sound

This article presents a theoretical analysis of optimally distinguishing among environmental parameters from ocean ambient sound. Recent approaches to this problem either focus on parameter estimation or attempt to classify the environment into one of many known types through machine learning. This classification problem is framed as one of hypothesis testing on the received ambient sound snapshots. The resulting test depends on the Kullback-Leibler divergence (KLD) between the distributions corresponding to different environments or sediment types. Analysis of the KLD shows the dependence on the signal-to-noise ratio, the underlying signal subspace, and the distribution of eigenvalues of the respective covariance matrices. This analysis provides insights into both when and why successful hypothesis testing is possible. Experiments demonstrate that our analysis provides insight as to why certain environmental parameters are more difficult to distinguish than others. Experiments on sediment types from the Naval Oceanographic Office Bottom Sediment type database show that certain types are indistinguishable for a given array configuration. Further, the KLD can be used to provide a quantitative alternative to examining bottom loss curves to predict array processing performance.

Reservoir concatenation and the spectrum distribution of concatenated reservoir state matrices

Reservoir computing, one of the state-of-the-art machine learning architectures, processes time-series data generated by dynamical systems. Nevertheless, we have realized that reservoir computing with the conventional single-reservoir structure suffers from capacity saturation. This leads to performance stagnation in practice. Therefore, we propose an extended reservoir computing architecture called reservoir concatenation to further delay such stagnation. Not only do we provide training error analysis and test error comparison of reservoir concatenation, but we also propose a crucial measure, which is the trace associated with a reservoir state matrix, that explains the level of responsiveness to reservoir concatenation. Two reservoir dynamics are compared in detail, one by using the echo state network and the other by using a synchronization model called an explosive Kuramoto model. The distinct eigenvalue distributions of the reservoir state matrices from the two models are well reflected in the trace values that are shown to account for the different reservoir capacity behaviors, determining the different levels of responsiveness.

A dimension expanded preconditioning technique for block two-by-two linear equations

Abstract In this article, we introduce a novel block preconditioner for block two-by-two linear equations by expanding the dimension of the coefficient matrix. Theoretical results on the eigenvalues distribution of the preconditioned matrix are obtained, and a feasible implementation is discussed. Some numerical examples, including the solution of the Navier-Stokes equations, are presented to support the theoretical findings and demonstrate the preconditioner’s efficiency.

Region stability analysis and precise tracking control of linear stochastic systems

This paper provides a novel region stability criterion on the basis of the pole (eigenvalue) distribution of the linear stochastic system (LSS). Corresponding stabilization control and tracking control are also discussed, which are more suitable for practical engineering applications. First, in the light of the connection between the eigenvalues of the system and the dynamic properties of LSSs, the concept of region stability is given to describe the finer dynamic behavior of LSSs. The criterion for the region stability is also presented. Second, the region stabilization method is investigated according to region stability, which can ensure that the system satisfies a certain dynamic performance by setting the eigenvalues in a certain convex region. Third, a precise tracking control method of LSSs is addressed as an application of region stabilization. It can regulate the steady state properties and transient response of the tracking signal more precisely. Two examples are given to demonstrate the effectiveness of the new methods.

Algebra preconditionings for 2D Riesz distributed‐order space‐fractional diffusion equations on convex domains

AbstractWhen dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and ‐preconditioners and of the generalized minimal residual with split circulant and ‐preconditioners and conduct related numerical experiments.