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The Laplacian eigenvalues and polynomials of the networks play an essential role in understanding the relations between the topology and the dynamic of networks. Generally, computation of the Laplacian spectrum of a network is a hard problem and there are just a few classes of graphs with the property that their spectra have been completely computed. Laplacian spectrum for n-prism networks was investigated in [Liu et al., Neurocomputing 198 (2016) 69-73]. In this paper, we give a method for calculating the eigenvalues and characteristic polynomial of the Laplacian matrix of a generalized n-prism network. We show how such large networks can be constructed from small graphs by using graph products. Moreover, our results are used to obtain the Kirchhoff index and the number of the spanning trees in the generalized n-prism networks. We also give some examples of applications, that explain the usefulness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper concerns a diffusive predation model with nonlinear growth, cross-diffusion and protection zone terms. The main purpose is to investigate the effects of nonlinear growth and cross-diffusion on the coexistent solution when protection zone is present. Firstly, a priori estimate and the existence of positive solutions are discussed, including local and global existence. Then, some asymptotic properties of coexistent solutions induced by the mortality rate, nonlinear growth of predator and cross-diffusion are analyzed. It is revealed that there exist critical values related to certain principal eigenvalues such that the nonlinear growth, cross-diffusion and protection zone all have significant effects on the coexistent solutions; as far as the nonlinear growth concerned, we find that it has important influences on the coexistence region of two species undoubtedly. Biologically, this implies that these critical values greatly affect the survival of species. [ABSTRACT FROM AUTHOR]

Let Aα(G) be the Aα-matrix of a digraph G and λα1,,λα2…,λαn be the eigenvalues of Aα(G). Let ρλ(G) be the Aλ spectral radius of G and Eλ(G) = Σi=1nλαi² be the Aα energy of G by using second spectral moment. Let Gnm be the set of non-strongly connected digraphs with n vertices containing a unique strong component with m vertices and some directed trees hanging on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal Aα spectral radius and the maximal (or minimal) Aα energy in Gnm. [ABSTRACT FROM AUTHOR]

Racah matrices of quantum algebras are of great interest at present time. These matrices have a relation with matrices, which are much simpler than the Racah matrices themselves. This relation is known as the eigenvalue conjecture. In this paper we study symmetries of Racah matrices which follow from the eigenvalue conjecture for multistrand braids. [ABSTRACT FROM AUTHOR]

In this paper, we consider matrix Schrödinger equation, dynamical Schrödinger equation and matrix KdV. We construct their explicit solutions using our GBDT version of Bäcklund–Darboux transformation and square roots of the generalised matrix eigenvalues. A separate section is dedicated to several examples including the case of strongly singular potentials. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the stability of a vortex with azimuthal velocity profile V ˉ = 1 âˆ' 1 âˆ' ε r 2 e âˆ' r 2 / r. When ε  = 0, the Lambâ€"Oseen vortex model is recovered. Although the Lambâ€"Oseen vortex supports propagating waves known as Kelvin waves, the flow is stable according to Rayleigh’s circulation criterion. In this paper, on the other hand, the modified vortex profile admits linearly unstable disturbances for ε  > 0 and we investigate their characteristics. These may be either axisymmetric or non-axisymmetric, but we find that the axisymmetric perturbations have the largest growth rates. When their growth rates are small, it becomes very difficult to solve the linear equation governing the axisymmetric perturbations because the eigenfunctions have a rapid exponential growth as one moves outward radially from the vortex center. To deal with such cases, a modified Riccati transformation was employed and found to be effective in solving the associated eigenvalue problem. [ABSTRACT FROM AUTHOR]

Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson–Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) and then root-finding a Lagrange interpolation polynomial. In this paper, we obtain expressions for the RDM elements directly in terms of the EBV. The final expressions can be computed at the same cost as the rapidity expressions. Therefore, except, in particular, circumstances, it is entirely unnecessary to compute rapidities at all. The RDM elements require numerically inverting a matrix, and while this is usually undesirable, we demonstrate that it is stable, except when there is degeneracy in the single-particle energies. In such cases, a different construction would be required. [ABSTRACT FROM AUTHOR]

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation L u = f ∈ L 2 (Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue λ p , s ( ω δ ^) with respect to the domain ( ω δ ^) . Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi–Perera–Squassina–Yang [The Brezis–Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192–199] to the fractional p-Laplacian setting. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following quadratic pencil of Schrödinger operators L(λ) generated in L 2 (ℝ +) by the equation with the boundary condition y ′ (0) y (0) = β 1 λ + β 0 α 1 λ + α 0 , where p(x)and q(x) are complex valued functions and α0, α1, β0, β1 are complex numbers with α 0 β 1 − α 1 β 0 ≠ 0 . It is proved that L(λ) has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions and sup 0 ≤ x < + ∞ { e ε x [ | p ′ (x) | + | q ′ ′ (x) | ] } < + ∞ hold, where ε> 0. [ABSTRACT FROM AUTHOR]

This paper aims to consider the extended Perron complements for the collection of Mmatrices. We first exhibit the connection between the extended Perron complements of M-matrices and nonnegative matrices. Moreover, we present some common inequalities involving extended Perron complements, Schur complements, and principal submatrices of irreducibleM-matrices by utilizing the properties of M-matrices. We also discuss the monotonicity of the extended Perron complements and minimum eigenvalue. For the collection of M-matrices, we demonstrate that all (extended) Perron complements are M-matrices. Especially, we deduce that M-matrices and their Perron complements share the same minimum eigenvalue. Finally, a simple example is presented to illustrate our findings. [ABSTRACT FROM AUTHOR]

Summary: The modular multilevel converter (MMC) is a novel prospective multilevel converter topology for high‐voltage or high‐power applications. This paper presents the 3‐ Φ model integrated into a linear model in transient analysis and the design of current controller for MMC. This paper analyzes the control interactions of MMC on nearby turbine generator. The eigenvalue analysis is used to evaluate the impact of different controller modes of MMC. The impact of converter control with nonlinear switches is analyzed through the transient simulation. The eigenvalue analysis and controller design are carried out on a linear model, and the transient simulation analysis is performed on both the D‐Q model and a three‐phase model of MMC using MATLAB‐Simulink. The results show that the transient simulation of three‐phase model on the proposed approach is consistent with the transient simulation of D‐Q model. Also, the transient simulation results show a satisfactory response with the current controller parameters of the MMC at various operating cases. To validate the results of MATLAB‐Simulink model, the proposed system is implemented in hardware‐in‐the‐loop (HIL) simulation with an integrated subsystem built in two real‐time simulators. The subsystem of mathematical model is implemented in dSPACE, and the subsystem of 3‐ Φ MMC with controller is implemented in Typhoon. The HIL simulation results show a satisfactory response and consistent with MATLAB‐Simulink model results. [ABSTRACT FROM AUTHOR]

The aim of the current paper is to study a partially described inverse eigenvalue problem of a specific symmetric matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by the minimal eigenvalue of all leading principal submatrices and eigenpair (λ (n) 2, x) such that λ (n) 2 is the maximal eigenvalue of the required matrix. We investigate conditions for the solvability of the problem, and finally an algorithm and its numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, a new geometric inversion method is proposed for obtaining the elastic parameters of an anisotropic focal regions More precisely, the eigenvalues of a vertical transversely isotropic elasticity tensor of a focal region are inverted up to a constant for a given only one moment tensor with the accuracy depending on the strength of anisotropy. The reason for using only one moment tensor is that although there might occur several earthquakes in the same focal region, the orientation of the sources is similar to each other. Hence, one cannot obtain independent equations from each earthquake in order to use them in the inversion of elastic parameters of the focal region. Moreover, this method can be applied for real‐time inversion once the moment tensor of an earthquake is evaluated. The inversion method relies on the geometric fact that a moment tensor can be written as a linear combination of the eigenvectors of the anisotropic focal region's elasticity tensor. Then, in the inversion, we use the fact that each coefficient of this unique decomposition is proportional to the eigenvalues of the focal region's elasticity tensor. Two approximations are used in this inversion method, in particular for the potency and the source orientations. The strength of anisotropy of the focal region determines how accurate these approximations are, and, hence, it also determines the resolutions of the inverted eigenvalues. Because of the anisotropy of the focal region, the errors in the inversion do depend on orientations of the dip and rake angles but not on the strike angle since the focal region is vertically transversely isotropic. The accuracy of the inversion for the five parameters of vertical transversely isotropic is shown for every 10°‐gridded orientation of the dip and rake angles on the stereographic projection. The results are very promising along some orientations as shown in the figures. The last section of the paper deals with the inversion of eigenvalues, up to a constant, for a given set of moment tensors; not by using only one moment tensor. It turns out that the best fit corresponds to the average of inversions obtained for different orientations of the sources. [ABSTRACT FROM AUTHOR]

This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral radius of signless p-Laplacian of a graph, which is a generalization of the quadratic form of the signless Laplacian matrix for p = 2. Let G n , β be the set of simple graphs of order n with a given matching number β. In this paper, the graphs maximizing the largest signless p-Laplacian eigenvalue among G n , β are obtained. [ABSTRACT FROM AUTHOR]

For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using Pk - Pk-1 element (k = 1) is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order O(do f -2k d). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the eigenvalues and the eigenfunctions of both the q -Durrmeyer operators Dn,q and the limit q -Durrmeyer operators D∞,q introduced by V. Gupta in the case 0 < q < 1. All moments for Dn,q and D∞,q are provided. The coefficients for the eigenfunctions of the operators are explicitly derived and the eigenfunctions of these operators are illustrated by graphical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study some comparisons of Dirichlet, Neumann and buckling eigenvalues on Riemannian manifolds. By introducing a new parameter, we provide some new relationships, which improve corresponding results of Ilias and Shouman in [Calc. Var. Partial Differential Equations, 2020, 59: Paper No. 127, 15 pp.] in some sense. [ABSTRACT FROM AUTHOR]

In this article we present our results from the SC19 Student Cluster Competition Reproducibility Challenge. The challenge entails reproducing the article entitled “Computing Planetary Interior Normal Modes with A Highly Parallel Polynomial Filtering Eigensolver” presented at SC’18, which proposes a parallel polynomial filtered Lanczos algorithm to directly calculate the planetary normal modes of heterogeneous planets. The proposed algorithm showed excellent performance with relatively low memory consumption and high parallel efficiency. In this work, we reproduce the scaling tests in that article on a cluster using Intel Cascade Lake architecture and use the proposed algorithm to illustrate specific normal modes of Mars. We compare the results obtained on our cluster with those in the original article. We also design a new metric to better analyze the results. In addition, we use the profiling tool Intel VTune Amplifier to explain our discoveries. Our results demonstrate that the given models show great scalability, which is similar to the original article. The required normal modes of Mars are also successfully calculated and visualized. [ABSTRACT FROM AUTHOR]

This paper presents a dynamic event data-based stability risk assessment method for power grids with high penetrations of inverter-based resources (IBRs). This method relies on obtaining the IBRs’ DQ admittance through dynamic event data and computing the system’s eigenvalues based on the admittance models. Two critical technologies are employed in this research, including time-domain and frequency-domain data fitting and $dq$ -frame voltage and current signal derivation. The first technology is key to obtaining the $s$ -domain expressions from the transient response data, and the $s$ -domain DQ admittance model from the frequency-domain measurements. The second technology is key to obtaining the $dq$ -frame voltage and current signals from either the three-phase instantaneous measurements or the phasor measurement unit (PMU) data. The method is illustrated using data generated from a Type-4 wind power plant modeled in PSCAD. This paper demonstrates the technical feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

It is well known that one-dimensional integral equations of the convolution type considered on a finite interval cannot generally be solved by quadratures in contrast to similar equations considered on the entire line or on the half-line. For this reason, when studying their spectrum, some asymptotic method must be used. The paper considers a convolution-type integral operator with a logarithmic kernel defined on a finite interval. Using the Fourier transform, the problem is successively reduced to a conjugation problem and to a singular integral equation on the half-line in which the integral operator is not a contraction. It is shown that the leading part of the integral equation thus obtained can be inverted explicitly. The cases of even and odd eigenfunctions are considered, in each of which the asymptotics of the eigenvalues and eigenfunctions of the original operator is found. [ABSTRACT FROM AUTHOR]

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In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment. [ABSTRACT FROM AUTHOR]

Power system often experiences the problem of low-frequency electromechanical oscillations which leads the system to unstable condition. The problem can be corrected by implementing power system stabilizers (PSSs) in the excitation control system of alternator. This paper provides a novel and efficient approach to design an Improved Grasshopper Optimization Algorithm (IGOA)-based dual-input controller to damp the inter-area-mode power system oscillations. A three-fold optimization criterion has been formulated to calculate the optimum values of the controllers required for power system stability. The damping performance of the proposed controller is compared with conventional PSS and genetic algorithm-based controllers to validate the better performance of the proposed IGOA-based controller under various system loading conditions and disturbances. [ABSTRACT FROM AUTHOR]

Background: Spatial complexity is always associated with spatial autocorrelation. Spatial autocorrelation coefficients including Moran's index proved to be an eigenvalue of the spatial correlation matrixes. An eigenvalue represents a kind of characteristic length for quantitative analysis. However, if a spatial correlation process is based on self-organized evolution, complex structure, and the distributions without characteristic scale, the eigenvalue will be ineffective. In this case, a scaling exponent such as fractal dimension can be used to compensate for the shortcoming of characteristic length parameters such as Moran's index. Method: This paper is devoted to finding an intrinsic relationship between Moran's index and fractal dimension by means of spatial correlation modeling. Using relative step function as spatial contiguity function, we can convert spatial autocorrelation coefficients into spatial autocorrelation functions. Result: By decomposition of spatial autocorrelation functions, we can derive the relation between spatial correlation dimension and spatial autocorrelation functions. As results, a series of useful mathematical models are constructed, including the functional relation between Moran's index and fractal parameters. Correlation dimension proved to be a scaling exponent in the spatial correlation equation based on Moran's index. As for empirical analysis, the scaling exponent of spatial autocorrelation of Chinese cities is Dc = 1.3623±0.0358, which is equal to the spatial correlation dimension of the same urban system, D2. The goodness of fit is about R2 = 0.9965. This fractal parameter value suggests weak spatial autocorrelation of Chinese cities. Conclusion: A conclusion can be drawn that we can utilize spatial correlation dimension to make deep spatial autocorrelation analysis, and employ spatial autocorrelation functions to make complex spatial autocorrelation analysis. This study reveals the inherent association of fractal patterns with spatial autocorrelation processes. The work may inspire new ideas for spatial modeling and exploration of complex systems such as cities. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact delay range making input-delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete-time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well-known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems. [ABSTRACT FROM AUTHOR]

This paper examines the distribution of eigenvalues for a 2 × 2 random confusion matrix used in machine learning evaluation. We also analyze the distributions of the matrix's trace and the difference between the traces of random confusion matrices. Furthermore, we demonstrate how these distributions can be applied to calculate the superiority probability of machine learning models. By way of example, we use the superiority probability to compare the accuracy of four disease outcomes machine learning prediction tasks. [ABSTRACT FROM AUTHOR]

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it. [ABSTRACT FROM AUTHOR]

In this paper, we solve the bound state problem for the Varshni-Hellmann potential via a useful technique. In our technique, we obtain the bound state solution of the Schrödinger equation for the Varshni-Hellmann potential via ansatz method. We obtain the energy eigenvalues and the corresponding eigenfunctions. Also, the behavior of the energy spectra for both the ground and the excited state of the two body systems is illustrated graphically. The similarity of our results to the accurate numerical values is indicative of the efficiency of our technique. [ABSTRACT FROM AUTHOR]

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops A G S will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix A G S − σ n I , where σ denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph G S is explored. In addition, the existence of a graph with self-loops for every odd energy is proved. [ABSTRACT FROM AUTHOR]

In this paper, we determine the variation formula for the first eigenvalue of (p , q) -biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R 0 + . More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences. [ABSTRACT FROM AUTHOR]

In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions. [ABSTRACT FROM AUTHOR]

The comaximal graph Γ (R) of a commutative ring R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if u R + v R = R. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring ℤ n and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of ℤ n . [ABSTRACT FROM AUTHOR]

Random matrix theory, particularly using matrices akin to the Wishart ensemble, has proven successful in elucidating the thermodynamic characteristics of critical behavior in spin systems across varying interaction ranges. This paper explores the applicability of such methods in investigating critical phenomena and the crossover to tricritical points within the Blume–Capel model. Through an analysis of eigenvalue mean, dispersion, and extrema statistics, we demonstrate the efficacy of these spectral techniques in characterizing critical points in both two and three dimensions. Crucially, we propose a significant modification to this spectral approach, which emerges as a versatile tool for studying critical phenomena. Unlike traditional methods that eschew diagonalization, our method excels in handling short timescales and small system sizes, widening the scope of inquiry into critical behavior. [ABSTRACT FROM AUTHOR]