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In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis. • A discrete analogue of an equality (1.3) by Grisvard [1] on H 2 functions is proved for the reduced rectangular Morley (RRM for short in the sequel) element functions. This discrete equality makes the RRM space usable for bi-Laplacian problems with inhomogeneous coefficients. • Based on piecewise quadratic polynomials, the RRM scheme is the lowest-degree finite element scheme for the inhomogeneous bi-Laplace problems. Compared to other kinds of methods, it does not need tuning parameter or using indirect differential operators. • As revealed by [3] , the RRM element space does not admit a locally-defined projective interpolator. In this paper, however, a locally-defined stable interpolator (not projective) is carefully constructed for the RRM element space, and an optimal approximation is proved rigorously on both convex and nonconvex domains. [ABSTRACT FROM AUTHOR]

The paper discusses systems with viscoelastic elements that exhibit repeated eigenvalues in the eigenvalue problem. The mechanical behavior of viscoelastic elements can be described using classical rheological models as well as models that involve fractional derivatives. Formulas have been derived to calculate first- and second-order sensitivities of repeated eigenvalues and their corresponding eigenvectors. A specific case was also examined, where the first derivatives of eigenvalues are repeated. Calculating derivatives of eigenvectors associated with repeated eigenvalues is complex because they are not unique. To compute their derivatives, it is necessary to identify appropriate adjacent eigenvectors to ensure stable control of eigenvector changes. The derivatives of eigenvectors are obtained by dividing them into particular and homogeneous solutions. Additionally, in the paper, a special factor in the coefficient matrix has been introduced to reduce its condition number. The provided examples validate the correctness of the derived formulas and offer a more detailed analysis of structural behavior for structures with viscoelastic elements when altering a single design parameter or simultaneously changing multiple parameters. [ABSTRACT FROM AUTHOR]

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. [ABSTRACT FROM AUTHOR]

Installing multi-camera systems and inertial measurement units (IMUs) in self-driving cars, micro aerial vehicles, and robots is becoming increasingly common. An IMU provides the vertical direction, allowing coordinate frames to be aligned in a common direction. The degrees of freedom (DOFs) of the rotation matrix are reduced from 3 to 1. In this paper, we propose a globally optimal solver to calculate the relative poses and scale of generalized cameras with a known vertical direction. First, the cost function is established to minimize algebraic error in the least-squares sense. Then, the cost function is transformed into two polynomials with only two unknowns. Finally, the eigenvalue method is used to solve the relative rotation angle. The performance of the proposed method is verified on both simulated and KITTI datasets. Experiments show that our method is more accurate than the existing state-of-the-art solver in estimating the relative pose and scale. Compared to the best method among the comparison methods, the method proposed in this paper reduces the rotation matrix error, translation vector error, and scale error by 53%, 67%, and 90%, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first present multiple numerical simulations of the anti-symmetric matrix in the stability criteria for fractional order systems (FOSs). Subsequently, this paper is devoted to the study of the admissibility criteria for descriptor fractional order systems (DFOSs) whose order belongs to (0, 2). The admissibility criteria are provided for DFOSs without eigenvalues on the boundary axes. In addition, a unified admissibility criterion for DFOSs involving the minimal linear matrix inequality (LMI) variable is provided. The results of this paper are all based on LMIs. Finally, numerical examples were provided to validate the accuracy and effectiveness of the conclusions. [ABSTRACT FROM AUTHOR]

When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily as computational tools, and their behavior is typically characterized using linear upper bounds, or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding of Krylov subspace methods, and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and the generalized minimal residual (GMRES) method, it presents mathematically important as well as practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods. The paper additionally addresses the motivation of preconditioning. [ABSTRACT FROM AUTHOR]

It is assumed that a sheet of paper performs transverse vibrations as a plate having additional stiffness due to static tension in its plane. The first eigenmodes are determined. The setup for experimental investigation of the nodal lines of the standing waves and for the determination of eigenmodes using projection moire is created for the analysis of unsymmetric loading of the paper. The comparison of the theoretical and experimental results of investigations is performed. The obtained results could be used in the process of design of the elements of the printing device. [ABSTRACT FROM AUTHOR]

In this communication we provide our answers to the comments by Usoskin (2017) on our recent paper (Popova et al, 2017a). We show that Principal Component Analysis (PCA) allows us to derive eigen vectors with eigen values assigned to variance of solar magnetic field waves from full disk solar magnetograms obtained in cycles 21–23 which came in pairs. The current paper (Popova et al, 2017a) adds the second pair of magnetic waves generated by quadruple magnetic sources. This allows us to recover a centennial cycle, in addition to the grand cycle, and to produce a closer fit to the solar and terrestrial activity features in the past millennium. [ABSTRACT FROM AUTHOR]

In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the Laplacian eigenvalues of the extension of the zero divisor graph, Γ e (Z n) , for some particular values of n. We characterize the values of n that give the equality of the spectral radius and the second-smallest eigenvalue of Γ e (Z n). Finding Wiener index of Γ e (Z n) in terms of its Laplacian eigenvalues is another objective of this paper. [ABSTRACT FROM AUTHOR]

The quantum system behaviour in the presence of a potential is best described by the Schrödinger equation. Maturation of analytical methods in solving the Schrödinger equation for various potentials provides a theorist with powerful tools for better understanding the behaviour of quantum systems. With this objective in mind, this paper introduces a novel potential, called the generalized hyperbolic cotangent potential. This potential is interesting because it encompasses a few exponential-type potentials. A quantum system interacting with such a potential, therefore, warrants careful study. The paper presents analytical solutions to the bound state problem of the generalized hyperbolic cotangent potential for arbitrary l states using a newly established quantization rule. Towards this end, the analytical formula is derived for the energy eigenvalue and respective eigenstate. Special cases of the analytical formula of the energy spectrum are discussed for the generalized hyperbolic cotangent potential. [ABSTRACT FROM AUTHOR]

Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10−3 and 10−9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports. [ABSTRACT FROM AUTHOR]

This paper discusses pseudospectra and stability radii for structured nonlinear matrix functions, such as Hermitian, skew‐Hermitian, H‐even, H‐odd, complex symmetric, and complex skew‐symmetric. To compute pseudospectra and stability radii, eigenvalue backward error is required. Hence, we initially present the structured eigenvalue backward error. Subsequently, we compute the structured pseudospectra using the obtained results for the eigenvalue backward error of a class of structured nonlinear matrix functions. Finally, we discuss the stability radii of the above‐structured problems arising in different applications. The paper also generalizes the results on the eigenvalue backward error of matrix polynomials in the literature for the above structures. [ABSTRACT FROM AUTHOR]

Laplacian matrix and its spectrum are commonly used for giving a measure in networks in order to analyse its topological properties. In this paper, Laplacian matrix of graphs and their spectrum are studied. Laplacian energy of a graph G of order n is defined as LE (G) = ∑ i = 1 n | λ i (L) - d ¯ | , where λ i (L) is the i-th eigenvalue of Laplacian matrix of G, and d ¯ is their average. If LE (G) = LE (K n) for the complete graph K n of order n, then G is known as L-borderenergetic graph. In the first part of this paper, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ 1 = { G b , j , k = [ (((j - 2) k - 2 j + 2) b + 1) K (j - 1) k - (j - 2) ] ∪ b (K j × K k) | b , j , k ∈ Z + } , Λ 2 = { G 2 , b = [ K 6 ∇ b (K 2 × K 3) ] ∪ (4 b - 2) K 9 | b ∈ Z + } , Λ 3 = { G 3 , b = [ b K 8 ∇ b (K 2 × K 4) ] ∪ (14 b - 4) K 8 b + 6 | b ∈ Z + } , where ∇ is join operator and × is direct product operator on graphs. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω 1 = { K 2 ∇ a K 2 r ¯ | a ∈ Z + } , Ω 2 = { a K 3 ∪ 2 (K 2 × K 3) ¯ | a ∈ Z + } and Ω 3 = { a K 5 ∪ (K 3 × K 3) ¯ | a ∈ Z + } , where G ¯ is the complement operator on G. [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]

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p -Laplacian for any p ∈ (1 , + ∞) (this paper covers the case p = 1 whereas the case p = + ∞ was already known). [ABSTRACT FROM AUTHOR]

We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ (Ω) of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ (Ω) of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ (Ω) or λ 1 ⁢ (Ω) when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ (Ω) and λ 1 ⁢ (Ω) among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ (Ω) and λ 1 ⁢ (Ω) interesting for themselves. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we study the topological effects of the global monopole spacetime on the energy eigenvalues of spherical symmetric potentials in the nonrelativistic regime. We deal with the radial equation by using the Wentzel, Kramers and Brillouim (WKB) approximation. In the cases where the energy levels of the ℓ -waves can be achieved, the WKB approximation is used based on the Langer transformation. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

In this paper, we concern the existence result of the following general eigenvalue problem: { 풜 ⁢ ( u ) = λ ⁢ ℬ ⁢ ( u ) in  ⁢ Ω , D α ⁢ ( u ) = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 풜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 ⁢ n {2n} and 2 ⁢ ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 풜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 풜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N-function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N-function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts. [ABSTRACT FROM AUTHOR]

AbstractThis paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to develop a formal foundation to design an adaptive output feedback predictor for a class of unknown systems where parameters and order are unknown or high-dimensional. We present a reduced-order adaptive output-predictor scheme based on modal reduction and Lyapunov's method. Moreover, the credibility of the proposed reduced-order adaptive output-predictor scheme is validated by mathematical proof and numerical studies. [ABSTRACT FROM AUTHOR]

In this paper, we study the lower and upper bounds for Stokes eigenvalues by finite element schemes. For the schemes studied here, roughly speaking, the loss of the local approximation property of the discrete velocity and pressure spaces may lead to different computed bounds of the eigenvalues. Formally theoretical analysis is constructed based on certain mathematical hypotheses, and numerical experiments are given to illustrate the validity of the theory. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the existence of a unique weak solution for a nonlocal Kirchhoff problem under Dirichlet boundary conditions, involving the fractional φ-Laplacian operator. Our major outcome is acquired by applying variational approaches and critical points theory. In addition, we analyse the spectrum and the eigenvalues associated to this problem. At the end and under some assumptions, we give an application to the previous problems. [ABSTRACT FROM AUTHOR]

Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of W 1 , 2 (Ω) into L 1 (∂ Ω). We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension. [ABSTRACT FROM AUTHOR]

For a nilpotent group G without π -torsion, and x , y ∈ G , if x n = y n for a π -number n , then x = y ; if x m y n = y n x m for π -numbers m , n , then x y = y x. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let m be a given positive integer and A a complex square matrix satisfying that (i) all eigenvalues of A are nonnegative, and (ii) rank ⁡ A 2 = rank ⁡ A ; then A has a unique m -th root X with rank ⁡ X 2 = rank ⁡ X , all eigenvalues of X are nonnegative, and moreover there is a polynomial f (λ) with X = f (A). In addition, let A and B be complex n × n matrices with all eigenvalues nonnegative, and rank ⁡ A 2 = rank ⁡ A , rank ⁡ B 2 = rank ⁡ B ; then (i) A = B when A r = B r for some positive integer r , and (ii) A B = B A when A s B t = B t A s for two positive integers s and t. [ABSTRACT FROM AUTHOR]

A graph G is called triangle-free if G does not contain any triangle as its induced subgraph. Let G n be the set of triangle-free graphs of order n each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in G n , each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of G n can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues. [ABSTRACT FROM AUTHOR]

This paper concentrates on the positive edge consensus in directed networks with spanning trees using output feedback protocols. First, a novel positive system observer is established, which introduces a parameter to enhance the design freedom of the observer. Furthermore, sufficient conditions are proposed, which are based solely on the network's edge count. Specifically, improved consensus and non-negative conditions are obtained by optimizing the constraints on the eigenvalue information. Subsequently, based on positive edge-consensus design conditions, a programming algorithm is developed. Finally, the effectiveness of the proposed control protocol is verified utilizing numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]

L ν operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of L ν 2 operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of L ν 2 operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the L I I operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate. [ABSTRACT FROM AUTHOR]

The energy of a graph is calculated by summing the absolute values of the eigenvalues found in its adjacency matrix. In this study, we present examples of line graphs with energy equivalent to the energy of a complete graph, which are called the borderenergetic graphs. As the examples of borderenergetic line graphs are rare, we introduce a new type of graphs called almost borderenergetic graphs whose energy differs from the borderenergetic energy by at most 1. In this paper, we begin by exploring the energy properties of line graphs derived from regular graphs and strongly regular graphs. Specifically, we establish a criterion for a line graph to exhibit borderenergetic characteristics when it consists of p many connected regular graphs and q many complete graphs, even when the original regular graph is disconnected. Additionally, we present a criterion for the line graph of a strongly regular graph to exhibit borderenergetic characteristics. To illustrate these concepts, we offer examples of connected borderenergetic graphs that are not necessarily complete. In the second part, we give the spectrum of the line graph of rK1∇K2, and show that it is an almost borderenergetic graph for r ≥ 5. [ABSTRACT FROM AUTHOR]

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems. [ABSTRACT FROM AUTHOR]

This paper aims to employ the Darboux transformation (DT) to discover the interaction solutions of the Zakharov equation (Eq. (1.2) for δ=1). Through partial degradation of eigenvalues, interaction solutions of the model are constructed on the basis of high-order breather solutions. The study derives interaction solutions involving breather and b-positon solutions through partial degradation of eigenvalues (λj→λ1). Further, interaction solutions comprising breather and lump solutions are obtained through partial double degradation of eigenvalues (λj→λ0). Then, several interaction solutions containing b-positon and lump solutions are extracted through mixed degradation of eigenvalues (λj→λ1 and λk→λ0). In particular, the dynamic evolution characteristics of these solutions are studied. It is believed that these studies make a significant contribution to the understanding of the Zakharov equation and its possible applications in physics. [ABSTRACT FROM AUTHOR]

We consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable. [ABSTRACT FROM AUTHOR]

Domain generalization (DG) aims to learn knowledge from multiple related domains to achieve a robust generalization performance in unseen target domains, which is an effective approach to mitigate domain shift in remote sensing image classification. Although the sharpness-aware minimization (SAM) method enhances DG capability and improves remote sensing image classification performance by promoting the convergence of the loss minimum to a flatter loss surface, the perturbation loss (maximum loss within the neighborhood of a local minimum) of SAM fails to accurately measure the true sharpness of the loss landscape. Furthermore, its variants often overlook gradient conflicts, thereby limiting further improvement in DG performance. In this paper, we introduce implicit sharpness-aware minimization (ISAM), a novel method that addresses the deficiencies of SAM and mitigates gradient conflicts. Specifically, we demonstrate that the discrepancy in training loss during gradient ascent or descent serves as an equivalent measure of the dominant eigenvalue of the Hessian matrix. This discrepancy provides a reliable measure for sharpness. ISAM effectively reduces sharpness and mitigates potential conflicts between gradients by implicitly minimizing the discrepancy between training losses while ensuring a sufficiently low minimum through minimizing perturbation loss. Extensive experiments and analyses demonstrate that ISAM significantly enhances the model's generalization ability on remote sensing and DG datasets, outperforming existing state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper investigates the coupled Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in two-dimensional bounded domains with a focus on global well-posedness and the formation of striking patterns, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time in the case of subcritical mass. Concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method. While studying the global existence and the concentration phenomenon of the chemotaxis–fluid model, we develop the global W 2 , p theory for the 2D stationary Stokes system subject to Navier boundary conditions with zero friction and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem. [ABSTRACT FROM AUTHOR]

Isospectral surfaces provide a rich family of benchmark problems. In this paper the efficacy of a hp -finite element method in such Laplace-Beltrami eigenvalue problems has been studied. In addition, for the p -version natural auxiliary space error estimators have been shown to be effective also in this context. As part of the numerical experiments further numerical evidence of the validity of the so-called Quarter Sphere Conjecture has been produced. For the isospectral surface pairs constructed via transplantation, the extension for perforated surfaces is derived. The lack of such constraints for surfaces with isothermal coordinates is also demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors. [ABSTRACT FROM AUTHOR]

Aiming to improve the robustness and adaptiveness of the recently investigated capped norm linear discriminant analysis (CLDA), this paper proposes a regularized linear discriminant analysis based on the generalized capped l 2 , q -norm (GCLDA). Compared to CLDA, there are two improvements in GCLDA. Firstly, GCLDA uses the capped l 2 , q -norm rather than the capped l 2 , 1 -norm to measure the within-class and between-class distances for arbitrary q > 0 . By selecting an appropriate q, GCLDA is adaptive to different data, and also removes extreme outliers and suppresses the effect of noise more effectively. Secondly, by taking into account a regularization term, GCLDA not only improves its generalization ability but also avoids singularity. GCLDA is solved through a series of generalized eigenvalue problems. Experiments on an artificial dataset, some real world datasets and a high-dimensional dataset demonstrate the effectiveness of GCLDA. [ABSTRACT FROM AUTHOR]

An eigenvalue λ of a signed graph S of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. Characterizing signed graphs with exactly k (1 ≤ k ≤ n) main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Z. Stanić has noticed that a signed graph has exactly one main eigenvalue if and only if it is net-regular, and in this paper, we study signed graphs with exactly two distinct main eigenvalues by studying (0 , 1 , 2) -multi-graphs with exactly two distinct main eigenvalues. We partially characterize the case when the basic graphs of the (0 , 1 , 2) -multi-graphs are trees, and propose some problems for further research. [ABSTRACT FROM AUTHOR]

A graph is claw-free if it does not contain a star of order 4 as an induced subgraph. In this paper, we characterize all claw-free connected graphs with all but four eigenvalues equal to 0 or − 1. [ABSTRACT FROM AUTHOR]

In this paper, the Riemann‐Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first‐ and second‐order dark soliton solutions on the nonzero background under the reflectionless condition. [ABSTRACT FROM AUTHOR]

Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution. [ABSTRACT FROM AUTHOR]

The hybrid active power filter (HAPF) can effectively suppress the 75–125 Hz low-frequency harmonic current generated by the EAST poloidal field power supply. However, the low-frequency harmonic current suppression performance of HAPF is greatly affected by its control parameters when the circuit parameters are determined. In this paper, the state–space equation of the HAPF system is established, and the relationship between the eigenvalues of the state–space matrix and the HAPF control parameters is analyzed using the Lyapunov method. Based on the variation law of the eigenvalues, by optimizing the control parameters of the HAPF system, the low-frequency harmonic current suppression performance of HAPF is improved by 41.9%. [ABSTRACT FROM AUTHOR]