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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]

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]

This paper presents the topological derivative of the first eigenvalue for the free vibration model of plane structures. We conduct a topological asymptotic analysis to account for perturbations in the domain caused by inserting a small inclusion. The paper includes a rigorous derivation of the topological derivative for the eigenvalue problem along with a proof of its existence. Additionally, we provide numerical examples that illustrate the application of the proposed methodology for maximizing the first eigenvalue in plane structures. The results demonstrate that multiple eigenvalues were not encountered. [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 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]

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 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 modern power systems incorporate high penetration of renewable is a large, composite, interconnected network with dynamic behavior. The small disturbances occurring in the system may induce low-frequency oscillations (LFOs) in the system. If the (LFOs) are not suppressed within a stipulated time, it may cause system islanding or even blackouts. Hence, it is essential to investigate the behavior of the system under various levels of disturbances and control action must be taken to damp these oscillations. The established approach to damping the LFOs is by installing power system stabilizers (PSS). PSS uses the local signals from generators to control the oscillations. The dominant source of inter-area oscillations in power systems is due to overloaded weak interconnected lines, converter-interfaced generation, and the action of the high gain exciter present in the system. Consequently, wide area control is needed to control the inter-area oscillations existent in the system. This paper developed a coordinated design of conventional PSS, static compensator, renewable converters, and wide area controller for damping the local and inter-area oscillations in renewable incorporated power systems. The performance of the developed controller is evaluated through the time domain analysis and eigenvalue analysis. A comparison of the introduced controller has been done with other standard conventional methods. The choice of input signals for the wide area controller from the wide-area measurement system is done based on the controllability index. Additionally, the location of the controller must be identified to dampen the inter-area oscillations in the system. In this paper, the controllability index is calculated to find out the highly affected wide area signals for considering it as the feedback signal to a developed controller. The location of the controller is recognized by computing the participation factor. The developed controller has experimented on renewable incorporated large study power systems when time delay and noise are present in wide area signals. [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]

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]

The Q -index of a graph G is the largest eigenvalue of the signless Laplacian matrix of G. A graph is leaf-free if it has no pendent vertices. In this paper, we give sharp upper bounds on the Q -index of leaf-free graphs with given size, and characterize the corresponding extremal graphs completely. As a consequence, the 2-edge-connected graph (resp. Euler graph) with given size having the largest Q -index is also characterized. [ABSTRACT FROM AUTHOR]

Let f (x , y) > 0 be a real symmetric function. For a connected graph G , the weight of edge v i v j is equal to the value f (d i , d j) , where d i is the degree of vertex v i. The degree-based weighted adjacency matrix is defined as A f (G) , in which the (i , j) -entry is equal to f (d i , d j) if v i v j is an edge of G and 0 otherwise. In this paper, we first give some bounds of the weighted adjacency eigenvalue λ 1 (A f (G)) in terms of λ 1 (A f (H)) , where H is obtained from G by some kinds of graph operations, including deleting vertices, deleting an edge and subdividing an edge, and examples are given to show that bounds are tight. Second, we obtain some bounds for the largest weighted adjacency eigenvalue λ 1 (A f (G)) of irregular weighted graphs. [ABSTRACT FROM AUTHOR]

Given a rooted tree T with vertices u 1 , u 2 , ... , u n , the level matrix L (T) of T is the n × n matrix for which the (i , j) -th entry is the absolute difference of the distances from the root to v i and v j. This matrix was implicitly introduced by Balaji and Mahmoud (2017) as a way to capture the overall balance of a random class of rooted trees. In this paper, we present various bounds on the eigenvalues of L (T) in terms of other tree parameters, and also determine the extremal structures among trees with a given order. Moreover, we establish bounds on the multiplicity of any eigenvalue in the level spectrum and show that the bounds are best possible. Furthermore, we provide evidence that the level spectrum can characterise some trees. In particular, we provide an affirmative answer to a very recent conjecture on the level energy (sum of absolute values of eigenvalues). [ABSTRACT FROM AUTHOR]

This paper focuses on the minimum eigenvalue involving the Fan product. By utilizing the Hölder inequality and the classic eigenvalue inclusion theorem, we introduce two novel lower bounds for τ (A 1 ⋆ A 2) , representing the minimum eigenvalue involving the Fan product of two M-matrices A 1 , A 2 . The newly derived lower bounds are then compared with the traditional findings. Numerical tests are presented to illustrate that the new lower bound formulas significantly enhance Johnson and Horn's results in certain scenarios and are more precise than other existing findings. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph G. In this paper, we refer to the i -nullity pair of a matrix A as (null (A) , null (A (i)) , where A (i) is the matrix obtained from A by removing the i -th row and column. The inverse i -nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair. [ABSTRACT FROM AUTHOR]

The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and J -Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators. [ABSTRACT FROM AUTHOR]

A complex unit gain graph, or T -gain graph, is a triple Φ = (G , T , φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T = { z ∈ C : | z | = 1 } , and a gain function φ : E → → T with the property that φ (e i j) = φ (e j i) − 1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n (G) − 2 m (G) + 2 c (G) − 1 , where n (G) , m (G) and c (G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. [ABSTRACT FROM AUTHOR]

For a connected graph G with order n , let e (G) be the number of its distinct eigenvalues and d be the diameter. We denote by m G (μ) the eigenvalue multiplicity of μ in G. It is well known that e (G) ≥ d + 1 , which shows m G (μ) ≤ n − d for any real number μ. A graph is called m i n i m a l if e (G) = d + 1. In 2013, Wong et al. characterize all minimal graphs with m G (0) = n − d. In this paper, by applying the star complement theory, we prove that if G is not a path and m G (μ) = n − d , then μ ∈ { 0 , − 1 }. Furthermore, we completely characterize all minimal graphs with m G (− 1) = n − d. [ABSTRACT FROM AUTHOR]

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n ≥ 49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6 / π 2 , whereas the average degree increases linearly with slope 6 / π 2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49 ≤ n ≤ 288 and it is at most 2 for n ≥ 289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r -length labeled cycles, which indicates the existence of a cycle of length at most O (log n). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization. [ABSTRACT FROM AUTHOR]

This paper concerns an age-structured juvenile-adult model incorporating harvesting pulse and moving boundaries in a heterogeneous environment, in which harvesting reflects human periodic pulse intervention on adults and the moving boundaries describe the natural expanding front of species. The principal eigenvalue is firstly defined and its properties involving the intensity of harvesting and length of habitat sizes are analyzed. Then the criteria to determine whether the species spread or vanish is discussed, and some relevant sufficient conditions characterized by pulse are established. Our results reveal that the co-extinction or coexistence of species is influenced by internal expanding capacity from species itself and external harvesting pulse from human intervention, in which the intensity and timing of harvesting play key roles. Our numerical simulations validate that the larger the harvesting rate and the shorter the harvesting period, the worse the survival of the species due to the cooperation among juveniles and adults, and such harvesting pulse can even alter the situation of species from persistence to extinction. In addition, expanding capacities also affect or alter the outcomes of spreading and vanishing. [ABSTRACT FROM AUTHOR]

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the ϵ -neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is O log n / n 1 / (m + 2) , where m and n denote the dimension of the manifold and the sample size, respectively. [ABSTRACT FROM AUTHOR]

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of such a space when the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer what is perhaps the most general problem of the kind as proposed by Loewy and Radwan, by solving their conjecture in the positive. We give the maximal dimension of a vector space of $ n\times n $ n × n matrices with no more than k

This paper presents an investigation into one- and three-dimensional harmonic oscillators with time-dependent mass and frequency, within the framework of the Dunkl formalism, which is constituted by replacing the ordinary derivative with the Dunkl derivative. To ascertain a general form of the wave functions the Lewis-Riesenfeld method was employed. Subsequently, an exponentially changing mass function in time was considered and the parity-dependent quantum phase, energy eigenvalues, and the corresponding wave functions were derived in one dimension. The findings revealed that the mirror symmetries affect the wave functions, thus the associated probabilities. Finally, the investigation was extended to the three-dimensional case, where it was demonstrated that, as with the solution of the radial equation, the solutions of the angular equation could be classified according to their mirror symmetries. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that 'slower diffuser prevails', which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the "weak competition" case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the "strong–weak competition" case, we give the expression of critical competition intensity and the weak competitor will be wiped out. [ABSTRACT FROM AUTHOR]

In the current paper, the free longitudinal vibration response of axially restrained functionally graded nanorods is presented for the first time based on the doublet mechanics theory. Size dependent nanorod is considered to be made of functionally graded material consist of ceramic and metal constituents. It is assumed that the material properties of the functionally graded nanorod are assumed to vary in the radial direction. The aim of this study is that to investigate the influences of various parameters such as functionally graded index, small size parameter, length of the nanorod, mode number and spring stiffness on vibration behaviors of functionally graded nanorod restrained with axial springs at both ends. For this purpose, Fourier sine series are used to define the axial deflection of the functionally graded nanorod. Then, an eigenvalue approach is established for longitudinal vibrational frequencies thanks to Stokes' transformation to deformable axial springs. Thus, the presented eigenvalue solution method is attributed to both rigid and deformable boundary conditions for the axial vibration of the functionally graded nanorod. With the help of the results obtained with the presented eigenvalue problem, it is observed that the parameters examined cause significant changes in the frequencies of the functionally graded nanorod. [ABSTRACT FROM AUTHOR]

We investigate a phase-field version of the Faber-Krahn theorem based on a phase-field optimization problem introduced by Garcke et al. in their 2023 paper formulated for the principal eigenvalue of the Dirichlet-Laplacian. The shape that is to be optimized is represented by a phase-field function mapping into the interval [0,1]. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to 0 and 1 except for a thin transition layer whose thickness is of order ε>0. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a Γ-convergence result which allows us to recover a variant of the Faber-Krahn theorem for sets of finite perimeter in the sharp interface limit. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (G) of a graph G is the largest eigenvalue of the adjacency matrix of G. For a graph G with maximum degree Δ (G) , it is known that ρ (G) ≤ Δ (G) with equality when G is connected if and only if G is regular. So the quantity β (G) = Δ (G) - ρ (G) is a spectral measure of irregularity of G. In this paper, we identify the trees of order n ≥ 12 with the first 15 largest β -values, the unicyclic graphs of order n ≥ 17 with the first 16 largest β -values, as well as the bicyclic graphs of order n ≥ 30 with the first 11 largest β -values. We also determine the graphs with the largest β -values among all connected graphs with given order and clique number. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized gapped k-mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers ℓ and k, with k ≤ ℓ , and an ℓ -tuple B = (b 1 , … , b ℓ) of integers b i ≥ 2 , i = 1 , … , ℓ . We introduce and study an incidence matrix A = A ℓ , k ; B . We develop a Möbius-like function ν B which helps us to obtain closed forms for a complete set of mutually orthogonal eigenvectors of A ⊤ A as well as a complete set of mutually orthogonal eigenvectors of A A ⊤ corresponding to nonzero eigenvalues. The reduced singular value decomposition of A and combinatorial interpretations for the nullity and rank of A, are among the consequences of this approach. We then combine the obtained formulas, some results from linear algebra, and combinatorial identities of elementary symmetric functions and ν B , to provide the entries of the Moore–Penrose pseudo-inverse matrix A + and the Gapped k-mer filter matrix A + A . [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping into H 1 of a small ball. [ABSTRACT FROM AUTHOR]

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are related to the normalized Laplacian eigenvalues for uniform hypergraphs. [ABSTRACT FROM AUTHOR]

For a simple graph on n vertices, any of its signless Laplacian eigenvalues is in the interval [ 0 , 2 n − 2 ]. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in [ 0 , 2 n − 2 ] and graph invariants including matching number and diameter. [ABSTRACT FROM AUTHOR]

A graph G = (V G , E G) is said to be a cograph if the path P 4 does not appear among its induced subgraphs. The vicinal preorder ≺ on the vertex set V G is defined in terms of inclusions between neighborhoods. The minimum number ∇ (G) of ≺-chains required to cover G is called the Dilworth number of G. In this paper it is proved that for a cograph G , the multiplicity of every Seidel eigenvalue λ ≠ ± 1 does not exceed ∇ (G). This bound turns out to be tight and can be further improved for threshold graphs. Moreover, it is shown that cographs with at least two vertices have no Seidel eigenvalues in the interval (− 1 , 1). [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]

This paper addresses the classification problem of integrable deformations of solutions of “degenerate” Riemann–Hilbert–Birkhoff (RHB) problems. These consist of those RHB problems whose initial datum has diagonal pole part with coalescing eigenvalues. On the one hand, according to theorems of Malgrange, Jimbo, Miwa, and Ueno, in the non-degenerate case, there exists a universal integrable deformation inducing (via a unique map) all other deformations [M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, Physica D 2 (1981) 306–352; B. Malgrange, Déformations de systèmes différentiels et microdifférentiels, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 351–379; B.Malgrange, Sur les déformations isomonodromiques, II, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 427–438; B. Malgrange, Deformations of differential systems, II, J. Ramanujan Math. Soc. 1 (1986) 3–15]. On the other hand, in the degenerate case, Sabbah proved, under sharp conditions, the existence of an integrable deformation of solutions, sharing many properties of the one constructed by Malgrange–Jimbo–Miwa–Ueno [C.Sabbah, Integrable deformations and degenerations of some irregular singularities, Publ. RIMS Kyoto Univ. 57(3–4) (2021) 755–794; arXiv:1711.08514v3]. Albeit the integrable deformation constructed by Sabbah is not, stricto sensu, universal, we prove that it satisfies a relative universal property. We show the existence and uniqueness of a maximal class of integrable deformations all induced (via a unique map) by Sabbah’s integrable deformation. Furthermore, we show that such a class is large enough to include all generic integrable deformations whose pole and deformation parts are locally holomorphically diagonalizable. In itinere, we also obtain a characterization of holomorphic matrix-valued maps which are locally holomorphically Jordanizable. This extends, to the case of several complex variables, already known results independently obtained by Thijsse and Wasow [Ph. G. A.Thijsse, Global holomorphic similarity to a Jordan form, Results Math. 8 (1985) 78–87; W.Wasow, Linear Turning Point Theory (Springer-Verlag, New York, 1985)]. [ABSTRACT FROM AUTHOR]

This paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite‐sized systems. We consider one‐dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large‐size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators. [ABSTRACT FROM AUTHOR]

Off-policy is a key setting for reinforcement learning algorithms. In recent years, the stability of off-policy learning for value-based reinforcement learning has been guaranteed even when combined with linear function approximation and bootstrapping. Convergence rate analysis is currently a hot topic. However, the convergence rates of learning algorithms vary, and analyzing the reasons behind this remains an open problem. In this paper, we propose an essentially simplified version of a convergence rate to generate general off-policy temporal difference learning algorithms. We emphasize that the primary determinant influencing convergence rate is the minimum eigenvalue of the key matrix. Furthermore, we conduct a comparative analysis of the influencing factor across various off-policy learning algorithms in diverse numerical scenarios. The experimental findings validate the proposed determinant, which serves as a benchmark for the design of more efficient learning algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan's first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity. [ABSTRACT FROM AUTHOR]