Nonzero Eigenvalues Research Articles

Lower estimates for the length of the second fundamental form of submanifolds

In a remarkable work [35], Wei established estimates for the eigenvalues of the Laplacian on closed submanifolds Mn embedded in a unit sphere Sn+m. In this study, we extend these results to the eigenvalues of the p-Laplacian. As a consequence, we provide new characterizations of the sphere Sn. Additionally, we derive integral inequalities in terms of the norm of the second fundamental form of M and the first non-zero eigenvalue of the p-Laplacian, thereby generalizing the results previously established by Santos and Soares [11] for hypersurfaces.

Tubes and Steklov Eigenvalues in Negatively Curved Manifolds

Abstract We consider the Steklov eigenvalue problem on a compact pinched negatively curved manifold $M$ of dimension at least three with totally geodesic boundaries. We obtain a geometric lower bound for the first nonzero Steklov eigenvalue in terms of the total volume of $M$ and the volume of its boundary. We provide examples illustrating the necessity of these geometric quantities in the lower bound. Our result can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalue on closed pinched negatively curved manifolds of dimension at least three proved by Schoen in 1982. The proof is composed of certain key elements. We provide a uniform lower bound for the first eigenvalue of the Steklov–Dirichlet problem on a neighborhood of the boundary of $M$ and show that it provides an obstruction to having a small first nonzero Steklov eigenvalue. As another key element of the proof, we give a tubular neighborhood theorem for totally geodesic hypersurfaces in a pinched negatively curved manifold. We give an explicit dependence for the width function in terms of the volume of the boundary and the pinching constant.

Low-rank decomposition on the antisymmetric product of geminals for strongly correlated electrons

We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.

CONTROL DESIGN FOR A MULTIDIMENSIONAL SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS WITH RELAY HYSTERESIS AND PERTURBATION

A multidimensional controllable system with a constant matrix, a significant nonlinearity of the twoposition relay type with hysteresis as a control and a continuous periodic perturbation function is considered. The system matrix has simple, real, non-zero eigenvalues, among which one can be positive. Conditions for the system parameters, including the nonlinearity ones, are established under which there is a single two-point oscillatory periodic solution with a period comparable to the period of the perturbation function in the case of a special type of the feedback vector. The asymptotic stability of the solution has been proven using the phase plane method. The results obtained are illustrated by examples for three-dimensional systems.

An upper bound for the first nonzero Steklov eigenvalue

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\Sect_g\leq \kappa$ for $\kappa\leq 0$ and $\Ric_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where $\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov eigenvalues of $\Omega$ and $\Omega^*$ respectively, and $C=C(n,\kappa, K, d)$ is an explicit constant. When $\kappa=K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.

Computing the Kirchhoff index of a family of phenylene chain networks

Abstract The Kirchhoff index is a fundamental topological metric that provides insights into the structural and electrical characteristics of networks. It is defined as the sum of resistance distances between all pairs of nodes, serving as a key factor in understanding the dynamics within networks. To investigate the impact of structural variations on the Kirchhoff index, we select a family of phenylene chain networks as our model and establish a methodology to explore the Kirchhoff index using the Laplacian spectrum. By analyzing the network structure, we introduce a parameter to control the number of iterations, providing a recursive relationship between the Laplacian matrix and its eigenvalues at intervals of generations. This approach enables the derivation of an analytical expression for both the sum of the reciprocals of all nonzero Laplacian eigenvalues and the Kirchhoff index.

Fully distributed event-triggered secondary voltage resilient tracking control for microgrids

This paper aims to design the fully distributed event-triggered secondary voltage resilient tracking control strategy for microgrids subject to denial-of-service (DoS) attacks. The fully distributed method is employed to remove the requirement of the smallest nonzero eigenvalue of the Laplacian matrix, which is the global information. In addition, an attack compensation with the latest successful transmission state information from neighbor's as the attack compensation term is designed for both the event-triggered scheme and the controller to save communication resources and mitigate the negative effects caused by DoS attacks. Finally, the effectiveness of the designed fully distributed event-triggered resilient control method subject to DoS attacks is demonstrated via a simulation example with comparisons.

Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces

Abstract Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär’s theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.

Continuity and minimization of spectrum related with the two-component Novikov equation

This article is concerned with the spectral problem{dϕ1(x)=ϕ2(x)dh(x),dϕ2(x)=ϕ3(x)dg(x),dϕ3(x)=−λϕ1(x)dx, x∈(−1,1),ϕ2(−1)=0, ϕ3(−1)=0, ϕ3(1)=0,which is associated with the two-component Novikov equation. The coefficients g and h are functions of bounded variation. The first aim is to prove the continuous dependence of the n-th eigenvalue on the coefficients g and h with different topologies; in addition, we discuss the Fréchet differentiability of the algebraically simple eigenvalues on the coefficients g, h with respect to the norm topology of total variations. Secondly, utilizing the theory of oscillatory (non-symmetric) kernels, conditions on g, h under which the spectrum is nonnegative and algebraically simple are obtained. Finally, we solve the extremal problems of the lowest nonzero eigenvalue, when the norm ‖⋅‖V of the coefficients g, h is given. The latter can be seen as an application of our first two main results and the trace formulas.

An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

Abstract Suppose that $\Sigma ^{n}\subset \mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $\lambda _{1}$ of the induced Laplace–Beltrami operator on $\Sigma $ satisfies $\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$, where $a_{n}$ and $b_{n}$ are explicit dimensional constants and $\Lambda $ is an upper bound for the length of the second fundamental form of $\Sigma $. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $\lambda _{1} \geq \frac{n}{2}$ without any further assumptions on $\Sigma $.

Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

AbstractLet be the simplicial complex whose vertices are the nontrivial subspaces of and whose simplices correspond to families of subspaces forming a flag. Let be the ‐dimensional weighted upper Laplacian on . The spectrum of was first studied by Garland, who obtained a lower bound on its nonzero eigenvalues. Here, we focus on the case. We determine the asymptotic behavior of the eigenvalues of as tends to infinity. In particular, we show that for large enough , has exactly distinct eigenvalues, and that every eigenvalue of tends to as goes to infinity. This solves the zero‐dimensional case of a conjecture of Papikian.

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

Property analysis and coherence dynamics for tree-symmetric networks with noise disturbance

Abstract In this paper, we investigate the leaderless and the leader–follower coherence of tree-symmetric networks. Firstly, the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues of tree-symmetric networks are computed. Secondly, comparing the leaderless and leader–follower coherence, the existence of leader nodes and network parameters have a great impact on the coherence in the noisy environment. Finally, the Laplacian eigenvalues of tree-symmetric networks were utilized to analyze other properties of the network. These research results have certain theoretical significance for the promotion of distributed system and block chain technology.

Eigenvalue estimates for p-Laplace problems on domains expressed in Fermi coordinates

We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if γ denotes a non-closed curve in R2 symmetric with respect to the y-axis, let D⊂R2 denote the domain of points that lie on one side of γ and within a prescribed distance δ(s) from γ(s) (here s denotes the arc length parameter for γ). Write μ1odd(D) for the lowest nonzero eigenvalue of the Neumann p-Laplacian with an eigenfunction that is odd with respect to the y-axis. For all p>1, we provide a lower bound on μ1odd(D) when the distance function δ and the signed curvature k of γ satisfy certain geometric constraints. In the linear case (p=2), we establish sufficient conditions to guarantee μ1odd(D)=μ1(D). We finally study the asymptotics of μ1(D) as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann p-Laplace problem.

Form-finding of thermal-adaptive pin-bar assemblies based on eigenvalue modification

Lattice structures with tunable expansion properties have been investigated in multidisciplinary fields to control the temperature effects of structures or materials. The expected thermal adaptivity can be achieved by optimizing the structural geometry. A novel method for the form-finding of thermal-adaptive pin-bar assemblies is developed in this paper by considering the control of structural temperature effects as the minimization of the potential energy of the system. Based on the stationarity condition of the potential energy with respect to the nodal coordinates, the compatibility relationship between the thermal elongations of members and the target nodal displacements is proven to be the sufficient and necessary condition for structural thermal adaptivity. The solvability of the compatibility equation is determined by the rank equality between the compatibility matrix and its augmented form, which can be measured by the number of nonzero eigenvalues of its Gramian matrix. The analytical relationship between the eigenvalues of the Gramian matrix and the nodal coordinates is established using the matrix perturbation theory. A numerical strategy based on Newton’s method is proposed in which the eigenvalues are gradually modified by adjusting the nodal coordinates until the rank equality is satisfied. To address the existence of multiple solutions with structural thermal adaptivity, structural symmetry and periodicity constraints are introduced to narrow the solution space. The thermal-adaptive configurations of three illustrative pin-bar assemblies are analyzed using the proposed form-finding method, and the expected thermal deformations are verified for the obtained configurations using the finite element software ABAQUS. Comparing the results obtained by the proposed method with those obtained by nonlinear programming and the genetic algorithm validates the advantages of the proposed method in terms of computational time, optimality of the obtained configuration and applicability to complex structural geometries.

Distributed Nash Equilibrium Seeking and Disturbance Rejection for High-Order Integrators Over Jointly Strongly Connected Switching Networks.

In this article, we study the problem of Nash equilibrium (NE) seeking of N -player games with high-order integrator agents over jointly strongly connected networks. The same problem was studied recently over static, undirected, and connected networks. Since a jointly strongly connected network can be directed and disconnected at every time instant, our result strictly includes the existing results as special cases. Moreover, in addition to some analytic conditions on the payoff functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. By adopting a different approach, our result does not rely on the satisfaction of this inequality. Furthermore, our result can also be extended to the case where the additive disturbances are imposed on the input of each agent. Our design is illustrated by two numerical examples.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

The upper bound of the harmonic mean of the Neumann eigenvalues in curved spaces

Considering an n-dimensional Riemannian manifold M whose sectional curvature is bounded above by κ and Ricci curvature is bounded below by (n−1)K, we obtain an upper bound for the harmonic mean of the first (n−1) non-zero Neumann eigenvalues of the Laplacian for domains contained in M. This can be viewed as certain isoperimetric inequality and generalizes the results for domains in space forms [5,15].

Singular graphs and the reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3], the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.