Laplacian Eigenvalues Research Articles

Cooperative Output Feedback Tracking Control of Stochastic Linear Heterogeneous Multiagent Systems

We study cooperative output feedback tracking control of stochastic linear heterogeneous leader- following multiagent systems. Each agent has a conti- nuous-time linear heterogeneous dynamics with incompletely measurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose a set of admissible distributed observation strategies for estimating the leader’s and the followers’ states and a set of admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show that for observable leader’s dynamics and stabilizable and detectable followers’ dynamics, if the intensity coefficient of multiplicative noises multiplied by the sum of real parts of the leader’s unstable modes is less than one-fourth of the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking, provided the associated output regulation equations are solvable. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation.

The Brézis–Nirenberg equation for the $ {(m,p)} $ Laplacian in the whole space

In this paper, we prove the existence of a nontrivial (weak) solution of the Brézis–Nirenberg equation for the $ (m, p) $ Laplacian in the whole space $ \mathbb R^N $. Critical problems have been intensively studied in the last decades, starting with the pioneering paper by Brézis and Nirenberg for the Dirichlet Laplacian problems in bounded domains of $ \mathbb R^N $. Even if we obtain existence of solutions for the $ (m, p) $ Laplacian Brézis–Nirenberg equation in the whole space via standard variational techniques, the last step of the proof is pretty involved and requires new ideas and a delicate use of the Talenti functions.

On graphs with a few distinct reciprocal distance Laplacian eigenvalues

<abstract><p>For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.</p></abstract>

Laplacian Eigenvalue Allocation Through Asymmetric Weights in Acyclic Leader-Follower Networks

Laplacian Eigenvalue Allocation Through Asymmetric Weights in Acyclic Leader-Follower Networks

Remarks on the sum of powers of normalized signless Laplacian eigenvalues of graphs

Let G = (V, E), V = {v1, v2,..., vn}, be a simple connected graph of order n and size m. Denote by ?+1 ? ?+2 ?...? ?+n ? 0 the normalized signless Laplacian eigenvalues of G, and by ??(G) the sum of ?-th powers of the normalized signless Laplacian eigenvalues of a connected graph. The paper deals with bounds of ??. Some special cases, when ? = 12 and ? = ?1, are also considered.

On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G) = RT(G) ? RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RDL(G) is a real symmetric matrix, we denote its eigenvalues as ?1(RDL(G)) ? ?2(RDL(G)) ?...? ?n(RDL(G)). The largest eigenvalue ?1(RDL(G)) of RDL(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RDL(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph Sn is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph Kn, Kn ? e, the star Sn, the complete balanced bipartite graph Kn/2 ,n/2 and the complete split graph CS(n, ?) are all determined from the RDL-spectrum.

Regular convergence and finite element methods for eigenvalue problems

Regular convergence, together with other types of convergence, have been studied since the 1970s for discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of a compact operator $T$ that can be written as an eigenvalue problem of a holomorphic Fredholm operator function $F(\eta) = T-\frac{1}{\eta} I$. Focusing on finite element methods (conforming, discontinuous Galerkin, non-conforming, etc.), we show that the regular convergence of the discrete holomorphic operator functions $F_n$ to $F$ follows from the compact convergence of the discrete operators $T_n$ to $T$. The convergence of the eigenvalues is then obtained using abstract approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to prove the convergence of various finite element methods for eigenvalue problems such as the Dirichlet eigenvalue problem and the biharmonic eigenvalue problem.

Shooting from singularity to singularity and a quasilinear <i>p</i>-Laplace-Beltrami equation with indefinite weight

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation.

An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, then the first Steklov-Dirichlet eigenvalue $\sigma_1(\Omega)$ has a maximum when $R_1$ and the measure of $\Omega$ are fixed. Moreover, if $\Omega_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

Sharp bounds for Dirichlet eigenvalue ratios of the Camassa–Holm equations

Sharp bounds for Dirichlet eigenvalue ratios of the Camassa–Holm equations

A smooth basis for atomistic machine learning.

Machine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighborhood of each atom in the system. Symmetry considerations support the use of spherical harmonics to expand the angular dependence of this density, but there is, as of yet, no clear rationale to choose one radial basis over another. Here, we investigate the basis that results from the solution of the Laplacian eigenvalue problem within a sphere around the atom of interest. We show that this generates a basis of controllable smoothness within the sphere (in the same sense as plane waves provide a basis with controllable smoothness for a problem with periodic boundaries) and that a tensor product of Laplacian eigenstates also provides a smooth basis for expanding any higher-order correlation of the atomic density within the appropriate hypersphere. We consider several unsupervised metrics of the quality of a basis for a given dataset and show that the Laplacian eigenstate basis has a performance that is much better than some widely used basis sets and competitive with data-driven bases that numerically optimize each metric. Finally, we investigate the role of the basis in building models of the potential energy. In these tests, we find that a combination of the Laplacian eigenstate basis and target-oriented heuristics leads to equal or improved regression performance when compared to both heuristic and data-driven bases in the literature. We conclude that the smoothness of the basis functions is a key aspect of successful atomic density representations.

Chain graphs with simple Laplacian eigenvalues and their Laplacian dynamics

Chain graphs with simple Laplacian eigenvalues and their Laplacian dynamics

Universal Bounds for Fractional Laplacian on a Bounded Open Domain in $${\mathbb {R}}^{n}$$

Let \(\Omega \) be a bounded open domain on the Euclidean space \({\mathbb {R}}^{n}\) and \({\mathbb {Q}}_{+}\) be the set of all positive rational numbers. Chen and Zeng (Cal Var Part Differ Equ 56:131, 2017) investigated the eigenvalues with higher order of the fractional Laplacian \(\left. (-\Delta )^{s}\right| _{\Omega }\) for \(s>0\) and \(s \in {\mathbb {Q}}_{+}\), and they obtained a universal inequality of Yang type. In the spirit of Chen and Zeng’s work, we study the eigenvalues of fractional Laplacian and establish an inequality of eigenvalues with lower order under the same condition. Also, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators given by Jost et al. (Trans Am Math Soc 363(4):1821–1854, 2011).

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Compressed hydrogen atoms confined within generic boxes

The ‘compressed hydrogen atom’ problem involves solving the Schrödinger equation for a hydrogen atom confined within an impenetrable (often, spherical) box, with the wave function vanishing on the boundary of this box. To better understand the role the structure of the box plays on the compressed hydrogen atom, we study the compressed hydrogen atom in boxes of myriad shapes and sizes, developing a theory for how the structure of a generic box influences the hydrogen ground state energy. Our theory predicts that the ground state energy increases with the first Dirichlet eigenvalue (which encodes information on both the shape and size of a box) of the box and decreases with the distance of the nucleus from the boundary of the box. This theoretical prediction is supported by numerical simulations and variational approximations. Consideration is also given to the problem of a shell-confined hydrogen atom, with the electron confined to a separate ‘cage’ away from the nucleus. The behaviour of the shell-confined hydrogen ground state is well-approximated by a perturbation theory we develop for generic cages. Our results provide a unified framework for understanding confined, high-pressure hydrogen with relevance to applications ranging from crystal lattices of hydrogen to metallic hydrogen.

Distributed Event-Triggered Synchronization of Interconnected Linear Two-Time-Scale Systems With Switching Topology.

This article investigates the synchronization problem of interconnected linear two-time-scale systems (TTSSs) with switching topology. By utilizing the Chang transformation, a distributed synchronization protocol is proposed with event-triggered communication. Static and dynamic event-triggered mechanisms are proposed successively, which both contain two separated event-triggering conditions corresponding to the slow and the fast subsystems. The existence of a strictly positive time period between any two successive transmissions is ensured regardless of the initial states. The main difficulty of this study lies in that the state jump and parametric uncertainty appear because of the system transformation. To overcome the difficulty, the system is first modeled as an uncertain hybrid system. Then, the control gain is properly designed by solving Riccati-like equations dependent on the rough bounds of the eigenvalues of communication graph Laplacians, and a piecewise quadratic Lyapunov function is proposed with which the jump caused by the switching topology is subtly evaluated. Sufficient conditions are thus established to achieve the event-triggered synchronization. Furthermore, the results are also extended to solve the synchronization problem of the interconnected impulsive linear TTSSs. Finally, three numerical examples are provided to demonstrate the effectiveness of the proposed theoretical results.

NORMALIZED LAPLACIAN ENERGY AND NORMALIZED LAPLACIAN-ENERGY-LIKE INVARIANT OF SOME DERIVED GRAPHS

For a connected graph $G$, the smallest normalized Laplacian eigenvalue is 0 while all others are positive and the largest cannot exceed the value 2. The sum of absolute deviations of the eigenvalues from 1 is called the normalized Laplacian energy, denoted by $\mathbb{LE}(G)$. In analogy with Laplacian-energy-like invariant of $G$, we define here the normalized Laplacian-energy-like as the sum of square roots of normalized Laplacian eigenvalues of $G$, denoted by $\mathbb{LEL}(G)$.

Risk-sensitive control for a class of diffusions with jumps

We consider a class of diffusions controlled through the drift and jump size, and driven by a jump Lévy process and a nondegenerate Wiener process, and we study infinite horizon (ergodic) risk-sensitive control problems for this model. We start with the controlled Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results in the literature on exit rate control problems. Then we consider the infinite horizon average risk-sensitive minimization and maximization problems on the whole domain. Under suitable hypotheses, we establish existence and uniqueness of a principal eigenfunction for the Hamilton–Jacobi–Bellman (HJB) operator on the whole space, and fully characterize stationary Markov optimal controls as the measurable selectors of this HJB equation.

Distance (Signless) Laplacian Eigenvalues of $k$-uniform Hypergraphs

The distance (signless) Laplacian eigenvalues of a connected hypergraph are the eigenvalues of its distance (signless) Laplacian matrix. For all $n$-vertex $k$-uniform hypertrees, we determine the $k$-uniform hypertree with minimum second largest distance (signless) Laplacian eigenvalue. For all $n$-vertex $k$-uniform unicyclic hypergraphs, we obtain the $k$-uniform unicyclic hypergraph with minimum largest distance (signless) Laplacian eigenvalue, and the $k$-uniform unicyclic hypergraph with minimum second largest distance Laplacian eigenvalue.

Small Scale CLTs for the Nodal Length of Monochromatic Waves

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).