Number Of Positive Eigenvalues Research Articles

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.

On the relation between the positive inertia index and negative inertia index of weighted graphs

Let Gw be an edge-weighted graph and let A(Gw) be its adjacency matrix. The positive inertia index (respectively, the negative inertia index) of Gw, denoted by pGw+ (resp. pGw−), is defined to be the number of positive eigenvalues (resp. negative eigenvalues) of A(Gw). Clearly, pGw++pGw− is the rank of Gw, whereas pGw+−pGw− is defined as the signature of Gw. In this paper, some upper and lower bounds on pGw+−pGw− are determined for the weighted chordal graphs (resp. bipartite graphs, complete graphs, cactus graphs, join graphs, bisplit graphs, power graphs).

Spectrum theory of second-order difference equations with indefinite weight

In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

Bounds for the positive and negative inertia index of a graph

Let G be a graph and let A(G) be adjacency matrix of G. The positive inertia index (respectively, the negative inertia index) of G, denoted by p(G) (respectively, n(G)), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of A(G). In this paper, we present the bounds for p(G) and n(G) as follows:m(G)−c(G)≤p(G)≤m(G)+c(G),m(G)−c(G)≤n(G)≤m(G)+c(G), where m(G) and c(G) are respectively the matching number and the cyclomatic number of G. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds respectively.

Characterization of graphs whose signature equals the number of odd cycles

Let G be a simple graph with vertex set V(G) and edge set E(G). The signature s(G) of G is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix A(G). In [20], it was proved that −c1(G)≤s(G)≤c1(G), where c1(G) denotes the number of odd cycles in G. A problem arises naturally: What graphs have signature attaining the upper bound c1(G) (resp., the lower bound −c1(G))? In this paper, we focus our attention on this problem, characterizing graphs G whose signature equals c1(G) (resp., −c1(G)).

The signature of k-cyclic graphs of ∞-type

Let be a simple graph with vertex set and edge set . The signature of is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix . In [22, Linear Algebra Appl. 2013;438:331–341], it was conjectured that , where denotes the number of cycles in of length , and denotes the number of cycles in of length . The authors of this paper have established the conjecture for trees, unicyclic graphs and bicyclic graphs. We prove the conjecture for -cyclic graphs of -type, that is, the graphs in which and in which no two cycles have a common vertex.

Using variants of zero forcing to bound the inertia set of a graph

Zero forcing is a combinatorial game played on a graph with a goal of changing the color of every vertex at minimal cost. This leads to a parameter known as the zero forcing number that can be used to give an upper bound for the maximum nullity of a matrix associated with the graph. A variation on the zero forcing game is introduced that can be used to give an upper bound for the maximum nullity of such a matrix when it is constrained to have exactly q negative eigenvalues. This constrains the possible inertias that a matrix associated with a graph can achieve and gives a method to construct lower bounds on the inertia set of a graph (which is the set of all possible pairs (p,q) where p is the number of positive eigenvalues and q is the number of negative eigenvalues).

Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues

We analyze the eigenstructure of the integral operator $\mathcal{K}_{l, \alpha, k}$ which arise naturally from the beam deflection equation on linear elastic foundation with finite beam. We show that $\mathcal{K}_{l, \alpha, k}$ has countably infinite number of positive eigenvalues approaching 0 as the limit, and give explicit upper and lower bounds on each of them. Consequently, we obtain explicit upper and lower bounds on the $L^{2}$ -norm of the operator $\mathcal{K}_{l, \alpha, k}$ . We also present precise approximations of the eigenvalues as they approach the limit 0, which describes the almost regular structure of the spectrum of $\mathcal{K}_{l, \alpha, k}$ . Additionally, we analyze the dependence of the eigenvalues, including the $L^{2}$ -norm of $\mathcal{K}_{l, \alpha, k}$ , on the intrinsic length $L = 2 l \alpha$ of the beam, and show that each eigenvalue is continuous and strictly increasing with respect to L. In particular, we show that the respective limits of each eigenvalue as L goes to 0 and infinity are 0 and $1/k$ , where k is the linear spring constant of the given elastic foundation. Using Newton’s method, we also compute explicitly numerical values of the eigenvalues, including the $L^{2}$ -norm of $\mathcal{K}_{l, \alpha, k}$ , corresponding to various values of L.

Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight

By virtue of the eigenvalues of discrete linear second-order Neumann eigenvalue problems, we study the eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight. We find that these two problems have T real eigenvalues (including the multiplicity) respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the order relation.

Spectrum of Discrete Second-Order Neumann Boundary Value Problems with Sign-Changing Weight

We study the spectrum structure of discrete second-order Neumann boundary value problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the NBVPs to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to thejth positive/negative eigenvalue changes its sign exactlyj-1times.

Fisher Difference Discriminant Analysis: Determining the Effective Discriminant Subspace Dimensions for Face Recognition

In most manifold learning based subspace discriminant analysis algorithms, how to construct the local neighborhood graphs and determine the effective discriminant subspace dimensions in applications are difficult but important problems. In this paper, we propose a novel supervised subspace learning method called Fisher Difference Discriminant Analysis (FDDA) for linear dimensionality reduction. FDDA introduces the local soft scatter to characterize the distributions of the data set. By combining Fisher criterion and difference criterion together, FDDA obtains the optimal discriminant subspace, on which a large margin between different classes is provided for classification. Eigenvalue analysis shows that the effective discriminant subspace dimensions of FDDA can be automatically determined by the number of positive eigenvalues and are robust to noise and invariant to rotations, rescalings and translations of the data. Comprehensive comparison and extensive experiments show that FDDA is superior to some state-of-the-art techniques in face recognition.

Exact boundary observability for a kind of second-order quasilinear hyperbolic system

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global classical solution, we establish the local exact boundary observability for a kind of second-order quasilinear hyperbolic system in which the number of positive eigenvalues and the number of negative ones are not equal. As an application, we obtain the one-sided local exact boundary observability and two-sided local exact boundary observability with fewer observed values for first-order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative ones are decoupled.

Spectral concentration phenomena for the Laplace operator with the Dirichlet boundary condition on a cavity

Let D1 ⊂ R3 be a non-empty simply connected open bounded set and Γ = ∂D1 be its boundary. We assume Γ to be a C2 surface and let (Γn)n∈N be a sequence of surfaces, such that Γn is closed, , and the surface measure of Γ\Γn is positive for n ∈ N. The set D1 is the cavity associated with (Γn)n∈N. Let HΓn, n ∈ N, HΓ be the self-adjoint operators on L2 associated with the minus Laplace operator with Dirichlet boundary condition on Γn, n ∈ N and on Γ, respectively. Note that, roughly speaking, HΓ is the operator associated with minus the Laplacian on R3\Γ with the Dirichlet boundary condition on Γ. A similar statement holds for HΓn, n ∈ N. We show that the spectral measure, dEΓn (λ), λ ∈ R, associated with the operator HΓn, n ∈ N, when n → +∞ converges to the spectral measure dEΓ (λ), λ ∈ R, associated with the operator HΓ. That is in a sense made precise later we prove that Σ (HΓn) → Σ (HΓ) when n → +∞, where Σ (HΓ) and Σ (HΓn), n ∈ N, are the spectra of HΓ and HΓn, n ∈ N, respectively. Moreover, we show that the point spectrum of HΓ is made of an infinite number of positive eigenvalues of finite multiplicity and that the point spectrum of HΓn, n ∈ N, is empty and that the essential spectrum of HΓn, n ∈ N, and of HΓ is continuous and is contained in [0, +∞). Under the extra hypothesis that Γ is a C4 surface and that the Gaussian curvature of Γ is positive at every point of Γ, we prove that the essential spectrum of HΓ is a continuous spectrum and is given by [0, +∞). So that the convergence of dEΓn (λ), λ ∈ R to dEΓ (λ), λ ∈ R, when n → +∞ can be interpreted as a spectral concentration phenomenon that consists in the fact that the limit of the spectrum of operators with purely continuous spectrum is the spectrum of an operator that is made of eigenvalues embedded in a continuous spectrum.

Dimensionality of Stable and Unstable Directions in the Gravitational N-Body Problem

The gravitational n—body problem is chaotic. Phase trajectories that start very near each other separate rapidly. The rate looks exponential over long times. At any instant, trajectories separated in certain directions move apart rapidly (unstable directions), while those separated in other directions stay about the same (stable directions). Unstable directions lie along eigenvectors that correspond to positive eigenvalues of the matrix of force gradients. The number of positive eigenvalues of that matrix gives the dimensionality of stable regions. This number has been studied numerically in a series of 100—body integrations. It continues to change as long as the integration continues because the matrix changes extremely rapidly. On average, there are about 1.2n unstable directions out of 3n. Issues of dimensionality arise when the tools of ergodic studies are brought to bear on the problem of trajectory separation. A method of estimating the rate of trajectory separation based on matrix descriptions is presented in this note. Severe approximations are required.

A Bootstrap Comparison of Genetic Covariance Matrices

SUMMARY An important problem in evolutionary biology is how population bottlenecks affect additive genetic variance (VA). When only additive effects are present, a decrease in VA is predicted, whereas with many forms of nonadditive interactions, an increase is predicted. We present an analysis of the changes in variance that occur as a result of a bottleneck in a population of the red flour beetle Tribolium castaneum. In this experiment, the additive genetic covariance matrix for four traits in two sexes was measured in the ancestral population. The population then went through a bottleneck and the genetic covariance matrix was again measured. These data were analyzed by first examining the univariate estimates for VA and, second, examining the covariance matrices. For the univariate measures, the statistic we used was the change in VA. For the multivariate comparisons, we used three measures: the difference in rank (the difference in the number of positive eigenvalues) to test changes in the number of heritable traits, a multivariate equivalent of Bartlett's statistic to test changes in the size of the matrices, and a modification of Mantel's correlation to test for changes in the correlation structure of the matrices. Because the distribution of all of these measures is complex and dependent on the specifics of the particular matrices being compared, we chose to use a bootstrap analysis to test for significance. For females, we found that, following the population bottleneck, VA was significantly decreased and that the correlation structure of the matrix was significantly changed. For males, we found that, following the population bottleneck, VA significantly increased but that the correlation structure was unchanged.

Reply by Author to J. L. Tietze

P RUSSING points out the dangers of trying to use the principal minor test for positive definite matrices as a test for a positive semidefinite matrix. Kerr agrees, but feels that modifying the principal minor test to suit the new task is too expensive computationally. He recommends using the singular value decomposition (SVD) of the matrix as a more efficient and more stable technique to evaluate definiteness. A Choleski factorization, although computationally faster, is rejected because it is not stable for a semidefinite matrix. As an indicator of computational speed, the flop count for a Choleski factorization is given as «/6, whereas the flop count for an SVD is given as 2w/3. This Comiiient suggests a different concept for this test, namely, the inertia of a matrix. The computational burden is similar to that of a Choleski factorization and software to perform this calculation is available in LINPACK. The inertia of a symmetric matrix is defined as the number triple (ir9p,£]9 where ir is the number of positive eigenvalues, p is the number of negative eigenvalues, and f is the number of zero eigenvalues. For an «th-order matrix, the statement that it is positive definite is equivalent to the statement that its inertia is {n,0,0]. Similarly, a positive semidefinite matrix would have an inertia of {m,Q,p}, m+p = n. The flop count for the matrix inertia calculation is «/6, nearly four times faster than the equivalent SVD. Although the matrix inertia calculation does not indicate the magnitude of the matrix eigenvalues, it can be used to discover eigenvalue bounds (see Slicing the Spectrum in Ref. 5 for a lucid commentary). In summary, using LINPACK software to calculate the inertia of a matrix is a computationally efficient way to check whether the matrix is positive definite, positive semidefinite, or indefinite. The SVD is more suitable if information on why the matrix is not positive definite is required.

Number of positive eigenvalues of the solution to the discrete Lyapunov matrix equation

It is known that the discrete Lyapunov matrix equation has a positive semidefinite symmetric solution under an appropriate condition. This note proves the fact that the number of positive eigenvalues of the solution is equal to the rank of the controllability matrix formed from the coefficient matrices.

On the number of positive eigenvalues in a class of two-body quantum systems with nonlocal interaction

Using the dilatation analytic techniques recently introduced in potential scattering, an upper bound on the number of positive eigenvalues is obtained for a class of two-body quantum systems with nonlocal interactions.