Null Eigenvalues Research Articles

THE ROLE OF UNBOUNDED TIME-SCALES IN GENERATING LONG-RANGE MEMORY IN ADDITIVE MARKOVIAN PROCESSES

Any additive stationary and continuous Markovian process described by a Fokker–Planck equation can also be described in terms of a Schrödinger equation with an appropriate quantum potential. By using such analogy, it has been proved that a power-law correlated stationary Markovian process can stem from a quantum potential that (i) shows an x-2 decay for large x values and (ii) whose eigenvalue spectrum admits a null eigenvalue and a continuum part of positive eigenvalues attached to it. In this paper we show that such two features are both necessary. Specifically, we show that a potential with tails decaying like x-μ with μ < 2 gives rise to a stationary Markovian process which is not power-law autocorrelated, despite the fact that the process has an unbounded set of time scales. Moreover, we present an exactly solvable example where the potential decays as x-2 but there is a gap between the continuum spectrum of eigenvalues and the null eigenvalue. We show that the process is not power law autocorrelated, but by decreasing the gap one can arbitrarily well approximate it. A crucial role in obtaining a power-law autocorrelated process is played by the weights [Formula: see text] giving the contribution of each time-scale contribute to the autocorrelation function. In fact, we will see that such weights must behave like a power-law for small energy values λ. This is only possible if the potential VS(x) shows a x-2 decay to zero for large x values.

A higher order thin-walled beam model including warping and shear modes

A thin-walled beam model that considers higher order effects is presented in this paper. The beam displacement field is approximated through a linear combination of products between a set of linear independent functions, which are defined over the beam cross-section, and the associated amplitudes that are only dependent on the beam axis. The beam model governing equations are then obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section approximation functions. A set of uncoupled beam deformation modes is obtained from a non-linear eigenvalue problem that stems directly from the general solution of the differential homogeneous equilibrium equations. The classic deformation modes are obtained without prior assumptions, being associated with null eigenvalues, which requires an adequate computation of a Jordan chain, whereas the higher order modes correspond to the non-null eigenvalues, which allows to measure the mode decay along the beam axis. A numerical example is presented in order to verify the model capabilities to simulate the non-classic effects associated with thin-walled beam higher order deformation modes.

Numerical evaluation of the stability of stationary points of index-2 differential-algebraic equations: Applications to reactive flash and reactive distillation systems

The dynamic behavior of many chemical processes can be represented by an index-2 system of differential-algebraic equations. This index can be reduced by differentiation, but unfortunately the index reduced systems are not guaranteed to possess the same stability characteristics as that of the original system. When the set of differential-algebraic equations can be written in Hessenberg form, the matrix pencil of the linearized system can be used to directly evaluate the stability of a steady state without the need for index reduction. Direct evaluations of stability of reactive flash and reactive distillation are presented. It is also shown that a commonly used index reduction will always result in null eigenvalues at steady state. Stabilization methods were successfully applied to this reduced system. An alternative index reduction method for a reactive flash is generalized and shown to be highly sensitive to minor changes in the jacobian.

The positivity and other properties of the matrix of capacitance: Physical and mathematical implications

We prove that the matrix of capacitance in electrostatics is a positive-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for such a matrix. Many properties are easily visualized by constructing a "potential space" isomorphic to the euclidean space. The problem of minimizing the internal energy of a system of conductors under constraints is considered, and an equivalent capacitance for an arbitrary number of conductors is obtained. Moreover, some properties of systems of conductors in successive embedding are examined. Finally, we discuss some issues concerning the gauge invariance of the formulation.

Stability of the Parisi solution for the Sherrington–Kirkpatrick model near T = 0

To test the stability of the Parisi solution near T = 0, we study the spectrum of the Hessian of the Sherrington–Kirkpatrick model near T = 0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit T ≪ 1, two regions can be identified. In the first region, for x close to 0, where x is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region T ≪ x ⩽ 1, as T → 0, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for T ≪ 1. In the limit T → 0, the width of the first region shrinks to 0 and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.

Analytical treatment of SUSY Quasi-normal modes in a non-rotating Schwarzschild black hole

We use the Fock-Ivanenko formalism to obtain the Dirac equation which describes the interaction of a massless 1/2-spin neutral fermion with a gravitational field around a Schwarzschild black hole (BH). We obtain approximated analytical solutions for the eigenvalues of the energy (quasi-normal frequencies) and their corresponding eigenstates (quasi-normal states). The interesting result is that all the excited states [and their supersymmetric (SUSY) partners] have a purely imaginary frequency, which can be expressed in terms of the Hawking temperature. Furthermore, as one expects for SUSY Hamiltonians, the isolated bottom state has a real null energy eigenvalue.

Characteristic behavior of prismatic anisotropic beam via generalized eigenvectors

Elastic, anisotropic, non-homogeneous, prismatic beams are solved through a semi-analytical formulation. The resulting variational formulation is solved with a finite element discretization over the cross-section, leading to a set of Hamiltonian ordinary differential equations along the beam. Such a formulation is characterized by a group of generalized eigenvectors associated to null eigenvalues, which are shown to combine rigid body motions and the classical De Saint-Venant’s beam solutions. The related generalized deformation parameters are identified through the amplitude of the deformable generalized eigenvectors. Results obtained from the analysis of both isotropic and composite beams are presented.

Instability of equilibrium points of some Lagrangian systems

In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n − 2 , then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues. The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces.

Current sheet formation and nonideal behavior at three-dimensional magnetic null points

The nature of the evolution of the magnetic field, and of current sheet formation, at three-dimensional (3D) magnetic null points is investigated. A kinematic example is presented that demonstrates that for certain evolutions of a 3D null (specifically those for which the ratios of the null point eigenvalues are time-dependent), there is no possible choice of boundary conditions that renders the evolution of the field at the null ideal. Resistive magnetohydrodynamics simulations are described that demonstrate that such evolutions are generic. A 3D null is subjected to boundary driving by shearing motions, and it is shown that a current sheet localized at the null is formed. The qualitative and quantitative properties of the current sheet are discussed. Accompanying the sheet development is the growth of a localized parallel electric field, one of the signatures of magnetic reconnection. Finally, the relevance of the results to a recent theory of turbulent reconnection is discussed.

Time‐domain properties of reactive dual circuits

AbstractThe present work explores some effects of the replacement of capacitors by inductors and vice versa in state and semistate models of lumped circuits. Such a replacement, when performed together with an inversion of the capacitance and inductance matrices, yields a transformation of the form λ→λ−1 in the system spectra. In the semistate context, this covers in particular extremal cases in which null eigenvalues or infinite ones with higher index appear in the matrix pencil associated with the model; these cases describe certain pathological circuit configurations. This approach leads to a discussion of new properties of strictly passive circuits; specifically, from the known fact that the index of strictly passive circuits does not exceed two, we derive that the index of null eigenvalues in this setting cannot exceed one. This precludes in particular Takens‐Bogdanov degeneracies, defined by an index‐two double‐zero eigenvalue, in strictly passive circuits. Although the results are addressed in a linear context, they can be extended via linearization to non‐linear problems, as it is the case in the transformation of singularity‐induced bifurcation phenomena into steady bifurcations discussed at the end of the paper. Copyright © 2006 John Wiley & Sons, Ltd.

A Riemannian Framework for Tensor Computing

Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D2 ƒ has at least n − r null eigenvalues everywhere.

Complexity of vector spin glasses.

We study the annealed complexity of the m-vector spin glasses in the Sherrington-Kirkpatrick limit. The eigenvalue spectrum of the Hessian matrix of the Thouless-Anderson-Palmer free energy is found to consist of a continuous band of positive eigenvalues in addition to an isolated eigenvalue and (m-1) null eigenvalues due to rotational invariance. Rather surprisingly, the band does not extend to zero at any finite temperature. The isolated eigenvalue becomes zero in the thermodynamic limit, as in the Ising case (m=1), indicating that the same supersymmetry breaking recently found in Ising spin glasses occurs in vector spin glasses.

Computing Cavity Resonances Using Eigenvalues Displacement

The numerical discretization of the field inside a cavity by means of edge elements results in a generalized algebraic eigenvalues problem that contains several undesired null eigenvalues. This occurrence prevents the effective use of iterative eigensolvers. To overcome this difficulty, a complementary eigenproblem has been proposed in the literature. This paper extends this method by introducing a family of algebraically built complementary eigenproblems, and determines, by numerical experiments and heuristics, which complementary eigenproblems are best suited for the preconditioned inverse iteration eigensolver and the Lanczos method.

Particle geodesics and spectra in the spacetime tangent bundle

A limiting curvature of worldlines in spacetime serves as a possible basis for the differential geometry of the tangent bundle of spacetime. I first review briefly the resulting differential geometric structure of the spacetime tangent bundle. Next, I examine the relations between bundle geodesies and their projections on the spacetime base manifold. Possible classical particle equations of motion result from geodesic motion in the bundle manifold. Additionally, the Laplace-Beltrami operator is constructed on the bundle manifold. Possible particle spectra are represented by quantum fields that have a null eigenvalue when acted upon by the Laplace-Beltrami operator of the bundle manifold. The fields are shown to be automatically regularized by the limiting curvature of worldlines in spacetime, and particle spectra are shown to be cut off at very high energy.

A Spectral-Element Method for Particulate Stokes Flow

A numerical procedure is presented for solving the equations of Stokes flow past a fixed bed of rigid particles and the equations describing the motion of a suspension of rigid particles upon which a specified force and torque is exerted, for general flow configurations and arbitrary particle shapes. The problem is formulated in terms of an integral equation of the first kind for the distribution of the boundary traction incorporating a Green function that observes the periodicity of the flow and the geometry of the boundaries of the flow, accompanied by appropriate boundary conditions and integral constraints. The integral equation is solved for two-dimensional flow by a spectral-element orthogonal-collocation method. Two important components of the numerical method are (a) preconditioning of the linear system that arises from the discretization of integral equation followed by reduction to remove the eigenfunctions corresponding to the null eigenvalue over each particle surface and (b) a physically motivated iterative solution of the master linear system based on particle clustering. It is found that, for the purpose of computing the force and torque exerted on fixed particles and the velocity of translation and angular velocity of rotation of freely suspended particles, the orthogonal collocation method has significant advantages over the trapezoidal discretization. The iterative solution of the integral equation converges even for closely spaced particles where each particle is treated as a cluster.

Shape ambiguities in structure from motion

This paper examines the fundamental ambiguities and uncertainties inherent in recovering structure from motion. By examining the eigenvectors associated with null or small eigenvalues of the Hessian matrix, we can quantify the exact nature of these ambiguities and predict how they affect the accuracy of the reconstructed shape. Our results for orthographic cameras show that the bas-relief ambiguity is significant even with many images, unless a large amount of rotation is present. Similar results for perspective cameras suggest that three or more frames and a large amount of rotation are required for metrically accurate reconstruction.

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, \(M \subset X\)a C2 submanifold, \(\Omega \subset M\) an openset with C2 boundary \(S = \partial \Omega \).Denote by \(\mu _M (\mathcal{O}_X )\) (resp. \(\mu _M (\mathcal{O}_X )\)) the microlocalization along M (resp. \(\Omega \)) of the sheaf \(\mathcal{O}_X \) of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups\(H^j \mu _M (\mathcal{O}_X )_p {\text{ for }}p \in \dot T_M^* X\).The most general gives the vanishing outside a range of indices j whose length is equal to\(s^0 (M,p)\) (with \(s^{ + , - ,0} (M,p)\) being the number of respectively positive, negative and null eigenvalues for the‘microlocal’ Levi form \(L_M (p)\)).The sharpest result gives the concentration in a single degree, provided that the difference\(s^ - (M,p\prime ) - \gamma (M,p\prime )\) is locally constant for \(p\prime \in T_M^* X\) near p (with\(\gamma (M,p) = {\text{ dim}}^C (T_M^* X \cap iT_M^* X)_z \) for z the base point of p).The first result was restated for the complex \(\mu _\Omega (\mathcal{O}_X )\)in [D'A-Z 2], in the case codim\(_M S = 1\)We extend it here to any codimension and moreover we also restate for \(\mu _\Omega (\mathcal{O}_X )\) the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.

A dynamic interpretation of the load-flow Jacobian singularity for voltage stability analysis

In voltage stability analysis, both static and dynamic approaches are used to evaluate the system critical conditions. The static approach is based on the standard load flow equations. For small-disturbance analysis, the dynamic approach is based on the eigenvalue computation of the linearized system, while for large-disturbance analysis a complete time-domain simulation is required. However, both the equilibrium point around which linearization is performed and the initial conditions for the simulation are computed by a procedure which uses the standard load-flow equations. The standard load-flow equations make some implicit assumptions on the steadystate behaviour of dynamic components (generator control systems, loads). These assumptions are not satisfied by the usual dynamic models, and this discrepancy leads to different results in the voltage stability assessment using static and dynamic methods. In the framework of bifurcation theory, this paper discusses the relationships between static and small-disturbance dynamic approaches to find the voltage stability critical condition, with emphasis on system component modelling. A set of hypotheses on generator control systems and load models is given for a multimachine system, according to which the same critical conditions are obtained both from the load flow equations and from the full eigenvalue analysis. These hypotheses are less restrictive than those previously proposed in the literature and make it possible to obtain equivalence between the singularity of the load flow Jacobian and a null eigenvalue of the linearized dynamic system. Following a dynamic argumentation based on small-disturbance analysis, this result may justify the use of simple and fast static methods for voltage stability assessment and shows that the smalldisturbance voltage stability limit depends only on the steady-state characteristics of the dynamic components of the system.

Orthogonality constraints in finite basis set calculations

The orthogonality-constrained one-body eigenvalue problem is formulated for complete, finite-dimensional spaces in terms of projection operators that partition the space into two orthogonal subspaces, one containing the constraints and one containing the desired solutions. We derive a Hermitian operator, associated with the Hamiltonian, having eigenvectors that are correspondingly partitioned, each subset of the eigenvectors providing a basis for one of the subspaces. This Hermitian operator advantageously replaces a non-Hermitian operator that was proposed in recent work of this group. The non-zero spectrum of the two operators is demonstrably the same, and the occurrence of null eigenvalues is clarified through characterization of the associated eigenvectors. The deviation from zero of these eigenvalues (apart from numerical inaccuracies) is related directly to the normalization of the constraint vectors. The formalism is extended to the case of non-orthonormal primitive basis sets, and a numerical application is carried out using B-spline functions.