The Sturm–Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (2011), 555–591] and includes the Korteweg–de Vries and the Camassa– Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm–Liouville potentials [Stoch. Dyn. 8 ...

Bisection is one of the most common methods used to compute the eigenvalues of symmetric tridiagonal matrices. Bisection relies on the Sturm count : for a given shift σ, the number of negative pivots in the factorization T − σI = LDLT equals the number of eigenvalues of T that are smaller than σ. In IEEE-754 arithmetic, the value ∞ permits the computation to continue past a zero pivot, producin...

In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The Chebyshev cardinal functions are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions. These functions defined by the Chebyshev function of the first kind. By using the operational matrix of derivative the problem ...

A higher order scheme is presented for the optimal correction method of Kanatani [5] for triangulation from two views and is compared with the method of Hartley and Sturm [3]. It is pointed out that the epipole is a singularity of the Hartley-Sturm method, while the proposed method has no singularity. Numerical simulation confirms that both compute identical solutions at other points. However, ...

We consider a singular Sturm-Liouville expression with the indefinite weight sgnx. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgnx)(−d2/dx2 + q) with non-real sp...

Let v be a hyperbolic equilibrium of a smooth finite-dimensional gradient or gradient-like dynamical system. Assume that the unstable manifold W of v is bounded, with topological boundary Σ = ∂W := (closW )\W . Then Σ need not be homeomorphic to a sphere, or to any compact manifold. However, consider PDEs ut = uxx + f(x, u, ux) of Sturm type, i.e. scalar reaction-advection-diffusion equations i...

We present an implemented algorithmic method for counting and isolating all p-adic roots of univariate polynomials f over the rational numbers. The roots of f are uniquely described by p-adic isolating balls, that can be refined to any desired precision; their p-adic distances are also computed precisely. The method is polynomial space in all input data including the prime p. We also investigat...

This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-t...

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered. MSC-cl...

We study global attractors Af of scalar partial differential equations ut = uxx + f(x, u, ux) on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call Af a Sturm global attractor. We assume all equilibria v to be hyperbolic. Due to a gradient-like structure we can then write (*) Af = ⋃