相关论文: The converse of Sturm's Separation Theorem

We show that Sturm's classical comparison theorem (SCT) on the interlacing of zeros of solutions of pairs of real second order two-term ordinary differential equations necessarily fails if the usual Sturmian-type conditions on the…

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase…

Sturm's Theorem is a fundamental 19th century result relating the number of real roots of a polynomial $f$ in an interval to the number of sign alternations in a sequence of polynomial division-like calculations. We provide a short direct…

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm oscillation theorem for systems of…

Sturm theory for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. Here we propose a Sturm oscillation theorem for indefinite systems of even…

The main result is that for lambda strong limit singular failing the continuum hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.

We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper,…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties which arise from the outer automorphism and non-split part of the torus. The outer…

In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(\mu_k^\nu,0)$ is a…

We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…

The title theorem is proved by example: an algebra of binary relations, closed under intersection and composition, that is not isomorphic to any such algebra on a finite set.

This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation…

Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm-Liouville problems. Although…

In this paper, we establish a generalization of Sturm--Picone comparison theorem for a pair of fractional nonlocal equations: \begin{eqnarray*} \begin{gathered} (-div. (A_1(x)\nabla))^{s} u = C_{1}(x) u \,\,\,\mbox{in}\,\,\Omega, u = 0…

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is…

We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict…

An uniqueness theorem for the inverse problem in the case of a second-order equation defined on the interval [0,1] when the boundary forms contain combinations of the values of functions at the points 0 and 1 is proved. The auxiliary…

In this study, we give the Sturm comparison theorems for discrete fractional Sturm-Liouville (DFSL) equations within Riemann-Liouville and Gr\"unwald-Letnikov sense. The emergence of Sturm-Liouville equations began as one dimensional…

The zeros of the eigenfunctions of self-adjoint Sturm-Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm-Liouville problem associated with the Schrodinger equation for…