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Search: All articles in the CJM digital archive with keyword Oscillation

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1. CJM 2012 (vol 65 pp. 808)

Grandjean, Vincent
On Hessian Limit Directions along Gradient Trajectories
Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$, we show that the limit $\nu$ of the secants at the limit point $\mathbf{0}$ of $|\gamma|$ along the trajectory $|\gamma|$ is an eigen-vector of the limit of the direction of the Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$ along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.

Keywords:gradient trajectories, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinity
Categories:34A26, 34C08, 32Bxx, 32Sxx

2. CJM 2012 (vol 65 pp. 222)

Sauer, N. W.
Distance Sets of Urysohn Metric Spaces
A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With $\operatorname{dist}(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous, universal, separable and complete. (It is not difficult to deduce that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds every separable metric space $\mathrm{M}$ with $\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.) The main results are: (1) A characterization of the sets $\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\mathrm{M}$ is homogeneous.

Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability
Categories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99

3. CJM 2010 (vol 62 pp. 261)

Chiang, Yik-Man; Ismail, Mourad E. H.
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
No abstract.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Problem
Categories:34M10, 33C15, 33C47

4. CJM 2006 (vol 58 pp. 726)

Chiang, Yik-Man; Ismail, Mourad E. H.
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above ``special function approach" can be described by a classical Heine problem for differential equations that admit polynomial solutions.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble
Categories:34M10, 33C15, 33C47

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