3,569 research outputs found
The CMV bispectral problem
A classical result due to Bochner classifies the orthogonal polynomials on
the real line which are common eigenfunctions of a second order linear
differential operator. We settle a natural version of the Bochner problem on
the unit circle which answers a similar question concerning orthogonal Laurent
polynomials and can be formulated as a bispectral problem involving CMV
matrices. We solve this CMV bispectral problem in great generality proving
that, except the Lebesgue measure, no other one on the unit circle yields a
sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear
differential operator of arbitrary order. Actually, we prove that this is the
case even if such an eigenfunction condition is imposed up to finitely many
orthogonal Laurent polynomials.Comment: 25 pages, final version, to appear in International Mathematics
Research Notice
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
- …