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Perturbation analysis of a matrix differential equation
Two complex matrix pairs and are contragrediently
equivalent if there are nonsingular and such that
. M.I. Garc\'{\i}a-Planas and V.V. Sergeichuk
(1999) constructed a miniversal deformation of a canonical pair for
contragredient equivalence; that is, a simple normal form to which all matrix
pairs close to can be reduced by
contragredient equivalence transformations that smoothly depend on the entries
of and . Each perturbation of defines the first order induced perturbation
of the matrix , which is the first order
summand in the product . We find all
canonical matrix pairs , for which the first order induced perturbations
are nonzero for all nonzero perturbations in
the normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises in
the theory of matrix differential equations , whose product of two
matrices: ; using the substitution , one can reduce by
similarity transformations and by contragredient equivalence
transformations
On weakly tight families
Using ideas from Shelah's recent proof that a completely separable maximal
almost disjoint family exists when , we construct a
weakly tight family under the hypothesis \s \leq \b < {\aleph}_{\omega}. The
case when \s < \b is handled in \ZFC and does not require \b <
{\aleph}_{\omega}, while an additional PCF type hypothesis, which holds when
\b < {\aleph}_{\omega} is used to treat the case \s = \b. The notion of a
weakly tight family is a natural weakening of the well studied notion of a
Cohen indestructible maximal almost disjoint family. It was introduced by
Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the
Kat\'etov order on almost disjoint families
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