738 research outputs found
Criteria for the strong regularity of J-inner functions and Îł-generating matrices
AbstractThe class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A2) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of Îł-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. AÂ determinental criterion is also presented
Szego asymptotics for matrix-valued measures with countably many bound states
Let be a matrix-valued measure with the essential spectrum a single
interval and countably many point masses outside of it. Under the assumption
that the absolutely continuous part of satisfies Szego's condition and
the point masses satisfy a Blaschke-type condition, we obtain the asymptotic
behavior of the orthonormal polynomials on and off the support of the measure.
The result generalizes the scalar analogue of Peherstorfer-Yuditskii and the
matrix-valued result of Aptekarev-Nikishin, which handles only a finite number
of mass points
Supporting students' key competences in visual art classes : the benefits of planning
This paper highlights the role of an art teacher in shaping the overall learning process and potential in supporting not only students’ artistic skills, but also skills to thrive in a contemporary world. The aim of this study is to develop and implement strategies that support students’ key competences in basic school visual art classes and to provide opportunities for students to be engaged with visual art in a more meaningful way. A preliminary questionnaire with 77 of Estonian basic school second level art teachers revealed that teachers support key competences rather implicitly and view these as a natural part of lessons that do not need extra planning. Basing on the action research cycle conducted with two 5th grade classes, we argue that explicit key competence support provides a more meaningful interaction between the teacher and students, and that planning plays a vital role in materialising key competences in the teaching.Peer reviewe
Bitangential interpolation in generalized Schur classes
Bitangential interpolation problems in the class of matrix valued functions
in the generalized Schur class are considered in both the open unit disc and
the open right half plane, including problems in which the solutions is not
assumed to be holomorphic at the interpolation points. Linear fractional
representations of the set of solutions to these problems are presented for
invertible and singular Hermitian Pick matrices. These representations make use
of a description of the ranges of linear fractional transformations with
suitably chosen domains that was developed in a previous paper.Comment: Second version, corrected typos, changed subsection 5.6, 47 page
Generalized solutions of Riccati equalities and inequalities
The Riccati inequality and equality are studied for infinite dimensional
linear discrete time stationary systems with respect to the scattering supply
rate. The results obtained are an addition to and based on our earlier work on
the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely
related to the results of Yu. M. Arlinski\u{\i} in [3]. The main difference is
that we do not assume the original system to be a passive scattering system,
and we allow the solutions of the Riccati inequality and equality to satisfy
weaker conditions.Comment: Published in Methods of Functional Analysis and Topology (MFAT),
available at http://mfat.imath.kiev.ua/article/?id=84
Multi-operator colligations and multivariate characteristic functions
In the spectral theory of non-self-adjoint operators there is a well-known
operation of product of operator colligations. Many similar operations appear
in the theory of infinite-dimensional groups as multiplications of double
cosets. We construct characteristic functions for such double cosets and get
semigroups of matrix-valued functions in matrix balls.Comment: 15p
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
Counting Berg partitions
We call a Markov partition of a two dimensional hyperbolic toral automorphism
a Berg partition if it contains just two rectangles. We describe all Berg
partitions for a given hyperbolic toral automorphism. In particular there are
exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same
connectivity matrix (k, l, m, n)
Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model
A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS)
system with a pseudo-exponential potential is introduced. For the corresponding
Weyl function the direct and inverse problem are solved explicitly in terms of
three parameter matrices. As an application explicit solutions are obtained for
the discrete integrable nonlinear equation corresponding to the isotropic
Heisenberg magnet model. State space techniques from mathematical system theory
play an important role in the proofs
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