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Of all publications in the section: 7
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Article
Zabrodin A., Zotov A. Constructive Approximation. 2015. Vol. 41. No. 3. P. 385-423.

In light of the quantum Painlevé–Calogero correspondence, we investigate the inverse problem. We imply that this type of the correspondence (classical-quantum correspondence) holds true, and we find out what kind of potentials arise from the compatibility conditions of the related linear problems. The latter conditions are written as functional equations for the potentials depending on a choice of a single function—the left-upper element of the Lax connection. The conditions of the correspondence impose restrictions on this function. In particular, it satisfies the heat equation. It is shown that all natural choices of this function (rational, hyperbolic, and elliptic) reproduce exactly the Painlevé list of equations. In this sense, the classical-quantum correspondence can be regarded as an alternative definition of the Painlevé equations. © 2015, Springer Science+Business Media New York.

Added: Sep 3, 2015
Article
Beckermann B., Kalyagin V. A. Constructive Approximation. 1997. Vol. 13. P. 481-510.
Added: Nov 3, 2009
Article
Beckermann B., Kalyagin V., Matos A. et al. Constructive Approximation. 2013. Vol. 37. No. 1. P. 101-134.

We prove existence and uniqueness of a solution to the problem of minimizing the logarithmic energy of vector potentials associated to a d-tuple of positive measures supported on closed subsets of the complex plane. The assumptions we make on the interaction matrix are weaker than the usual ones, and we also let the masses of the measures vary in a compact subset of ℝ+ d. The solution is characterized in terms of variational inequalities. Finally, we review a few examples taken from the recent literature that are related to our results.

Added: Feb 3, 2013
Article
Aptekarev A. I., Kalyagin V. A., Van Iseghem J. Constructive Approximation. 2000. Vol. 16. P. 487-454.
Added: Nov 6, 2009
Article
Kalyagin V. A., Saff E., Aptekarev A. I. Constructive Approximation. 2009. Vol. 30. No. 2. P. 175-223.

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Padé approximants as well as to vector continued fractions.


 

Added: Dec 26, 2012
Article
Iorgov N., Lisovyy O., Tykhyy Y. et al. Constructive Approximation. 2014. Vol. 39. No. 1. P. 255-272.

We outline recent developments relating Painlev ́e equations and 2D conformal field theory. Generic tau functions of Painlev ́e VI and Painlev ́e III_3 are written as linear combinations of c= 1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish connection formula for the tau function in the Painlev ́e VI case.

Added: Aug 14, 2015
Article
Beckermann B., Kalyagin V. A., Matos A. et al. Constructive Approximation. 2012. No. 1.

We prove existence and uniqueness of a solution to the problem of minimizing the logarithmic energy of vector potentials associated to a d-tuple of positive measures supported on closed subsets of the complex plane. The assumptions we make on the interaction matrix are weaker than the usual ones, and we also let the masses of the measures vary in a compact subset of \mathbbR+dRd+ . The solution is characterized in terms of variational inequalities. Finally, we review a few examples taken from the recent literature that are related to our results. 

Added: Dec 13, 2012