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## Quantum Painlevé-Calogero correspondence for Painlevé VI

This paper is a continuation of our previous paper where the Painlevé-Calogero correspondence has been extended to auxiliary linear problems associated with Painlevé equations. We have proved, for the first five equations from the Painlevé list, that one of the linear problems can be recast in the form of the non-stationary Schrödinger equation whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian for the corresponding Painlevé equation. In the present paper we establish the quantum Painlevé-Calogero correspondence for the most general case, the Painlevé VI equation. We also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding Schlesinger system with rational spectral parameter.

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We apply methods from Space Power Geometry to the fifth Painlevé equation. Near infinity we obtained 2 families of elliptic asymptotic forms and 4 families of periodic asymptotic forms of its solutions. All of these families are 2-parameter.

Applying methods of plane Power Geometry we are looking for the asymptotic expansions of solutions to the fifth Painleve ́ equation in the neighbourhood of its singular and nonsingular points.

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary time, ∂ tΨ=(1/2∂ 2 x +V (X,t))Ψ whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian H = 1/2p 2+V(x,t) for the corresponding Painlevé equation. In present paper, we present explicit constructions for the first five equations from the Painlevé list.

By means of Power Geometry we obtained all asymptotic expansions of solutions to the equation P5 of the following five types: power, power-logarithmic, complicated, exotic and half-exotic for all values of 4 complex parameters of the equation. They form 16 and 30 families in the neighbourhood of singular points z = infty and z = 0 correspondingly. There exist 10 families in the neighbourhood of nonsingular point. Over 20 families are new.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

In this work, the methods of power geometry are used to find asymptotic expansions of solutions to the fifth Painlevй equation as *x * 0 for all values of its four complex parameters. We obtain 30 families of expansions, of which 22 are obtained from published expansions of solutions to the sixth Painlevй equation. Among the other eight families, one was previously known and two can be obtained from the expansions of solutions to the third Painlevй equation. Three families of half-exotic expansions and two families of complicated expansions are new.

By using superconducting quantum interference device (SQUID) magnetometry, we investigated anisotropic high-field (H less than or similar to 7T) low-temperature (10 K) magnetization response of inhomogeneous nanoisland FeNi films grown by rf sputtering deposition on Sitall (TiO2) glass substrates. In the grown FeNi films, the FeNi layer nominal thickness varied from 0.6 to 2.5 nm, across the percolation transition at the d(c) similar or equal to 1.8 nm. We discovered that, beyond conventional spin-magnetism of Fe21Ni79 permalloy, the extracted out-of-plane magnetization response of the nanoisland FeNi films is not saturated in the range of investigated magnetic fields and exhibits paramagnetic-like behavior. We found that the anomalous out-of-plane magnetization response exhibits an escalating slope with increase in the nominal film thickness from 0.6 to 1.1 nm, however, it decreases with further increase in the film thickness, and then practically vanishes on approaching the FeNi film percolation threshold. At the same time, the in-plane response demonstrates saturation behavior above 1.5-2T, competing with anomalously large diamagnetic-like response, which becomes pronounced at high magnetic fields. It is possible that the supported-metal interaction leads to the creation of a thin charge-transfer (CT) layer and a Schottky barrier at the FeNi film/Sitall (TiO2) interface. Then, in the system with nanoscale circular domains, the observed anomalous paramagnetic-like magnetization response can be associated with a large orbital moment of the localized electrons. In addition, the inhomogeneous nanoisland FeNi films can possess spontaneous ordering of toroidal moments, which can be either of orbital or spin origin. The system with toroidal inhomogeneity can lead to anomalously strong diamagnetic-like response. The observed magnetization response is determined by the interplay between the paramagnetic-and diamagnetic-like contributions.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.