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On the exact Gevrey order of formal Puiseux series solutions to the third Painlevé equation
Journal of Dynamical and Control Systems. 2019. Vol. 25. No. 4. P. 681-690.
Parusnikova A., Vasilyev A. V.
In this paper, we study the third Painlevé equation with parameters γ = 0, αδ ≠ 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters δ = −β^2/2 ≠ 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.
Bershtein M., Shchechkin A., / Cornell University. Series math "arxiv.org". 2018.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blow-up relations on Nekrasov partition functions.
We also study series of q-deformed c=−2 ...
Added: November 22, 2018
Gavrylenko P., Lisovyy O., / arXiv.org. Series arXiv.org "math-ph". 2017. No. 1705.01869.
We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, ...
Added: May 5, 2017
Bobrova I., Mazzocco M., Journal of Geometry and Physics 2021 Vol. 166 Article 104271
In this paper we study the so-called sigma form of the second Painleve hierarchy. To obtain this form, we use some properties of the Hamiltonian structure of the second Painleve hierarchy and of the Lenard operator. ...
Added: September 25, 2021
V. A. Poberezhny, Journal of Mathematical Sciences 2013 Vol. 195 No. 4 P. 533-540
We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement ...
Added: February 14, 2014
Yanovich Yury Aleksandrovich, Journal of Mathematics and Statistics 2016 Vol. 12 No. 3 P. 157-175
In many applications, the real high-dimensional data occupy only a very small part in the high dimensional ‘observation space’ whose intrinsic dimension is small. The most popular model of such data is Manifold model which assumes that the data lie on or near an unknown manifold Data Manifold, (DM) of lower dimensionality embedded in an ...
Added: November 24, 2016
Yanovich Y., Proceedings of Machine Learning Research 2017 Vol. 60 P. 18-38
In many applications, the real high-dimensional data occupy only a very small part in the high dimensional ‘observation space’ whose intrinsic dimension is small. The most popular model of such data is Manifold model which assumes that the data lie on or near an unknown manifold (Data Manifold, DM) of lower dimensionality embedded in an ...
Added: June 15, 2017
Ulyanov V. V., Goetze F., A. Naumov, Journal of Theoretical Probability 2017 Vol. 30 No. 3 P. 876-897
In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed. ...
Added: March 17, 2016
Levin A., Olshanetsky M., Zotov A., Journal of Physics A: Mathematical and Theoretical 2016 Vol. 49 No. 39 P. 1-24
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its ...
Added: September 17, 2016
Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073507-1-073507-19
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 ...
Added: September 19, 2012
Anoshin V. I., Beketova A., Parusnikova A. et al., Computational Mathematics and Mathematical Physics 2023 Vol. 63 No. 1 P. 86-95
The second member of the fourth Painlevé hierarchy is considered. Convergence of certain power asymptotic expansions in a neighborhood of zero is proved. New families of power asymptotic expansions are found. Computations are carried out using a computer algebra system. Reference to a code that can be used for computing the Gevrey order of the ...
Added: March 30, 2023
Minabutdinov A., / Cornell University. Series arXiv "math". 2015. No. 1508.07421.
The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic ...
Added: September 12, 2015
Anastasia V. Parusnikova, Opuscula Mathematica 2014 Vol. 34 No. 3 P. 591-599
The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlevй equations near infinity. We consider the fifth Painleve equation in two cases: when αβγδ \neq 0 and when αβγ \neq 0, δ = 0 and the third Painlevé equation when all the parameters of the equation are ...
Added: February 28, 2014
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359-2402
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of 𝑐=1 Virasoro conformal blocks. We study similar series of 𝑐=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima–Yoshioka blowup relations on Nekrasov partition functions. We also study series of ...
Added: October 21, 2019
Minabutdinov A., Записки научных семинаров ПОМИ РАН 2015 Т. Том 436 С. 174-189
The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expan- sion. The research is motivated by the study of ergodic sums of the Pascal ...
Added: October 14, 2015
Parusnikova A., Vasilyev A., Journal of Mathematical Sciences 2019 Vol. 241 No. 3 P. 318-326
We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic ...
Added: October 26, 2019
Parusnikova A., / Cornell University. Series "Working papers by Cornell University". 2014. No. 1412.6690.
In the first section of this work we introduce 4-dimensional Power Geometry for second-order ODEs of a polynomial form. In the next five sections we apply this construction to the first five Painlev ́e equations. ...
Added: March 28, 2015
Parusnikova A., / ИПМ им. М.В. Келдыша РАН. Серия :: "ИПМ им. М.В. Келдыша РАН". 2013. № 97.
В данной работе рассматривается вопрос о суммируемости по Жевре степенных разложений решений четвёртого уравнения Пенлеве в окрестности бесконечности в случае общего положения αβ ≠ 0. Для анализа используются методы французской и японской школ, алгоритмы сравниваются с алгоритмами степенной геометрии. ...
Added: October 25, 2013
Bobrova I., Sokolov V., Journal of Nonlinear Mathematical Physics 2023 Vol. 30 No. 2 P. 646-662
All Hamiltonian non-abelian Painlevé systems of P1−P6 type with constant coefficients are found. For P1−P5 systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new P′3 and P5 systems thus obtained, we find isomonodromic Lax pairs for them. ...
Added: December 23, 2022
Брюно А. Д., Parusnikova A., / Институт прикладной математики им. М.В. Келдыша Российской академии наук. 2012. № 61.
Рассматривается пятое уравнение Пенлеве в окрестности бесконечности. Методами двумерной степенной геометрии вычисляются все экспоненциальные разложения его решений. Методами трёхмерной степенной геометрии вычисляются некоторые степенно-эллиптические и степенно-периодические асимптотики его решений. ...
Added: March 24, 2013
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359-2402
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima–Yoshioka blowup relations on Nekrasov partition functions. We also study series of ...
Added: August 31, 2020
Levin A., Ольшанецкий М. А., Зотов А. В., Успехи математических наук 2014 Т. 69 № 1(415) С. 39-124
В данной работе изомонодромные задачи описываются в терминах плоских G-расслоений на проколотых эллиптических кривых Σ_τ и связностей с регулярными особенностями в отмеченных точках. Расслоения классифицируются по их характеристическим классам, которые являются элементами группы вторых когомологий H^2(Σ_τ,Z(G)), где Z(G) – центр G. По каждой простой комплексной группе Ли G и произвольному характеристическому классу определяется пространство модулей ...
Added: January 21, 2015
Parusnikova A., Vasilyev A. V., / Cornell University. Series arXiv "math". 2017. No. 1702.05758.
In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of z^{2/3} -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them ...
Added: February 21, 2017
Bruno A., Parusnikova A., Доклады Академии наук 2012 Т. 442 № 5 С. 583-588
В работе методами степенной геометрии найдены все асимптотические разложения решений пятого уравнения Пенлеве в окрестности его не особой точки для всех значений четырех комплексных параметров уравнения. Получено 10 семейств разложений решений уравнения, одно из которых не было известно раньше. Три разложения являются рядами Лорана, а остальные семь – рядами Тейлора. Все они сходятся в (проколотой) ...
Added: November 30, 2012
Gavrylenko P., Lisovyy O., / Cornell University. Series math-ph "arXiv". 2016. No. 1608.00958.
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL(N,ℂ). The corresponding operator acts in the direct sum of N(n−3) copies of L2(S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of ...
Added: September 20, 2016