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## Разложения решений пятого уравнения Пенлеве в окрестности его неособой точки

Доклады Академии наук. 2012. Т. 442. № 5. С. 583-588.

Bruno A., Parusnikova A.

Bruno A., Parusnikova A., Доклады Академии наук 2011 Т. 438 № 4 С. 439-443

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 ...

Added: April 12, 2012

Parusnikova A., , in : International Conference “Painlevґe Equations and Related Topics”. : St. Petersburg : The Euler International Mathematical Institute, 2011. P. 126-131.

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 = ...

Added: April 16, 2012

Parusnikova A., / Cornell University. Series math "arxiv.org". 2014.

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 these construction to the first five Painleve equations. The seventh section of this work contains results on convergence of formal power series solutions to the fifth Painleve equation near ...

Added: May 11, 2014

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

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

Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073508-1-073508-19

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 ...

Added: September 19, 2012

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

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

Takeuchi K., Esterov A. I., Lemahieu A., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1309.0630v4.

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One ...

Added: September 18, 2017

Kiritchenko V., Smirnov E., Timorin V., Snapshots of modern mathematics from Oberwolfach (Germany) 2015

In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion. ...

Added: July 10, 2015

Parusnikova A., , in : Banach Center Publications. Vol. 97: Formal and Analytic Solutions of Differential and Difference Equations,.: Warsz. : Polish Academy of Sciences, 2012. P. 113-124.

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. ...

Added: March 24, 2013

Esterov A. I., Takeuchi K., Nagoya Mathematical Journal 2018 Vol. 231 P. 1-22

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al. [Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990), 255–271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves ...

Added: October 31, 2018

Parusnikova A., Vasilyev A. V., Journal of Dynamical and Control Systems 2019 Vol. 25 No. 4 P. 681-690

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 ...

Added: June 4, 2019

Parusnikova A., / Cornell University Library. 2013. No. 1310.5345.

The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlev\'{e} equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal power series solutions. The results obtained are compared with the ones obtained by means of Power Geometry. ...

Added: October 20, 2013

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1-707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

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

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

Berlin : De Gruyter, 2012

http://www.degruyter.com/view/books/9783110275667/9783110275667.v/9783110275667.v.xml ...

Added: February 16, 2013

Казарновский Б. Я., Хованский А. Г., Esterov A. I., Успехи математических наук 2021 Т. 76 № 1 С. 95-190

The notions of Newton polytope, toric variety, tropical geometry and Groebner basis established fundamental relations between the algebraic and convex geometries. This survey presents the state of the art of the interfaces between these notions. ...

Added: October 27, 2020

Esterov A. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 229-245

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.
In particular, our result proves the multivariate ...

Added: February 5, 2019

Esterov A. I., Takeuchi K., Lemahieu A., Journal of the European Mathematical Society 2021

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide ...

Added: November 28, 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

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

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