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## Convergence of Formal Solutions to the Second Member of the Fourth Painlevé Hierarchy in a Neighborhood of Zero

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 formal expansion of the solution to the second-order differential equation in a symbolic computation packet is given.

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

Koroleva Y., Chechkin G. A., Persson L. -. et al., Eurasian Math. Journal 2011 Vol. 2 No. 1 P. 81-103

We derive a new three-dimensional Hardy-type inequality for a cube for the class of functions from the Sobolev space H1 having zero trace on small holes distributed periodically along the boundary. The proof is based on a careful analysis of the asymptotic expansion of the first eigenvalue of a related spectral problem and the best constant of the corresponding Friedrichs-type inequality. ...

Added: October 18, 2021

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

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

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

Minabutdinov A., Journal of Mathematical Sciences 2016 Vol. 215 No. 6 P. 738-747

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. ...

Added: July 8, 2016

NY : ACM, 2018

The International Symposium on Symbolic and Algebraic Computation (ISSAC) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2018, to be held at the City University of New York, New York City, USA, is the 43rd meeting in this series. The series has been held annually since 1981. ISSAC is sponsored ...

Added: November 1, 2019

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

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

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

Bobrova I., Sokolov V., Nonlinearity 2022 Vol. 35 No. 12 P. 6528-6556

Using the Painleve–Kovalevskaya test, we find several polynomial matrix
systems, which can be regarded as non-commutative generalisations of the
Painleve-4 equation. For these systems isomonodromic Lax pairs are presented.
Limiting transitions that reduce them to known matrix Painleve-2 equations are
found. ...

Added: October 29, 2022

NY : ACM, 2017

Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation ...

Added: September 15, 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

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

Association for Computing Machinery (ACM), 2021

The International Symposium on Symbolic and Algebraic Computation (ISSAC) is the premier conference for research in symbolic computation and computer algebra. ISSAC is the continuation of conferences with the names SYMSAC, EUROCAL, EUROSAM, and EUROCAM, which have alternated between North America and Europe before merging into ISSAC. The whole series started in 1966 with the ...

Added: September 27, 2021

Bychkov A., Pogudin G., ACM Communications in Computer Algebra 2021 Vol. 54 No. 3 P. 119-123

Transformation of a polynomial ODE system to a special quadratic form has been successfully used recently as a preprocessing step for model order reduction methods. However, to the best of our knowledge, there has been no practical algorithm for performing this step automatically with any optimality guarantees.
We present an algorithm that, given a system of ...

Added: October 19, 2021

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

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

Borisov D. I., Cardone G., Chechkin G. A. et al., Calculus of Variations and Partial Differential Equations 2021 Vol. 60 Article 48

We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the ...

Added: September 20, 2021

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

Stukopin V., Linear Algebra and its Applications 2019 Vol. 580 No. 1 P. 292-335

This paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric (in general non-Hermitian) Toeplitz matrix. The symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane. The main result describes the asymptotic ...

Added: March 1, 2020

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

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

I. A. Bobrova, Sokolov V. V., Journal of Geometry and Physics 2023 Vol. 191 Article 104885

We find all non-abelian generalizations of P1 - P6 Painleve systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations. ...

Added: June 21, 2023