Painlevé scheme

Yusuke Sasano

Home/Archive/arXiv

arXiv · 0709.0597

Painlevé scheme

Yusuke Sasano

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we review the notion of Painlevé scheme of the sixth Painlevé equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${Σ_2}$. The key method is Painlevé $α$-method for each accessible singular point. Giving a Painlevé scheme in the differential system satisfying certain conditions, we can recover the Painlevé VI system with the polynomial Hamiltonian. We also consider the case of the Painlevé V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${Σ_n}$. This equation has symmetry of symmetric group of degree $n+2$.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yusuke Sasano. 2016-04-20. Painlevé scheme. https://arxiv.org/abs/0709.0597

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Geometric Duality Between Constraints and Gauge Fields: Mirror Realization and Reduction Geometry on Principal Bundles

A connection and a nonzero parallel adjoint field determine an invariant hyperplane constraint on a principal bundle. Its sign mirror preserves the hyperplane and reverses its coorientation; global gauge realization is controlled by a twisted stabilizer reduction. For regular fields we identify the normalizing gauge extension as a pushout of the torus-normalizer extension, giving exact lift orders and simultaneous-splitting criteria. In singular rank-two block families, reductions on a fixed trivial bundle form an affine second-Chern lattice whose Weyl stabilizers and finite-order lift spectra detect topology invisible to paired curvature. The reduction framework also determines the structure group and second cohomology of the matched-flag diagonalization space of Friedman and Park, and gives a first- and second-Chern criterion for normal matrices with fixed separated spectrum on four-complexes; every integral solution of their three-eigenline equation on $S^2\times S^2$ is realized. For moving reductions, the projected circle curvature differs from the ambient paired curvature by a covariant-derivative term. Full fatness on a closed four-manifold forces a nontrivial sign-mirror obstruction for every circle reduction; hyperbolic self-dual-form bundles also provide circle reductions in the $y$-fat setting of Florit and Ziller. Contact transgression, bundle automorphism twists, and the natural first-jet Spencer operator complete the geometric picture.

math.GM↗

arXiv2026

A proof of the irrationality of $π$ motivated by nested radicals with roots of $2$

In this work, we prove the irrationality of $π$ motivated by nested radicals with roots of $2$ of the kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $π$ as the integer $k$ increases are presented.

math.GM↗

arXiv2026

Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions

We provide an explicit analytical evaluation of the known rational Ramanujan-type series for the theory of signature 2. Focusing on the singular moduli $k_r$ for $r \in \{2, 3, 4, 7\}$, we demonstrate that the underlying elliptic identities can be established through modular transformations of degree 2. In particular, we showcase a family of rational harmonic Ramanujan-type series for $1/π$ involving higher-degree polynomials

math.GM↗