Kumamoto journal of mathematics最新文献

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Polynomial Hamiltonian structure for the A_3 system A_3系统的多项式哈密顿结构

Kumamoto journal of mathematics

Pub Date : 1997-03-01 DOI: 10.2206/KYUSHUJM.52.455

Deming Liu, Kazuo Okamoto

引用次数: 3

On Some Hamiltonian Structures of Painleve Systems, III 关于疼痛级系统的若干hamilton结构，第三章

Kumamoto journal of mathematics

Pub Date : 1997-03-01 DOI: 10.2969/JMSJ/05140843

Atushi Matumiya

This is the second part of the series of our papers. In the preceding paper([11]), we studied a Hamiltonian structure of the sixth Painlevé system (H V I) equivalent to the sixth Painlevé equation P V I. In this paper, we continue the study for Painlevé systems (H J) or Painlevé equations P J for are the equations given by P V : d 2 x dt 2 = 1 2x + 1 x − 1 dx dt 2 − 1 t dx dt + (x − 1) 2 t 2 αx + β x +γ x t + δ x(x + 1) x − 1 , P IV : d 2 x dt 2 = 1 2x dx dt 2 + 3 2 x 3 + 4tx 2 + 2(t 2 − α)x + β x P III : d 2 x dt 2 = 1 x dx dt 2 − 1 t dx dt + 1 t (αx 2 + β) + γx 3 + δ x , P II : d 2 x dt 2 =2x 3 + tx + α, where x and t are complex variables, α, β, γ, and δ are complex constants([4]). It is known that each P J is equivalent to a Hamiltonian system (H J) : dx/dt = ∂H J /∂y, dy/dt = −∂H J /dx, where H V (x, y, t) = 1 t [x(x − 1) 2 y 2 − {κ 0 (x − 1) 2 + κ t x(x − 1) − ηtx}y + κ(x − 1)] (κ := 1 4 {(κ 0 + κ t) 2 − κ 2 ∞ }), H IV (x, y, t) =2xy 2 − {x 2 + 2tx + 2κ 0 }y + κ ∞ x, H III (x, y, t) = 1 t [2x 2 y 2 − {2η ∞ tx 2 + (2κ 0 + 1)x − 2η 0 t}y + η ∞ (κ 0 + κ ∞)tx], H II (x, y, t) = 1 2 y 2 − (x 2 + t 2)y − (α + 1 2)x. Here the relations between the constants in the equations P J and those in the Hamiltonians are given by α = κ ∞ 2 /2, β = −κ 0 2 /2, γ = −η(1 + κ t), δ = −η 2 /2

这是我们系列论文的第二部分。在前面的文章([11]),我们学习第六Painleve系统的哈密顿结构(H V I)第六Painleve方程等效P V。在本文中,我们继续研究Painleve系统(H J)或Painleve方程是P J方程由P V: d 2 x dt 2 = 1 2 x + 1−1 dx dt 2−1 t dx dt + x (x−1)2 t 2α+βx t + x +γδx (x + 1) x−1,P IV: d 2 x dt 2 = 1 2 x dx dt 2 + 3 2 x 3 + 4 tx 2 + 2 (t 2−α)x +βx P III:d 2x dt 2 = 1 x dx dt 2 - 1 t dx dt + 1 t (α x2 + β) + γ x3 + δ x, P II: d 2x dt 2 =2x 3 + tx + α，其中x和t是复变量，α， β， γ和δ是复常数([4])。众所周知,每个P J等效哈密顿系统(H J): dx / dt =∂H J /∂y, dy / dt =−∂H J / dx,在H V (x, y, t) = 1 t [x (x−1)2 y 2−{κ0 (x−1)2 +κt x (x−1)−ηtx} y +κ(x−1)](κ:= 1 4{(κ0 +κt) 2−κ2∞}),H IV (x, y, t) = 2 xy 2−{x 2 + 2 tx + 2κ0}y +κ∞x,第三H (x, y, t) = 1 t [2 x 2 y 2−{2η∞tx 2 +(2κ0 + 1)x−2η0 t} y +η∞(κ0 +κ∞)tx), H II (x, y, t) = 1 2 y 2−(x 2 + t 2) y−(α+ 1 2)x。方程P J常数与哈密顿常数之间的关系为:α = κ∞2 /2，β = - κ 0 2 /2， γ = - η(1 + κ t)， δ = - η 2 /2

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引用次数: 77

On the Proportion of Quadratic Twists for Non-vanishing and Vanishing Central Values of L-Functions Attached to Primitive Forms 附于原始形式的l -函数中心值不消失与消失的二次扭转的比例

Kumamoto journal of mathematics

Pub Date : 1900-01-01 DOI: 10.1007/978-3-319-11352-4_13

K. Makiyama

引用次数: 0

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