Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy

The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlev\'e equation. Moreover we prove that these are unique in certain sectors near infinity.Comment: 13 pages, Late

Ahlâk ve Demokrasi

Ahlâk nedir? Sokrates buna kendine hâkim olma ve kendini yönetme sanatı diyordu. Öğrettiği tek temel erdem egemenlikti (éyKpâTela): kendi üzerinde egemenlik. Demokrasi de halkın kendi kendini yönetmesi anlamına gelir; halkın halk tarafından yönetilmesi. Sonuç olarak, demokrasi ahlâkî düşüncenin siyasete açık bir şekilde uygulanması değil midir? Demokratik yönetim ve ahlâka dayalı yönetim tek ve aynı şey gibi görünmektedir. Tarihsel gerçeklik etimolojiye dayanan bu çıkarımı doğruluyor mu? Gözümüze çarpan ilk şey, demokrasi olarak adlandırılan yönetimlerin aşırı çeşitliliğidir. Demokrasi kavramının iki unsuru iki kavramdan oluşur: halk ve hükümet. Bunların her birine çok farklı yorumlar getirilmiştir

Quasi-linear Stokes phenomenon for the Painlev\'e first equation

Using the Riemann-Hilbert approach, the $\Psi$-function corresponding to the solution of the first Painleve equation, $y_{xx}=6y^2+x$, with the asymptotic behavior $y\sim\pm\sqrt{-x/6}$ as $|x|\to\infty$ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the power-like expansion to the latter are found.Comment: version accepted for publicatio

Stable Non--Perturbative Minimal Models Coupled to 2D Quantum Gravity

A generalisation of the non--perturbatively stable solutions of string equations which respect the KdV flows, obtained recently for the $(2m-1,2)$ conformal minimal models coupled to two--dimensional quantum gravity, is presented for the $(p,q)$ models. These string equations are the most general string equations compatible with the $q$--th generalised KdV flows. They exhibit a close relationship with the bi-hamiltonian structure in these hierarchies. The Ising model is studied as a particular example, for which a real non-singular numerical solution to the string susceptibility is presented.Comment: (35 pp; two figures not included; plain TEX

A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

We present some observations on the tau-function for the fourth Painlev´e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlev´e equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero

Quasi-linear Stokes phenomenon for the second Painlev\'e transcendent

Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. The precise description of the exponentially small jump in the dominant solution approaching $\alpha/x$ as $|x|\to\infty$ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is found.Comment: 19 pages, LaTe