International Journal of Mathematics

We study the underlying relationship between Painlevé equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g. Lax formalism, Hirota bilinear relations for τ-functions, Weyl group symmetry and algebraic solutions in terms of the character polynomials, i.e. the Schur function and the universal character.

In this paper we describe a simply connected component of the complex manifold Stab(X) of stability conditions on a generic complex torus X. A generic complex torus is a complex torus X with H^p,p(X) ∩ H^2p(X, ℤ) = 0 for all 0 < p < dim X.

The purpose of this paper is twofold. The first is to show a second main theorem with truncated counting function for holomorphic curves from punctured disks into semi-Abelian varieties. The second is to give an alternative proof of a Big Picard's theorem for algebraically non-degenerate holomorphic curves by Nevanlinna theory.

In this paper, we study the regularity of positive solutions to an integral system associated with Bessel potentials. It is a natural extension of the work of Chen, Li, and Ma in which regularity of solutions to a system of Wolff potentials was proved by using the method of regularity lifting. We will fully use the decay estimate for the Bessel kernel both at the origin and the infinity. This integral system is of independent interest since it coincides with the famous Schrödinger type system of two semi-linear fractional partial differential equations.

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M_Sp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_Sp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where , together with the automorphisms induced by automorphisms of X.

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

In the present paper, we consider the generalized Hyers–Ulam stability for fractional differential equations of the form:

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e. a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.

In this paper, we investigate the initial boundary value problem of the nonlinear fourth-order dispersive-dissipative wave equation. By using the concavity method, we establish a blow-up result for certain solutions with arbitrary positive initial energy.

This paper studies chambers cut out by a special kind of hyperplane arrangements in general position, the Veronese arrangements, in the real projective spaces.