The dependence of the sixth equation of Painlevé on its four parameters (2α,−2β, 2γ, 1− 2δ) = (θ ∞ , θ 0 , θ 2 1 , θ 2 x) is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar “Lax” pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on θ∞ (and a holomorphic dep...

By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed m...

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a wellknown Lax representation – the three-particle open Toda lattice – is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certai...

We study the conformal properties of the multi-constraint KP hierarchy and its nonstandard partner by covariantizing their corresponding Lax operators. The associated second Hamiltonian structures turn out to be nonlocal extension of Wn algebra by some integer or half-integer spin fields depending on the order of the Lax operators. In particular, we show that the complicated second Hamiltonian ...

We provide a proof of Nagata’s compactification theorem: any separated map of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism. This is a detailed exposition of private notes of Deligne that translate Nagata’s method into modern terms, and includes some applications of general interest in the theo...

Popular statistical software packages do not have the proper procedures for determining the number of components in factor and principal components analyses. Parallel analysis and Velicer's minimum average partial (MAP) test are validated procedures, recommended widely by statisticians. However, many researchers continue to use alternative, simpler, but flawed procedures, such as the eigenvalue...

The ability of lymphokine-activatedkiller (LAK) cells to mediate antibody-dependent cellular cytotoxicity and its efficacy against a LAX resistant tumor were investigated. Cells ofthe MH134 murine hepatoma line are scarcely lysed by LAX cells generated in vitro by incubatiOn of G3H/HeN mousespleen cells with humanrecombinantinterleukin2 (rIL 2). However, the splenic LAX cells potently lysed the...

1. INTRODUCTION. Suppose that Alice wants to color a planar map using four colors in a proper way, that is, so that any two adjacent regions get different colors. Despite the fact that she knows for certain that it is eventually possible, she may fail in her first attempts. Indeed, there are usually many proper partial colorings not extend-able to proper colorings of the whole map. Thus, if she...