Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces

We prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Besov spaces $B^{r}_{p,q}(\mathbb{R}^n)$, $r>n/2p$. When $p=2$ and $n\geq 3$, we obtain global solutions, provided the parameters $r,q$ and $n$ satisfy certain inequalities. This is an improvement over known analogous Sobolev space results, which required $n=3$

Anglo-American Studies for In-training Teachers

The students who take the class英米地域研究（Eibei Chiiki Kenkyuu）or Anglo-American Area Studies have a high interest in English and Western culture but few of them know anything about the origins of English or the events that resulted in English taking on the role of a global language used by more than 2 billion people throughout the world. In class, we cover three areas. First, the three major stages of the development of the English language, Old English, Middle English, and Modern English are studied. The second stage deals with the development of English as a global language. In the third and final part, we study the recent concept of World Englishes.研究ノー

Illuminating hadron structure by scattering light on light

The results of an Amplitude Analysis of the world data on integrated and differential cross-sections on $\gamma\gamma\to\pi\pi$ are presented, following the publication of the Belle charged pion results.Comment: 5 pages, 7 figures. To appear in the Proceedings of the International Workshop on $e^+e^-$ collisions from $\phi$ to $\psi$, Frascati, April 2008. This version rewords the reference to the work of Achasov and Shestakov, with apologies to them for any misunderstandin

The six-point remainder function to all loop orders in the multi-Regge limit

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.Comment: 25 pages, 1 figur

Pennington to Earlham College, July 20, 1947

Pennington writing to Earlham College endorsing Ruanna Jean Gulley as a student.https://digitalcommons.georgefox.edu/levi_pennington/1176/thumbnail.jp

Levi Pennington Writing to Bertha May, October 15, 1946

Levi Pennington writing a heart-felt letter to his daughter Bertha May about the difficulties they have had with each other and about how she has been handling her health and finances.https://digitalcommons.georgefox.edu/levi_pennington/1108/thumbnail.jp