Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $\pi_k O(f)=\pi_k M$ for $k\geq 3$, and $\pi_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $\pi_1 O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$. We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tame Lie group on tame Frechet manifold is a tame Frechet manifold itsel

B553: Consumer Packages for Maine Mcintosh Apples

Three kinds of consumer packages for apples were developed for testing in the 1955-56 marketing season. In developing these packages, the authors modified the jumble-pack, polyethylene package in a way that would protect the fruit from most of the bruising and still maintain almost complete visibility of the fruit. One consumer package developed was a long narrow polyethylene bag, another was a polyethylene bag with a divider insert, and the third package had a cell partition placed in a similar plastic bag. All three packages were well accepted by consumers in the Portland market.https://digitalcommons.library.umaine.edu/aes_bulletin/1085/thumbnail.jp

New empirical fits to the proton electromagnetic form factors

Recent measurements of the ratio of the elastic electromagnetic form factors of the proton, G_Ep/G_Mp, using the polarization transfer technique at Jefferson Lab show that this ratio decreases dramatically with increasing Q^2, in contradiction to previous measurements using the Rosenbluth separation technique. Using this new high quality data as a constraint, we have reanalyzed most of the world e-p elastic cross section data. In this paper, we present a new empirical fit to the reanalyzed data for the proton elastic magnetic form factor in the region 0 < Q^2 < 30 GeV^2. As well, we present an empirical fit to the proton electromagnetic form factor ratio, G_Ep/G_Mp, which is valid in the region 0.1 < Q^2 < 6 GeV^2