Hamiltonian G-Spaces with Regular Momenta

Let G be a compact connected non-Abelian Lie group and let (P, w, G, J) be a Hamiltonian G-space. Call this space a G-space with regular momenta if J(P) ⊂ g*reg, here g*reg⊂g* denotes the regular points of the co-adjoint action of G. Here problems involving a G-space with regular momenta are reduced to problems in an associated lower dimensional Hamiltonian T-space, where T ⊂ G is a maximal torus. For example two such G-spaces are shown to be equivalent if and only if they have equivalent associated T-spaces. We also give a new construction of a normal form due to Marle (1983), for integrable G-spaces with regular momenta. We show that this construction, which is a kind of non-Abelian generalization of action-angle coordinates, can be reduced to constructing conventional action-angle coordinates in the associated T-space. In particular the normal form applies globally if the action-angle coordinates can be constructed globally. We illustrate our results in concrete examples from mechanics, including the rigid body. We also indicate applications to Hamiltonian perturbation theory

Pseudogroups via pseudoactions: Unifying local, global, and infinitesimal symmetry

A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold $M$. A pseudoaction generates a pseudogroup of transformations of $M$ in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the action of a Lie algebra on $M$, possibly twisted. A global converse to Lie's third theorem proven here states that every twisted Lie algebra action is integrated by a pseudoaction. When the twisted Lie algebra action is complete it integrates to a twisted Lie group action, according to a generalization of Palais' global integrability theorem.Comment: 31 pages; minor revision

Cartan Connections on Lie Groupoids and their Integrability

A multiplicatively closed, horizontal $n$-plane field $D$ on a Lie groupoid $G$ over $M$ generalizes to intransitive geometry the classical notion of a Cartan connection. The infinitesimalization of the connection $D$ is a Cartan connection $\nabla$ on the Lie algebroid of $G$, a notion already studied elsewhere by the author. It is shown that $\nabla$ may be regarded as infinitesimal parallel translation in the groupoid $G$ along $D$. From this follows a proof that $D$ defines a pseudoaction generating a pseudogroup of transformations on $M$ precisely when the curvature of $\nabla$ vanishes. A byproduct of this analysis is a detailed description of multiplication in the groupoid $J^1 G$ of one-jets of bisections of $G$

Using bone measurements to estimate the original sizes of bluefish (Pomatomus saltatrix) from digested remains

The ability to estimate the original size of an ingested prey item is an important step in understanding the community and population structure of piscivorous predators (Scharf et al., 1998). More specifically, knowledge of original prey size is essential for deriving important biological information, such as predator consumption rates, biomass of the prey consumed, and selectivity of a predator towards a specific size class of prey (Hansel et al., 1988; Scharf et al., 1997; Radke et al., 2000). To accurately assess the overall “top-down” pressure a predator may exert on prey community structure, prey size is crucial. However, such information is often difficult to collect in the field (Trippel and Beamish, 1987). Stomach-content analyses are the most common methods for examining the diets of piscivorous fish, but the prey items found are often thoroughly digested and sometimes unidentifiable. As a result, obtaining a direct measurement of prey items is frequently impossible

Gargoyles on Glatfelter Hall

When one walks around the campus of Gettysburg College, Glatfelter Hall towers above them, as one of the College’s most commanding edifices. One takes notice of the arched doorways, sunken windows, and the giant bell tower whose occupant chimes on the hour. What one may not notice are the eyes watching from the brownstone; faces and creatures at home in the stone, surveying your every move. Grotesques and gargoyles sit in the moldings, on the window sills and at the junction where roof and wall meet, hidden from the eye that does not have the compulsion to look. These architectural ornaments are not noticed outright because they tend to blend in with the stonework on the building. However, once you have seen them, you never cease to feel their eyes upon you. [excerpt] Course Information: Course Title: HIST 300: Historical Method Academic Term: Spring 2006 Course Instructor: Dr. Michael J. Birkner \u2772 Hidden in Plain Sight is a collection of student papers on objects that are hidden in plain sight around the Gettysburg College campus. Topics range from the Glatfelter Hall gargoyles to the statue of Eisenhower and from historical markers to athletic accomplishments. You can download the paper in pdf format and click View Photo to see the image in greater detail.https://cupola.gettysburg.edu/hiddenpapers/1001/thumbnail.jp