Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds

Given a finite dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$ where $M$ is a compact manifold with $\dim M<\dim N$. For a right invariant weak Riemannian metric on $\operatorname{Diff}_{\mathcal S}(N)$ induced by a quite general operator $L:\frak X_{\mathcal S}(N)\to \Gamma(T^*N\otimes\operatorname{vol}(N))$, we consider the induced weak Riemannian metric on $B(M,N)$ and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O'Neill's formula very transparent, and we use it finally to compute sectional curvature on $B(M,N)$.Comment: 28 pages. In this version some misprints correcte

Possible canted antiferromagnetism in UCu9_9Sn4_4

We report on the new compound UCu${}_9$Sn${}_4$ which crystallizes in the tetragonal structure \emph{I}4/\emph{mcm} with lattice parameters $a = 8.600{\rm\AA}$ and $c = 12.359{\rm\AA}$. This compound is isotyp to the ferromagnetic systems RECu${}_9$Sn${}_4$ (RE = Ce, Pr, Nd) with Curie temperatures $T{}\rm_C$ = 5.5 K, 10.5 K and 15 K, respectively. UCu${}_9$Sn${}_4$ exhibits an uncommon magnetic behavior resulting in three different electronic phase transitions. Below 105 K the sample undergoes a valence transition accompanied by an entropy change of 0.5 Rln2. At 32 K a small hump in the specific heat and a flattening out in the susceptibility curve probably indicate the onset of helical spin order. To lower temperatures a second transition to antiferromagnetic ordering occurs which develops a small ferromagnetic contribution on lowering the temperature further. These results are strongly hinting for canted antiferromagnetism in UCu${}_9$Sn${}_4$.Comment: 2 pages, 3 figures, SCES0

Construction of completely integrable systems by Poisson mappings

Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail.Comment: AmsTeX, 9 page

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1+1)-dimensional completely integrable soliton equations of differential-difference type.Comment: 47 page

Completely integrable systems: a generalization

We present a slight generalization of the notion of completely integrable systems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page

Towards a Lagrange-Newton approach for PDE constrained shape optimization

The novel Riemannian view on shape optimization developed in [Schulz, FoCM, 2014] is extended to a Lagrange-Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example.Comment: 16 pages, 4 figures, 1 tabl

A quantum-group-like structure on noncommutative 2-tori

In this paper we show that in the case of noncommutative two-tori one gets in a natural way simple structures which have analogous formal properties as Hopf algebra structures but with a deformed multiplication on the tensor product

People in Nature: Environmental History of the Kennebec River, Maine

The quality of a river affects the tributaries, lakes, and estuary it feeds; it affects the wildlife and vegetation that depend on the river for energy, nutrients, and habitat, and also affects the human community in the form of use, access, pride, and sustainability. In an age of mass consumerism and materialism, dwindling natural resources and wild spaces, and advanced technology, the ability to make a living and at the same time enjoy the benefits of rural living is increasingly difficult. Using the entire Kennebec River watershed as the scale of investigation with particular focus on the river corridor itself, my project looks at the interconnectedness of the river and the surrounding human community in a whole ecosystem analysis. Through coordinated efforts in the 1960s and 1970s in pollution abatement control and natural resource management, the communities of central Maine improved water quality in the Kennebec River from what was once described as an open sewer to conditions that provide for thriving aquatic life and improved access, enjoyment and economic health for the people. Vibrant commercial districts appeared, and tourism, fishing, boating and swimming all increased as a result of the improved river quality. Based on Mainer\u27s values of economy, rural living, and environmental health, management of the Kennebec met the needs and values of the whole ecosystem (social, economic, biogeophysical). Providing the river with conditions necessary for clean water, the people were in turn, sustained by the it as a natural resource; this resembles cutting-edge ecological theory called supply-side sustainability: maintaining, or fostering the development of, the systemic contexts that produce the goods, services, and amenities that people need or value, at an acceptable cost, for as long as they are needed or valued. My project provides an example of how people can translate their values of economic well being, ecological integrity, and the enjoyment of nature in their everyday lives into a sustainable system which provides for their every value

Poisson structures on double Lie groups

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gau{\ss}-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.Comment: AmSTeX, 37 page