Gert Heckman

On the variation in the cohomology of the symplectic form of the reduced phase space

is called the momentum mapping of the Hamiltonian T-action. Given (1.1), the condition (1.2) just means that T acts along the fibers of J. For the basic definitions and properties of non-commutative Hamiltonian group actions, see [AM]. The results of this paper can easily be extended to Hamiltonian actions of arbitrary compact connected Lie groups, by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping. For some more details, see the remarks at the end of Sect. 2. We will assume throughout this paper that the momentum map is proper, that is J I(U) is compact for each compact subset U of t*. Now let r be a regular value of J, that is T,,J: TmM-~t* is surjective for all meYe=J 1(4 ). Then Ye is a smooth submanifold of M, compact because

Addendum to “on the variation in the cohomology of the symplectic form of the reduced phase space”

Here we have used the notation (~rlX) for the inner product of the vector field X with the form a, and ,Ix(m)= (X, J(m)) for the X-component of a. In an earlier paper [4] it was shown that the push forward J,(dm) of the Liouville measure dm on M under the momentum mapping J is a piecewise polynomial measure on !.*. Moreover, in case X has isolated isolated zeros on M an explicit formula for the integral

The volume of hyperbolic Coxeter polytopes of even dimension

Let H denote hyperbolic space of dimension n, and let S be an index set for a finite collection of open half spaces H s in H n bounded by codimension one hyperplanes Hs. We assume that for all distinct s, t ∈ S either Hs ∩ Ht is not empty and the (interior) dihedral angle of H s ∩ H + t along Hs ∩ Ht has size π mst for certain integers mst = mts ≥ 2, or Hs ∩Ht is empty while H + s ∩H + t is not empty. In the latter case we put mst = mts = ∞ and we also put mss = 1. Under these assumptions the intersection C = ⋂ s H + s is not empty, and its closure D is called a hyperbolic Coxeter polytope. By abuse of notation let s ∈ S also denote the reflection of H in the hyperplane Hs. Now the group W of motions of H n generated by the reflections s ∈ S is discrete, and D is a strict fundamental domain for the action of W on H. Moreover (W,S) is a Coxeter group with Coxeter matrix M = (mst), i.e. W has a presentation with generators s ∈ S and relations (st)s,t = 1 for s, t ∈ S. Let l(w) denote the length of w ∈ W with respect to the generating set S, and let PW (t) ∈ Z[[t]] be the Poincare series of W defined by PW (t) = ∑ w t .

On the regularization of the Kepler problem

We show that for the Kepler problem the canonical Ligon-Schaaf regularization map can be understood in a straightforward manner as an adaptation of the Moser regularization. In turn this explains the hidden symmetry in a geometric way.

Quantum integrability for the Kovalevsky top

LetN be a differentiable manifold of dimension n. LetM = T ∗N be the cotangent bundle of N , and denote by σ the natural symplectic form on M . In local coordinates q = (q1, . . . qn) on N and pi = ∂ ∂qi (viewed as function on M) we have σ = ∑ dpi∧dqi. For f ∈ C ∞(M) the hamiltonian vector field vf on M is defined by df(·) = −σ(vf , ·). For this sign convention we have [vf , vg] = v{f,g} (1.1) with the Poisson bracket {f, g} of f, g ∈ C∞(M) defined by {f, g} = vf (g) = σ(vf , vg), or in local coordinates {f, g} = ∑ (

High-precision spatial localization of mouse vocalizations during social interaction

Mice display a wide repertoire of vocalizations that varies with age, sex, and context. Especially during courtship, mice emit ultrasonic vocalizations (USVs) of high complexity, whose detailed structure is poorly understood. As animals of both sexes vocalize, the study of social vocalizations requires attributing single USVs to individuals. The state-of-the-art in sound localization for USVs allows spatial localization at centimeter resolution, however, animals interact at closer ranges, involving tactile, snout-snout exploration. Hence, improved algorithms are required to reliably assign USVs. We develop multiple solutions to USV localization, and derive an analytical solution for arbitrary vertical microphone positions. The algorithms are compared on wideband acoustic noise and single mouse vocalizations, and applied to social interactions with optically tracked mouse positions. A novel, (frequency) envelope weighted generalised cross-correlation outperforms classical cross-correlation techniques. It achieves a median error of ~1.4 mm for noise and ~4–8.5 mm for vocalizations. Using this algorithms in combination with a level criterion, we can improve the assignment for interacting mice. We report significant differences in mean USV properties between CBA mice of different sexes during social interaction. Hence, the improved USV attribution to individuals lays the basis for a deeper understanding of social vocalizations, in particular sequences of USVs.

Angular momenta of relative equilibrium motions and real moment map geometry

Chenciner and Jiménez-Pérez (Mosc Math J 13(4):621–630, 2013) showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general Euclidean space form a convex polytope. In this note we explain how this result follows from a general convexity theorem of O’Shea and Sjamaar in real moment map geometry (Math Ann 31:415–457, 2000). Finally, we provide a representation-theoretic description of the pushforward of the normalized measure under the real moment map for Riemannian symmetric pairs.

An Odd Presentation for W(E6)

The Weyl group W(E6) has an odd presentation due to Christopher Simons as factor group of the Coxeter group on the Petersen graph by deflation of the free hexagons. The goal of this paper is to give a geometric meaning for this presentation, coming from the action of W(E6) on the moduli space of marked maximally real cubic surfaces and its natural tessellation as seen through the period map of Allcock, Carlson and Toledo.

Chapter 2 – The periodic Calogero-Moser system

This chapter discusses the quantum integrability for the periodic Calogero-Moser system. H as an affine algebraic variety the algebra C[P] is just the ring of regular functions on H, or equivalently the ring of holomorphic functions on H with moderate growth at infinity. The operator L(k) is the standard second-order hypergeometric operator. The periodic Calogero-Moser potential with coupling constant g2 Є K (the 2 is a square) is the function. Calogero subsequently studied the quantum scattering problem for an arbitrary number of particles on the line. Moser proved the classical integrability (still in case R of type An) by giving a Lax representation. Generalizing the method of Moser partial results on the classical integrability were obtained by Olshanetsky and Perelomov for classical root systems R.

Invariant differential operators

This chapter discusses the invariant differential operators. The description of D(G/K) is based on the Iwasawa decomposition. The description of D(G/H) is given for the general semisimple symmetric space G/H. Viewing functions on G/H as right H-invariant functions on G it follows that there is a natural action of the elements of U (g)H on C∞ (G/H). The Harish-Chandra isomorphism is discussed. The proof of the Wo-invariance is presented. A Caftan subspace for G/H is a maximal abelian subspace of q, consisting of semisimple elements. If G is a classical Lie group, or if the rank of G/H is one, then Z (G/H) = D (G/H).