Geometrical properties of Maslov indices in periodic-orbit theory

Maslov indices in periodic-orbit theory are investigated using phase space path integral. Based on the observation that the Maslov index is the multi-valued function of the monodromy matrix, we introduce a generalized monodromy matrix in the universal covering space of the symplectic group and show that this index is uniquely determined in this space. The stability of the orbit is shown to determine the parity of the index, and a formula for the index of the n-repetition of the orbit is derived.Comment: 18pages, 8figures, typos correcte

Note on the smallest root of the independence polynomial

One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real

A Bayesian approach to the estimation of maps between riemannian manifolds

Let \Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \gamma be a smooth map \Theta into a riemannian manifold \Lambda. An unknown state \theta \in \Theta is observed via X=\theta+\epsilon \xi where \epsilon>0 is a small parameter and \xi is a white Gaussian noise. For a given smooth prior on \Theta and smooth estimator g of the map \gamma we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \Theta and \Lambda, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \gamma is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.Comment: 20 pages, no figures published version includes correction to eq.s 31, 41, 4