From Bayesian Inference to MCMC and Convex Optimisation in Hadamard Manifolds

Analytic properties of the scattering amplitude for Coulomb potential on the background of the space of constant negative curvature are studied. Special attention is given to the comparison of the Regge trajectories for curved and flat spaces. We show that there exist considerably differences in the behavior of the Regge trajectories in these spaces.

The leaky tori model for scattering theory in mesoscopic systems was introduced by Gutzwiller in 1983. Muller’s class of admissable surfaces subsumes the leaky tori models of Gutzwiller. In addition, they form a natural generalization of locally symmetric spaces of constant negative curvature and finite volume. In this chapter we review Muller’s spaces in Section 1. In Section 2 scattering operators are developed. In Section 3 Weyl’s law for mesoscopic systems is discussed. In Section 4 Miiller’s trace formula is presented. In Sections 5 to 8 work of Guillope and Zworski (1994) on scattering theory on hyperbolic half-cylinders is developed. Their work uses some classical results of Poschl and Teller which allows one to explicitly deduce the resonance set for the Laplacian on the hyperbolic half-cylinder. In Section 9 we briefly review related results on scattering theory for two strictly convex bodies. And in Section 10 we return to the Gutzwiller trace formula and look at including diffraction terms.

The metrics of visual and haptic space based on parallelity judgements

A noncompact, nonintegrable billiard of importance in cosmology---the infinite equilateral triangle on a space of constant negative curvature---gives rise to a natural noncompact integrable approximation, which is studied here quantum mechanically. The Weyl formula for the averaged spectral staircase contains a nonstandard logarithmic term due to the noncompactness of the billiard. For two symmetry classes we determine numerically the spectral staircase, the level-spacing distribution p(s), and the averaged ${\mathrm{\ensuremath{\Delta}}}_{3}$ statistic, and compare with analytical expressions that follow from Poisson statistics and semiclassical theory. The Poissonian result for the correlation function of ${\mathrm{\ensuremath{\Delta}}}_{3}$ is derived in closed form and also compared with the data.

Visual space is, according to Luneburg, a non-Euclidean hyperbolic space of constant negative curvature. Two personal constants enter into the geometrical description of binocular subjective space. One is σ indicating the degree of depth perception characteristic to the observer. The second is K, the curvature; a value which denotes by how much the visual space of an individual departs from Euclidean space. The constants σ and K were determined experimentally for 30 untrained observers by Luneburg's 3- and 4-point tests. All values found for K were negative and support the hypothesis that the geometry of binocular space is hyperbolic. Some discrepancies between the theory and findings were observed, while the sensitivity of the experiments to the lack of training and even a slight measure of aniseikonia is evident.

The two-dimensional one-component logarithmic Coulomb gas is mapped onto a non-hermitian fermionic field theory. At $\beta=2$, the field theory is free. Correlation functions are calculated and a perturbation theory is discussed for extending to other $\beta$. A phase transition is found at the mean-field level at large $\beta$. Some results are extended to spaces of constant negative curvature.

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R − r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient R/r. 1. Preliminaries and the main results In [1] A. Borisenko and V. Miquel proved that a closed hypersurface with normal curvatures kn satisfying the inequality kn > 1 in the Lobachevsky space H(−1) can be put into a spherical shell between two concentric spheres of radii R and r such that the width R−r of the shell satis esR−r 6 ln 2. A similar estimate holds in Hadamard manifolds (see [2]). In [3] these results were extended for Riemannian manifolds of constant-signed sectional curvatures and hypersurfaces with normal curvatures bounded below. In the present paper we re ne some results from [3]. For this purpose we consider hypersurfaces with normal curvatures at any point and in any direction pinched by two positive constants. Such restriction allows us to obtain sharper estimates for the width R− r than in [3]. Furthermore, for such surfaces we are able to derive an upper bound on the quotient R/r, which can be arbitrarily large for a hypersurface with normal curvatures just bounded below. Let us denote by M(c) a complete simply connected m-dimensional Riemannian manifold of constant sectional curvatures equal to c. In order to state the main results, we need the following de nition. De nition 1.1. A hypersurface F ⊂ M(c) is said to be κ1, κ2-convex (with κ2 > κ1, and for c = 0 we assume that κ1 > 0, for c > 0 we assume that κ1 > 0 and for c √ −c), if for any point P ∈ F there exist two nested geodesic spheres S2 ⊂ S1 ⊂ M(c) of constant normal curvatures equal to, respectively, κ1 and κ2, and passing through P such that locally near P the hypersurface F lies inside S1 and outside S2. 2010 Mathematics Subject Classi cation. 53C40.

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function of a special convex set. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive, which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas--Rachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods such as the cyclic proximal point algorithm and half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric, which does not have a constant curvature.

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses move on a circle of not necessarily constant size has to be either a regular cocircular homographic orbit in flat space, or a regular polygonal rotopulsator in curved space, under the constraint that the minimal distance between point masses attains its minimum in finite time. Additionally, we prove that the same holds true if we add an extra mass at the center of that circle and find an explicit formula for the mass of each point particle in terms of the radius of the circle. Finally, we prove that for each order of the masses there is at most one cocircular homographic orbit for the case that the masses need not be constant.

It is shown that no non-flat empty space-time can be embedded locally and isometrically in a five-dimensional space of nonzero constant curvature.

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds Fl of dimension l ≥ 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H ≠ 0, ∇ ⊥ H ≠ 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S1 × S1 ⊂ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic Sl+1 × Sl+1 ⊂ G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold Fl ⊂ El+2 (l ≥ 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].

An old result of Stevenson [Phys. Rev.59, 842 (1941)] concerning the Kepler–Coulomb quantum problem on the three-dimensional (3-D) hypersphere is considered from the perspective of the radial Schrödinger equations on 3-D spaces of any (either positive, zero or negative) constant curvature. Further to Stevenson, we show in detail how to get the hypergeometric wave function for the hydrogen atom case. Finally, we make a comparison between the "space curvature" effects and minimal length effects for the hydrogen spectrum.