The pure state space of quantum mechanics as Hermitian symmetric space

The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kähler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

Riemannian geometry of quantum computation

A review is given of some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group, SU(2ⁿ). Using the Lie algebra su(2ⁿ), detailed derivations are given of a useful Riemannian geometry of SU(2ⁿ), including the connection and the geodesic equation for minimal complexity quantum computations.     

Collars and partitions of hyperbolic cone-surfaces

For compact Riemann surfaces, the collar theorem and Bers’ partition theorem are major tools for working with simple closed geodesics. The main goal of this article is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone angles to be strictly less than π to be able to consider partitions. Emily B. Dryden—partially supported by the US National Science Foundation grant DMS-0306752. Hugo Parlier—supported by the Swiss National Science Foundation grants 21-57251.99 and 20-68181.02.     

Necessary and Sufficient Conditions for Mcshane’s Identity and Variations

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane’s identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch’s variation of McShane’s identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.     

Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one —IV

This paper is the first of a series of papers in which we generalize our results in (Asian J. of Math. 4, 817–830 (2000); J. Geom. Anal. 12, 63–79 (2002); Intern. J. Math. 14, 259–287 (2003)) to the general complex compact almost homogeneous manifolds of real cohomogeneity one. In this paper we deal with the exceptional case of the G ₂ action (Cf. Intern. J. Math. 14, 259–287 (2003), p. 285). In particular, we prove the existence of Kähler-Einstein metric on this manifold.     

Properties of reachable sets in the sub-Lorentzian geometry

The aim of this paper is to develop local theory of future timelike, nonspacelike and null reachable sets from a given point _q0$q_0$ in the sub-Lorentzian geometry. In particular, we prove that if U$U$ is a normal neighbourhood of _q0$q_0$ then the three reachable sets, computed relative to U$U$, have identical interiors and boundaries with respect to U$U$. Further, among other things, we show that for Lorentzian metrics on contact distributions on R²ⁿ⁺¹${\mathbb{R}}^{2n+1}$, n≥1$n\ge 1$, the boundary of reachable sets from _q0$q_0$ is, in a neighbourhood of _q0$q_0$, made up of null future directed curves starting from _q0$q_0$. Every such curve has only a finite number of non-smooth points; smooth pieces of every such curve are Hamiltonian geodesics. For general sub-Lorentzian structures, contrary to the Lorentzian case, timelike curves may appear on the boundary. It turns out that such curves are always Goh curves. We also generalize a classical result on null Lorentzian geodesics: every null future directed Hamiltonian sub-Lorentzian geodesic initiating at _q0$q_0$ is contained, at least to a certain moment of time, in the boundary of the reachable set from _q0$q_0$.     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

Projective structure in 4-dimensional manifolds with positive definite metrics

This paper considers the situation on a 4-dimensional manifold admitting two metric connections, one of which is compatible with a positive definite metric, and which have the same unparametrised geodesics. It shows how, in many cases, the relationship between these connections and metrics can be found. In many of these cases, the connections are found to be necessarily equal. The general technique used is that based on a certain classification of the curvature tensor together with holonomy theory.