A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general.Comment: 36 page

Routing on the Visibility Graph

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the \emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {\em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the 28th International Symposium on Algorithms and Computation (ISAAC 2017). Final version appeared in the Journal of Computational Geometr

Top quark mass measurement with ATLAS

The top quark mass measurement with ATLAS in the lepton plus jets channel is summarized from the perspective of the early data. Using the invariant mass of the three jets arising from the hadronic side as the estimator of the top quark mass, a precision of the order of 1 to 3.5 GeV on the top quark mass measurement should be achievable, assuming a jet energy scale uncertainty of 1 to 5%.Comment: Poster session at ICHEP08, Philadelphia, USA, July 2008. 3 pages, LaTeX, 3 eps figure

Observation of D0→ρ0γD^0\to \rho^0\gamma and search for CPCP violation in radiative charm decays

We report the first observation of the radiative charm decay $D^0 \to \rho^0 \gamma$ and the first search for $CP$ violation in decays $D^0 \to \rho^0 \gamma$, $\phi\gamma$, and $\overline{K}{}^{*0} \gamma$, using a data sample of 943 fb$^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. The branching fraction is measured to be $\mathcal{B}(D^0 \to \rho^0 \gamma)=(1.77 \pm 0.30 \pm 0.07) \times 10^{-5}$, where the first uncertainty is statistical and the second is systematic. The obtained $CP$ asymmetries, $\mathcal{A}_{CP}(D^0 \to \rho^0 \gamma)=+0.056 \pm 0.152 \pm 0.006$, $\mathcal{A}_{CP}(D^0 \to \phi \gamma)=-0.094 \pm 0.066 \pm 0.001$, and $\mathcal{A}_{CP}(D^0 \to \overline{K}{}^{*0} \gamma)=-0.003 \pm 0.020 \pm 0.000$, are consistent with no $CP$ violation. We also present an improved measurement of the branching fractions $\mathcal{B}(D^0 \to \phi \gamma)=(2.76 \pm 0.19 \pm 0.10) \times 10^{-5}$ and $\mathcal{B}(D^0 \to \overline{K}{}^{*0} \gamma)=(4.66 \pm 0.21 \pm 0.21) \times 10^{-4}$

Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect

We calculate numerically the localization length critical index within the Chalker-Coddington (CC) model for plateau-plateau transitions in the quantum Hall effect. Lyapunov exponents have been calculated with relative errors on the order $10^{-3}$. Such high precision was obtained by considering the distribution of Lyapunov exponents for large ensembles of relatively short chains and calculating the ensemble average values. We analyze thoroughly finite size effects and find the localization length critical index $\nu= 2.517\pm 0.018$.Comment: 4 pages, 4 figure