Evaluating the cylindricity of a nominally cylindrical point set

International audienceThe minimum zone cylinder of a set of points in three dimensions is the cylindric crown defined by a pair of coaxial cylinders with minimal radial separation (width). In the context of tolerancing metrology, the set of points is nominally cylindrical, i.e., the points are known to lie in close proximity of a known reference cylinder. Using approximations which are valid only in the neighborhood of the reference cylinder, we can get a very good approximation of the minimum zone cylinder. The process provides successive approximations, and each iteration involves the solution of a linear programming problem in six dimensions. The error between the approximation and the optimal solution converges very rapidly (typically in three iterations in practice) down to a limit error of (8 omega^2)/R ( where omega is the width and R is the external radius of the zone cylinder)

The Efficiency of Globular Cluster Formation

(Abridged): The total populations of globular cluster systems (GCSs) are discussed in terms of their connection to the efficiency of globular cluster formation---the mass fraction of star-forming gas that was able to form bound stellar clusters rather than isolated stars or unbound associations---in galaxy halos. Observed variations in GCS specific frequencies (S_N=N_gc/L_gal), both as a function of galactocentric radius in individual systems and globally between entire galaxies, are reviewed in this light. It is argued that trends in S_N do not reflect any real variation in the underlying efficiency of cluster formation; rather, they result from ignoring the hot gas in many large ellipticals. This claim is checked and confirmed in each of M87, M49, and NGC 1399, for which existing data are combined to show that the volume density profile of globular clusters, rho_cl, is directly proportional to the sum of (rho_gas+rho_stars) at large radii. The constant of proportionality is the same in each case: epsilon=0.0026 +/- 0.0005 in the mean. This is identified with the globular cluster formation efficiency. The implication that epsilon might have had a universal value is supported by data on the GCSs of 97 early-type galaxies, on the GCS of the Milky Way, and on the ongoing formation of open clusters. These results have specific implications for some issues in GCS and galaxy formation, and they should serve as a strong constraint on more general theories of star and cluster formation.Comment: 36 pages with 11 figures; accepted for publication in The Astronomical Journa

Chandra measurements of gas homogeneity and turbulence at intermediate radii in the Perseus Cluster

We present a Chandra study of surface brightness fluctuations in the diffuse intracluster medium of the Perseus Cluster. Our study utilizes deep, archival imaging of the cluster core as well as a new mosaic of 29 short 5 ks observations extending in 8 different directions out to radii of r_500 ~ 2.2r_2500. Under the assumption that the distribution of densities at a given radius is log-normally distributed, two important quantities can be derived from the width of the log-normal density distribution on a given spatial scale: the density bias, which is equal to the square root of the clumping factor C; and the one-component turbulent velocity, v_(k, 1D). We forward-model all contributions to the measured surface brightness, including astrophysical and particle background components, and account for the Poisson nature of the measured signal. Measuring the distribution of surface brightness fluctuations in 1 arcmin^2 regions, spanning the radial range 0.3-2.2 r_2500 (7.8-57.3 arcmin), we find a small to moderate average density bias of around 3% at radii below 1.6r_2500. We also infer an average turbulent velocity at these radii of v_1D <400 km s^-1. Direct confirmation of our results on turbulent velocities inferred from surface brightness fluctuations should be possible using the X-ray calorimeter spectrometers to be flown aboard the XRISM and Athena. observatories.Comment: 17 pages, 11 figures. to be published in MNRA

Multistable Shell Structures

Multistable structures, which possess by definition more than one stable equilibrium configuration, are capable of adapting their shape to changing loading or environmental conditions and can further improve multi-purpose ultra-lightweight designs. Whilst multiple methods to create bistable shells have been proposed, most studies focussed on free-standing ones. Considering the strong influence of support conditions on related stability thresholds, surprisingly little is known about their influence on multistable behaviour. In fact, the lack of analytical models prevents a full understanding and constitutes a bottle-neck in the development process of novel shape-changing structures. The relevance becomes apparent in a simple example: whilst an unsupported sliced tennis ball can be stably inverted without experiencing a reversion, fixing its edge against rotation erodes bistability by causing an instantaneous snap-back to the initial configuration. This observation reveals the possibility to alter the structural response dramatically by a simple change of the support conditions. This dissertation explores the causes of this behaviour by gaining further insight into the promoting and eschewing factors of multistability and aims to point out methods to exploit this feature in optimised ways. The aforementioned seemingly simple example requires a geometrically nonlinear perspective on shells for which analytical solutions stay elusive unless simplifying assumptions are made. In order to captures relevant aspects in closed form, a novel semi-analytical Ritz approach with up to four degrees of freedom is derived, which enforces the boundary conditions strongly. In contrast to finite element simulations, it does not linearise the stiffness matrix and can thus explore the full solution space spanned by the assumed polynomial deflection field. In return, this limits the method to a few degrees of freedom, but a comparison to reference calculations demonstrated an excellent performance in most cases. First, the level of influence of the boundary conditions on the critical shape for enabling a bistable inversion is formally characterised in rotationally symmetric shells. Systematic insight is provided by connecting the rim to ground through sets of extensional and rotational linear springs, which allows use of the derived shell model as a macro-element that is connected to other structural elements. It is demonstrated that bistability is promoted by an increasing extensional stiffness, i.e. bistable roller-supported shells need to be at least twice as tall compared to their fixed-pinned counterparts. The effect of rotational springs is found to be multi-faceted: whilst preventing rotation has the tendency to hinder bistable inversions, freeing it can even allow for extra stable configurations; however, a certain case is emphasised in which an increasing rotational spring stiffness causes a mode transition that stabilises inversions. In a second step, a polar-orthotropic material law is employed to study variations of the directional stiffness of the shell itself. A careful choice of the basis functions is required to accurately capture stress singularities in bending that arise if the radial Young’s modulus is stiffer than its circumferential equivalent. A simple way to circumvent such singularities is to create a central hole, which is shown not to hamper bistable inversions. For significantly stiffer values of the radial stiffness, a strong coupling with the support conditions is revealed: whilst roller-supported shells do not show a bistable inversion at all for such materials, fixed-pinned ones feel the most disposed to accommodate an alternative equilibrium configuration. This behaviour is explained via simplified beam models that suggest a new perspective on the influence of the hoop stiffness: based on observations in free-standing shells, it was thought to promote bistability, but it is only insofar stabilising, as it evokes radial stresses; if these are afforded by immovable supports, it becomes redundant and even slightly hindering. Finally, combined actuation methods in stretching and bending that prescribe non-Euclidean target shapes are considered to emphasise the possibility of multifarious structural manipulations. When both methods are geared to each other, stress-free synclastic shape transformations in an over-constrained environment, or alternatively, anticlastic shape-changes with an arbitrary wave number, are achievable. Considering nonsymmetric deformations offers a richer buckling behaviour for certain in-plane actuated shells, where a secondary, approximately cylindrical buckling mode as well as a ‘hidden’ stable configuration of a higher wave number is revealed by the presented analytical model. Additionally, it is shown that the approximately mirror-symmetric inversion of cylindrical or deep spherical shells can be accurately described by employing a simpler, geometrically linear theory that focusses on small deviations from the mirrored shape. The results of this dissertation facilitate a versatile practical application of multistable structures via an analytical description of more realistic support conditions. The understanding of effects of the internal stiffness makes it possible to use this unique structural behaviour more efficiently by making simple cross-sectional adjustments, i.e. by adding appropriate stiffeners. Eventually, the provided theoretical framework of emerging actuation methods might inspire novel morphing structures.Friedrich Ebert Foundation (Friedrich-Ebert-Stiftung) Corpus Christi College, Cambridge Department of Engineering, Cambridg

Search for the lepton-family-number nonconserving decay \mu -> e + \gamma

The MEGA experiment, which searched for the muon- and electron-number violating decay \mu -> e + \gamma, is described. The spectrometer system, the calibrations, the data taking procedures, the data analysis, and the sensitivity of the experiment are discussed. The most stringent upper limit on the branching ratio of \mu -> e + \gamma) < 1.2 x 10^{-11} was obtained

High-Dimensional Geometric Streaming in Polynomial Space

Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.Comment: Abstract shortened to meet arXiv limits; v2 fix statements concerning online condition numbe