Quicksort with unreliable comparisons: a probabilistic analysis

We provide a probabilistic analysis of the output of Quicksort when comparisons can err.Comment: 29 pages, 3 figure

Self-forces on extended bodies in electrodynamics

In this paper, we study the bulk motion of a classical extended charge in flat spacetime. A formalism developed by W. G. Dixon is used to determine how the details of such a particle's internal structure influence its equations of motion. We place essentially no restrictions (other than boundedness) on the shape of the charge, and allow for inhomogeneity, internal currents, elasticity, and spin. Even if the angular momentum remains small, many such systems are found to be affected by large self-interaction effects beyond the standard Lorentz-Dirac force. These are particularly significant if the particle's charge density fails to be much greater than its 3-current density (or vice versa) in the center-of-mass frame. Additional terms also arise in the equations of motion if the dipole moment is too large, and when the `center-of-electromagnetic mass' is far from the `center-of-bare mass' (roughly speaking). These conditions are often quite restrictive. General equations of motion were also derived under the assumption that the particle can only interact with the radiative component of its self-field. These are much simpler than the equations derived using the full retarded self-field; as are the conditions required to recover the Lorentz-Dirac equation.Comment: 30 pages; significantly improved presentation; accepted for publication in Phys. Rev.

High energy neutrino yields from astrophysical sources II: Magnetized sources

We calculate the yield of high energy neutrinos produced in astrophysical sources for arbitrary interaction depths $\tau_0$ and magnetic field strengths $B$. We take into account energy loss processes like synchrotron radiation and diffusion of charged particles in turbulent magnetic fields as well as the scattering of secondaries on background photons and the direct production of charm neutrinos. Meson-photon interactions are simulated with an extended version of the SOPHIA model. Diffusion leads to an increased path-length before protons leave the source of size R_s and therefore magnetized sources lose their transparency below the energy $E\sim 10^{18}{\rm eV} (R_s/{\rm pc}) (B/{\rm mG}) \tau_0^{1/\alpha}$, with $\alpha=1/3$ and 1 for Kolmogorov and Bohm diffusion, respectively. Moreover, the neutrino flux is suppressed above the energy where synchrotron energy losses become important for charged particles. As a consequence, the energy spectrum and the flavor composition of neutrinos are strongly modified both at low and high energies even for sources with \tau_0\lsim 1.Comment: 15 pages, 16 figure

Eddy Impacts on the Florida Current

The Gulf Stream in the Atlantic carries warm water northwards and forms both the return closure of the subtropical gyre as well as the upper limb of the meridional overturning circulation. Recent time series recorded east of the Bahamas at 26°N indicate that from May 2009 to April 2011, in contrast with past observations, the northward flowing Antilles Current covaried with the Gulf Stream in the Florida Straits—the Florida Current—even though the Florida and Antilles Currents are separated by banks and islands spanning 150?km. The peak-to-trough amplitude of transport variations during this period was 15?×?106?m3?s?1 for the Florida Current and 12?×?106?m3?s?1 for the Antilles Current, at time scales of 50?days to a year. From satellite observations, we show that the fluctuations in both the Florida and Antilles Currents between May 2009 and April 2011 are driven by eddy activity east of the Bahamas. Since the Florida Current time series is a critical time series for the state of the oceans, and often compared to climate models, this newly identified source of variability needs careful consideration when attributing the variability of the Florida Current to changes in the larger-scale circulations (e.g., gyre and overturning) or wind forcing.<br/

Self-forces from generalized Killing fields

A non-perturbative formalism is developed that simplifies the understanding of self-forces and self-torques acting on extended scalar charges in curved spacetimes. Laws of motion are locally derived using momenta generated by a set of generalized Killing fields. Self-interactions that may be interpreted as arising from the details of a body's internal structure are shown to have very simple geometric and physical interpretations. Certain modifications to the usual definition for a center-of-mass are identified that significantly simplify the motions of charges with strong self-fields. A derivation is also provided for a generalized form of the Detweiler-Whiting axiom that pointlike charges should react only to the so-called regular component of their self-field. Standard results are shown to be recovered for sufficiently small charge distributions.Comment: 21 page

Kaspar Schott’s “encyclopedia of all mathematical sciences”

In 1661, Kaspar Schott published his comprehensive textbook “Cursus mathematicus” in Würzburg for the first time, his “Encyclopedia of all mathematical sciences”. It was so successful that it was published again in 1674 and 1677. In its 28 books, Schott gave an introduction for beginners in 22 mathematical disciplines by means of 533 figures and numerous tables. He wanted to avoid the shortness and the unintelligibility of his predecessors Alsted and Hérigone. He cited or recommended far more than hundred authors, among them Protestants like Michael Stifel and Johannes Kepler, but also Catholics like Nicolaus Copernicus. The paper gives a survey of this work and explains especially interesting aspects: The dedication to the German emperor Leopold I., Athanasius Kircher’s letter of recommendation as well as Schott’s classification of sciences, explanations regarding geometry, astronomy, and algebra

A comprehensive analysis of methods for assessing polygenic burden on Alzheimer’s disease pathology and risk beyond APOE

Genome-wide association studies have identified dozens of loci that alter the risk to develop Alzheimer’s disease. However, with the exception of the APOE-ε4 allele, most variants bear only little individual effect and have, therefore, limited diagnostic and prognostic value. Polygenic risk scores aim to collate the disease risk distributed across the genome in a single score. Recent works have demonstrated that polygenic risk scores designed for Alzheimer’s disease are predictive of clinical diagnosis, pathology confirmed diagnosis and changes in imaging biomarkers. Methodological innovations in polygenic risk modelling include the polygenic hazard score, which derives effect estimates for individual single nucleotide polymorphisms from survival analysis, and methods that account for linkage disequilibrium between genomic loci. In this work, using data from the Alzheimer’s disease neuroimaging initiative, we compared different approaches to quantify polygenic disease burden for Alzheimer’s disease and their association (beyond the APOE locus) with a broad range of Alzheimer’s disease-related traits: cross-sectional CSF biomarker levels, cross-sectional cortical amyloid burden, clinical diagnosis, clinical progression, longitudinal loss of grey matter and longitudinal decline in cognitive function. We found that polygenic scores were associated beyond APOE with clinical diagnosis, CSF-tau levels and, to a minor degree, with progressive atrophy. However, for many other tested traits such as clinical disease progression, CSF amyloid, cognitive decline and cortical amyloid load, the additional effects of polygenic burden beyond APOE were of minor nature. Overall, polygenic risk scores and the polygenic hazard score performed equally and given the ease with which polygenic risk scores can be derived; they constitute the more practical choice in comparison with polygenic hazard scores. Furthermore, our results demonstrate that incomplete adjustment for the APOE locus, i.e. only adjusting for APOE-ε4 carrier status, can lead to overestimated effects of polygenic scores due to APOE-ε4 homozygous participants. Lastly, on many of the tested traits, the major driving factor remained the APOE locus, with the exception of quantitative CSF-tau and p-tau measures

Electromagnetic self-forces and generalized Killing fields

Building upon previous results in scalar field theory, a formalism is developed that uses generalized Killing fields to understand the behavior of extended charges interacting with their own electromagnetic fields. New notions of effective linear and angular momenta are identified, and their evolution equations are derived exactly in arbitrary (but fixed) curved spacetimes. A slightly modified form of the Detweiler-Whiting axiom that a charge's motion should only be influenced by the so-called "regular" component of its self-field is shown to follow very easily. It is exact in some interesting cases, and approximate in most others. Explicit equations describing the center-of-mass motion, spin angular momentum, and changes in mass of a small charge are also derived in a particular limit. The chosen approximations -- although standard -- incorporate dipole and spin forces that do not appear in the traditional Abraham-Lorentz-Dirac or Dewitt-Brehme equations. They have, however, been previously identified in the test body limit.Comment: 20 pages, minor typos correcte

A Precision Measurement of Nuclear Muon Capture on 3He

The muon capture rate in the reaction mu- 3He -> nu + 3H has been measured at PSI using a modular high pressure ionization chamber. The rate corresponding to statistical hyperfine population of the mu-3He atom is (1496.0 +- 4.0) s^-1. This result confirms the PCAC prediction for the pseudoscalar form factors of the 3He-3H system and the nucleon.Comment: 13 pages, 6 PostScript figure