The geometry of the Barbour-Bertotti theories II. The three body problem

We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for $N\neq 3$, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there.Comment: 16 pages, 2 eps figures, to appear in Classical and Quantum Gravit

A Course on Economic Justice: The intersection of philosophy and economics

The process of teaching a topic that inhabits the upper reaches of both philosophy and economic theory, while swooping as near the earth as political policy, is both exhilarating and terrifying. To do it well is indeed rare. We present our approach, some of the characteristics and thoughts from our students, and some of the insights that we developed along the way.economics and philosophy; economic justice; interdisciplinary teaching

Stein factors for negative binomial approximation in Wasserstein distance

The paper gives the bounds on the solutions to a Stein equation for the negative binomial distribution that are needed for approximation in terms of the Wasserstein metric. The proofs are probabilistic, and follow the approach introduced in Barbour and Xia (Bernoulli 12 (2006) 943-954). The bounds are used to quantify the accuracy of negative binomial approximation to parasite counts in hosts. Since the infectivity of a population can be expected to be proportional to its total parasite burden, the Wasserstein metric is the appropriate choice.Comment: Published at http://dx.doi.org/10.3150/14-BEJ595 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The geometry of the Barbour-Bertotti theories I. The reduction process

The dynamics of $N\geq 3$ interacting particles is investigated in the non-relativistic context of the Barbour-Bertotti theories. The reduction process on this constrained system yields a Lagrangian in the form of a Riemannian line element. The involved metric, degenerate in the flat configuration space, is the first fundamental form of the space of orbits of translations and rotations (the Leibniz group). The Riemann tensor and the scalar curvature are computed by a generalized Gauss formula in terms of the vorticity tensors of generators of the rotations. The curvature scalar is further given in terms of the principal moments of inertia of the system. Line configurations are singular for $N\neq 3$. A comparison with similar methods in molecular dynamics is traced.Comment: 15 pages, to appear in Classical and Quantum Gravit

Description and evaluation of the Acoustic Profiling of Ocean Currents (APOC) system used on R. V. Oceanus cruise 96 on 11-22 May 1981

The underway current profiling system which consists of a microprocessor controlled data logger that collects and formats data from a four beam Ametek-Straza 300 kHz acoustic Doppler current profiler, heading from the ship's gyrocompass, and navigation information from a Loran-C receiver and a satellite navigation unit is discussed. Data are recorded on magnetic tape and real time is calculated. Time averaging is required to remove effects of ship motion. An intercomparison is made with a moored vector measuring current meter (VMCM). The mean difference in hourly averaged APOC and VMCM currents over the four hour intercomparison is a few mm s minus including: two Gulf Stream crossings, a warm core ring survey, and shallow water in a frontal zone to the east of Nantucket Shoals

Hydrographic data from R/V endeavor cruise #90

The final cruise of the NSF sponsored Warm Core Rings Program studied a Warm Core Ring (WCR) in the Fall of 1982 as it formed from a large northward meander of the Gulf Stream. This ring, known as 82-H or the eighth ring identified in 1982, formed over the New England Seamounts near 39.5 deg N, 65 deg W. Surveys using Expendable Bathythermographs, Conductivity-Temperature-Depth-Oxygen stations and Doppler Current Profiling provide a look at the genesis of a WCR. These measurements reveal that WCR 82-H separated from the Gulf Stream sometime between October 2-5. This ring was a typical WCR with a diameter of about 200 km and speeds in the high velocity core of the 175 cm/sec. Satellite imagery of 82-H following the cruise showed that it drifted WSW in the Slope Water region at almost 9 km/day, had at least one interaction with the Gulf Stream and was last observed on February 8, 1983 at 39 deg N, 72 deg W

A law of large numbers approximation for Markov population processes with countably many types

When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for rather general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm.Comment: revised version in response to referee comments, 34 page

The Definition of Mach's Principle

Two definitions of Mach's principle are proposed. Both are related to gauge theory, are universal in scope and amount to formulations of causality that take into account the relational nature of position, time, and size. One of them leads directly to general relativity and may have relevance to the problem of creating a quantum theory of gravity.Comment: To be published in Foundations of Physics as invited contribution to Peter Mittelstaedt's 80th Birthday Festschrift. 30 page

New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split

I show how there is an ambiguity in how one treats auxiliary variables in gauge theories including general relativity cast as 3 + 1 geometrodynamics. Auxiliary variables may be treated pre-variationally as multiplier coordinates or as the velocities corresponding to cyclic coordinates. The latter treatment works through the physical meaninglessness of auxiliary variables' values applying also to the end points (or end spatial hypersurfaces) of the variation, so that these are free rather than fixed. [This is also known as variation with natural boundary conditions.] Further principles of dynamics workings such as Routhian reduction and the Dirac procedure are shown to have parallel counterparts for this new formalism. One advantage of the new scheme is that the corresponding actions are more manifestly relational. While the electric potential is usually regarded as a multiplier coordinate and Arnowitt, Deser and Misner have regarded the lapse and shift likewise, this paper's scheme considers new {\it flux}, {\it instant} and {\it grid} variables whose corresponding velocities are, respectively, the abovementioned previously used variables. This paper's way of thinking about gauge theory furthermore admits interesting generalizations, which shall be provided in a second paper.Comment: 11 page

Quenched QCD at finite density

Simulations of quenched $QCD$ at relatively small but {\it nonzero} chemical potential $\mu$ on $32 \times 16^3$ lattices indicate that the nucleon screening mass decreases linearly as $\mu$ increases predicting a critical chemical potential of one third the nucleon mass, $m_N/3$, by extrapolation. The meson spectrum does not change as $\mu$ increases over the same range, from zero to $m_\pi/2$. Past studies of quenched lattice QCD have suggested that there is phase transition at $\mu = m_\pi/2$. We provide alternative explanations for these results, and find a number of technical reasons why standard lattice simulation techniques suffer from greatly enhanced fluctuations and finite size effects for $\mu$ ranging from $m_\pi/2$ to $m_N/3$. We find evidence for such problems in our simulations, and suggest that they can be surmounted by improved measurement techniques.Comment: 23 pages, Revte