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

Einstein gravity as a 3D conformally invariant theory

We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is irrelevant. The dual theory is invariant under foliation preserving 3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume (for the spatially compact case). Locally, this symmetry is identical to that of Horava-Lifshitz gravity in the high energy limit but our theory is equivalent to Einstein gravity. Specifically, we find that the solutions of general relativity, in a gauge where the spatial hypersurfaces have constant mean extrinsic curvature, can be mapped to solutions of a particular gauge fixing of the dual theory. Moreover, this duality is not accidental. We provide a general geometric picture for our procedure that allows us to trade foliation invariance for conformal invariance. The dual theory provides a new proposal for the theory space of quantum gravity.Comment: 27 pages. Published version (minor changes and corrections

Poisson approximations for the Ising model

A $d$-dimensional Ising model on a lattice torus is considered. As the size $n$ of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field $a=a(n)$ tends to $-\infty$ and the pair potential $b$ remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case.Comment: 25 pages, 1 figur

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

On the Phase Structure of the Schwinger Model with Wilson Fermions

We study the phase structure of the massive one flavour lattice Schwinger model on the basis of the finite size scaling behaviour of the partition function zeroes. At $\beta = 0$ we observe and discuss a possible discrepancy with results obtained by a different method.Comment: 3 pages (2 figures), POSTSCRIPT-file (174 KB), Contribution to Lattice 93, preprint UNIGRAZ-UTP 19-11-9

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

Covariant quantization of membrane dynamics

A Lorentz covariant quantization of membrane dynamics is defined, which also leaves unbroken the full three dimensional diffeomorphism invariance of the membrane. Among the applications studied are the reduction to string theory, which may be understood in terms of the phase space and constraints, and the interpretation of physical,zero-energy states. A matrix regularization is defined as in the light cone gauged fixed theory but there are difficulties implementing all the gauge symmetries. The problem involves the non-area-preserving diffeomorphisms which are realized non-linearly in the classical theory. In the quantum theory they do not seem to have a consistent implementation for finite N. Finally, an approach to a genuinely background independent formulation of matrix dynamics is briefly described.Comment: Latex, 21 pages, no figure

Restoration of chiral symmetry in quark models with effective one gluon exchange

The restoration of chiral symmetry at finite density and/or temperature is investigated in a wide class of one-gluon exchange models in the instantaneous approximation. If the effective quark interaction is less divergent than $1/k^2$ for small momentum transfer $k$, we obtain Gaussian critical exponents for the chiral phase transitions at finite temperature and density, respectively. In the opposite case, for an interaction diverging faster than $1/k^2$ in the infrared region, a qualitative different behavior of the quark self-energy near the critical Fermi momentum $k_c$ and the critical temperature $T_c$, respectively, is observed. In the first scenario, we find $k_c \approx 2 \ln 2 \; T_c$, which compares well with recent data from QCD lattice simulations.Comment: 12 pages LaTeX, no figure