Central limit approximations 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 there is usually no obvious natural upper limit on the number of individuals in a patch, this leads to systems in which there are countably infinitely many possible types of entity. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove central limit theorems for quite general systems of this kind, together with bounds on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm.Comment: 24 page

Interacting vector fields in Relativity without Relativity

Barbour, Foster and \'{O} Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang--Mills fields minimally coupled to gravity.Comment: Replaced with final version accepted by Classical and Quantum Gravity (14 pages, no figures

Towards the Unification of Gravity and other Interactions: What has been Missed?

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space ${\cal C}$ associated with matter configurations. We consider general relativity in ${\cal C}$. In spacetime, which is a 4-dimensional subspace of ${\cal C}$, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.Comment: 11 pages; Talk presented at "First Mediterranean Conference on Classical and Quantum Gravity", Kolymbari, Crete, Greece, 14-18 September 200

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

Leptons, quarks, and their antiparticles from a phase-space perspective

It is argued that antiparticles may be interpreted in macroscopic terms without explicitly using the concept of time and its reversal. The appropriate framework is that of nonrelativistic phase space. It is recalled that a quantum version of this approach leads also, alongside the appearance of antiparticles, to the emergence of `internal' quantum numbers identifiable with weak isospin, weak hypercharge and colour, and to the derivation of the Gell-Mann-Nishijima relation, while simultaneously offering a preonless interpretation of the Harari-Shupe rishon model. Furthermore, it is shown that - under the assumption of the additivity of canonical momenta - the approach entails the emergence of string-like structures resembling mesons and baryons, thus providing a different starting point for the discussion of quark unobservability.Comment: Talk given at Fifth Int. Workshop DICE2010 Space-Time-Matter, Castiglioncello, Italy, September 13-17, 201

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

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

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