Cubic structures, equivariant Euler characteristics and lattices of modular forms

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.Comment: 40pp, Final version, to appear in the Annals of Mathematic

Dispersion of biased swimming microorganisms in a fluid flowing through a tube

Classical Taylor-Aris dispersion theory is extended to describe the transport of suspensions of self-propelled dipolar cells in a tubular flow. General expressions for the mean drift and effective diffusivity are determined exactly in terms of axial moments, and compared with an approximation a la Taylor. As in the Taylor-Aris case, the skewness of a finite distribution of biased swimming cells vanishes at long times. The general expressions can be applied to particular models of swimming microorganisms, and thus be used to predict swimming drift and diffusion in tubular bioreactors, and to elucidate competing unbounded swimming drift and diffusion descriptions. Here, specific examples are presented for gyrotactic swimming algae.Comment: 20 pages, 4 figures. Published version available at http://rspa.royalsocietypublishing.org/content/early/2010/02/09/rspa.2009.0606.short?rss=

Identifying Nearby UHECR Accelerators using UHE (and VHE) Photons

Ultra-high energy photons (UHE, E > 10^19 eV) are inevitably produced during the propagation of 10^20 eV protons in extragalactic space. Their short interaction lengths (<20 Mpc) at these energies, combined with the impressive sensitivity of the Pierre Auger Observatory detector to these particles, makes them an ideal probe of nearby ultra-high-energy cosmic ray (UHECR) sources. We here discuss the particular case of photons from a single nearby (within 30 Mpc) source in light of the possibility that such an object might be responsible for several of the UHECR events published by the Auger collaboration. We demonstrate that the photon signal accompanying a cluster of a few > 6x10^19 eV UHECRs from such a source should be detectable by Auger in the near future. The detection of these photons would also be a signature of a light composition of the UHECRs from the nearby source.Comment: 4 pages, 2 figures, accepted for publication in PR

Analysis of enhanced diffusion in Taylor dispersion via a model problem

We consider a simple model of the evolution of the concentration of a tracer, subject to a background shear flow by a fluid with viscosity $\nu \ll 1$ in an infinite channel. Taylor observed in the 1950's that, in such a setting, the tracer diffuses at a rate proportional to $1/\nu$, rather than the expected rate proportional to $\nu$. We provide a mathematical explanation for this enhanced diffusion using a combination of Fourier analysis and center manifold theory. More precisely, we show that, while the high modes of the concentration decay exponentially, the low modes decay algebraically, but at an enhanced rate. Moreover, the behavior of the low modes is governed by finite-dimensional dynamics on an appropriate center manifold, which corresponds exactly to diffusion by a fluid with viscosity proportional to $1/\nu$

The Intergalactic Propagation of Ultra-High Energy Cosmic Ray Nuclei: An Analytic Approach

It is likely that ultra-high energy cosmic rays contain a significant component of heavy or intermediate mass nuclei. The propagation of ultra-high energy nuclei through cosmic radiation backgrounds is more complicated than that of protons and its study has required the use of Monte Carlo techniques. We present an analytic method for calculating the spectrum and the composition at Earth of ultra-high energy cosmic rays which start out as heavy nuclei from their extragalactic sources. The results obtained are in good agreement with those obtained using numerical methods.Comment: accepted for publication in Phys Rev