Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global $L^\infty$ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou's small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space.Comment: 20 page

Multipole structure of current vectors in curved spacetime

A method is presented which allows the exact construction of conserved (i.e. divergence-free) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric charge inside some classical extended body. Several applications are discussed. In particular, it is shown how to easily write down the class of all smooth and spatially-bounded currents with a given total charge. This implicitly provides restrictions on the moments arising from the smoothness of physically reasonable vector fields. We also show that requiring all of the moments to be constant in an appropriate sense is often impossible; likely limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid motion. A simple condition is also derived that allows currents to exist in two different spacetimes with identical sets of multipole moments (in a natural sense).Comment: 13 pages, minor changes, accepted to J. Math. Phy

From cellular properties to population asymptotics in the Population Balance Equation

Proliferating cell populations at steady state growth often exhibit broad protein distributions with exponential tails. The sources of this variation and its universality are of much theoretical interest. Here we address the problem by asymptotic analysis of the Population Balance Equation. We show that the steady state distribution tail is determined by a combination of protein production and cell division and is insensitive to other model details. Under general conditions this tail is exponential with a dependence on parameters consistent with experiment. We discuss the conditions for this effect to be dominant over other sources of variation and the relation to experiments.Comment: Exact solution of Eq. 9 is adde

On multiplicative functions which are small on average

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if \sum_{n2, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $\mu(n)n^{it}$ for some $t$.Comment: 51 pages. Slightly strengthened Theorem 1.2 and simplified its statement. Removed Remark 1.3. Other minor changes and corrections. To appear in Geom. Funct. Ana

Bose-Einstein condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons

Local scalar QFT (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the $C^*$-algebra of observables with respect to the conformal group PSL(2,\bR) are studied.It is shown that, in addition to the state studied by Guido, Longo, Roberts and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of PSL(2,\bR) symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are non equivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of PSL(2,\bR) associated with the Killing flow. The KMS property is valid for the two local sub algebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity (necessary to describe the whole PSL(2,\bR) action).Each of the found states can be interpreted as a different thermodynamic phase, containing Bose-Einstein condensate,for the considered quantum field. It is finally suggested that the found states could describe different black holes.Comment: 36 pages, 1 figure. Formula of condensate energy density modified. Accepted for pubblication in Journal of Mathematical Physic

A search for nuclear disintegrations produced by slow negative heavy mesons

This paper describes the preliminary results of a search for evidence of the nuclear interactions of negative heavy mesons. A qualitative analysis is given of the possible characteristics of their interactions and the appearance these might be expected to have in photographic emulsions. 37 ml. of emulsion, in which are recorded 10 000 stars and 1200 slow π-mesons, have been completely examined. In the conditions of exposure, such a volume should contain six examples, with good geometry, of the decay of heavy mesons. Mass measurements have been carried out, by the range/scattering method, on 417 tracks of σ-mesons. In addition, 1800 σ-mesons, observed in 42 ml. of emulsion, have been examined. No disintegrations which can be attributed to heavy mesons have been found. The results suggest that some of the negative heavy mesons, on being brought to rest in photographic emulsions, behave in a manner qualitatively different from that of negative π-particles. Possible explanations for this result are suggested

The origin and propagation of VVH primary cosmic ray particles

Several source spectra were constructed from combinations of 4- and s-process nuclei to match the observed charge spectrum of VVH particles. Their propagation was then followed, allowing for interactions and decay, and comparisons were made between the calculated near-earth spectra and those observed during high altitude balloon flights. None of the models gave good agreement with observations

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(\lambda))$ as an $L_J$-module where $L(\lambda)$ is a simple $G$-module with high weight $\lambda$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.Comment: 18 pages. Some proofs streamlined over previous version. Additional details added to some proofs in Section

"Peeling property" for linearized gravity in null coordinates

A complete description of the linearized gravitational field on a flat background is given in terms of gauge-independent quasilocal quantities. This is an extension of the results from gr-qc/9801068. Asymptotic spherical quasilocal parameterization of the Weyl field and its relation with Einstein equations is presented. The field equations are equivalent to the wave equation. A generalization for Schwarzschild background is developed and the axial part of gravitational field is fully analyzed. In the case of axial degree of freedom for linearized gravitational field the corresponding generalization of the d'Alembert operator is a Regge-Wheeler equation. Finally, the asymptotics at null infinity is investigated and strong peeling property for axial waves is proved.Comment: 27 page