Ultra high energy neutrinos: the key to ultra high energy cosmic rays

We discuss the relation between the acceleration spectra of extragalactic cosmic ray protons and the luminosity and cosmological evolution of their sources and the production of ultra high energy cosmogenic neutrinos in their propagation from the sources to us.Comment: 12 pages, 5 figures, Talk presented at the Vulcano06 workshop. Second version with some typos remove

The beginning of cosmic ray astronomy

We discuss the anisotropic arrival directions of the ultra high energy cosmic rays detected by Auger which I consider one of the biggest discoverie in astrophysics during the last year.Comment: 5 pages, 2 figures, Concluding remarks at the Vulcano2008 worksho

The additive group of the rationals does not have an automatic presentation

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $p$-divisible for infinitely many primes $p$, or $Q/Z$. The proof is combinatorial and uses most notably Freiman's theorem on sets with small doubling.Comment: 11 page

The Eynard--Orantin recursion for the total ancestor potential

It was proved recently that the correlation functions of a semi-simple cohomological field theory satisfy the so called Eynard--Orantin topological recursion. We prove that in the settings of singularity theory, the relations can be expressed in terms of periods integrals and the so called phase forms. In particular, we prove that the Eynard-Orantin recursion is equivalent to $N$ copies of Virasoro constraints for the ancestor potential, which follow easily from the definition of the potential.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1203.341

Computing parabolically induced embeddings of semisimple complex Lie algebras in Weyl algebras

An arbitrary proper parabolic subalgebra ${\mathfrak p}$ of a simple complex Lie algebra ${\mathfrak g}$ induces an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n$, and more generally an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n\otimes \operatorname{End}{} V$, where $\mathbb W_n$ is the Weyl algebra in $n$ variables, $n$ is the dimension of the nilradical of ${\mathfrak p}$, and $V$ is an arbitrary ${\mathfrak p}$-module. We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings $G_2 \hookrightarrow \mathbb W_5$, $F_4 \hookrightarrow \mathbb W_{15}$, $E_6 \hookrightarrow \mathbb W_{16}$, $E_7 \hookrightarrow\mathbb W_{27}$, $E_8 \hookrightarrow \mathbb W_{57}$ arising in this fashion

Pre-Plactic Algebra and Snakes

We study a factor Hopf algebra $\mathfrak{PP}$ of the Malvenuto-Reutenauer convolution algebra of functions on symmetric groups ${\mathfrak{S}}=\oplus_{n\geq 0} \mathbb C[{\mathfrak{S}}_n]$ that we coined pre-plactic algebra. The pre-plactic algebra admits the Poirier-Reutenauer algebra based on Standard Young Tableaux as a factor and it is closely related to the quantum pseudo-plactic algebra introduced by Krob and Thibon in the non-commutative character theory of quantum group comodules. The connection between the quantum pseudo-plactic algebra and the pre-plactic algebra is similar to the connection between the Lascoux-Sch\"utzenberger plactic algebra and the Poirier-Reutenauer algebra. We show that the dimensions of the pre-plactic algebra are given by the numbers of alternating permutations (coined snakes after V.I. Arnold). Pre-plactic algebra is instrumental in calculating the Hilbert-Poincar\'e series of the quantum pseudo-plactic algebra.Comment: 12 pages. Comments are welcome

Ultra high energy cosmic rays and neutrinos after Auger

We discuss the main results that were recently published by the Auger Collaboration and their impact on our knowledge of the ultra high energy cosmic rays and neutrinos.Comment: 12 pages, 4 figures, Talk at the Vulcano 2008 meeting "Frontier objects in astrophysics and particle physics", Vulcano, Ital

The IceTop experiment in 2010

We present the current status of the IceTop air shower array on top of the IceCube neutrino detector that IceTop can use as a huge detector of TeV muons. We laos give a brief discussion of different types of air shower events that contain information on the spectrum and composition of the cosmic rays in a wide energy range.Comment: 6 pages, 2 figures, presented at the Vulcano 2010 workshop, Vulcano, Ital

Atmospheric neutrino challenges

We briefly review the improvements in the predictions of atmospheric neutrino fluxes since the NOW2000 workshop. In spite of the great progress in calculational technique the predictions are still not exact because of the uncertainties in the two major sets of input - cosmic ray flux and hadronic interactions on light nuclei.Comment: 6 pages, 7 postscript figures, prepared for the NOW 2004 Proceeding

Possible tau appearance experiment with atmospheric neutrinos

We suggest an experimental measurement that could detect the appearance of the tau neutrinos in muon to tau neutrino oscillations of atmospheric neutrinos by measuring the energy spectra of neutrino induced showers. Tau neutrinos deposit a large fraction of their energy in showers generated in CC interactions and the subsequent tau--lepton decay. The appearance of tau neutrinos will enhance the spectrum of neutrino induced showers in energy ranges corresponding to the neutrino oscillation parameters. A shower rate lower than the `no oscillation' prediction is an indication for oscillations into sterile neutrinos.Comment: RevTeX, four pages with 4 postscript figure