Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product \B\times\{1,...,m\}, where \B is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians \;H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A + \lambda \Vv on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, \Vv is a random matrix potential, and $\lambda$ is the disorder parameter. Given any closed interval $I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})$, where $a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $\lambda$ small the random Schr\"odinger operator $\;H_\lambda$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$

Poincare submersions

We prove two kinds of fibering theorems for maps X --> P, where X and P are Poincare spaces. The special case of P = S^1 yields a Poincare duality analogue of the fibering theorem of Browder and Levine.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-2.abs.html Version 5: Statement of Theorem B corrected, see footnote p2

Agmon-type estimates for a class of jump processes

In the limit epsilon to 0 we analyze the generators H_epsilon of families of reversible jump processes in R^d associated with a class of symmetric non-local Dirichlet-forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being C^2 or just Lipschitz. Our estimates are analog to the semi-classical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice epsilon Z^d. Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques

Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I

Several N-body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (``acceleration equal force;'' in most cases, the forces are velocity-dependent) and are amenable to exact treatment (``solvable'' and/or ``integrable'' and/or ``linearizable''). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider ``few-body problems'' (with, say, \textit{N}=1,2,3,4,6,8,12,16,...) as well as ``many-body problems'' (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N-body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases

The Dualizing Spectrum, II

To an inclusion topological groups H->G, we associate a naive G-spectrum. The special case when H=G gives the dualizing spectrum D_G introduced by the author in the first paper of this series. The main application will be to give a purely homotopy theoretic construction of Poincare embeddings in stable codimension.Comment: Fixed an array of typo

A psychoanalytic concept illustrated: Will, must, may, can — revisiting the survival function of primitive omnipotence

The author explores the linear thread connecting the theory of Freud and Klein, in terms of the central significance of the duality of the life and death instinct and the capacity of the ego to tolerate contact with internal and external reality. Theoretical questions raised by later authors, informed by clinical work with children who have suffered deprivation and trauma in infancy, are then considered. Theoretical ideas are illustrated with reference to observational material of a little boy who suffered deprivation and trauma in infancy. He was first observed in the middle of his first year of life while he was living in foster care, and then later at the age of two years and three months, when he had been living with his adoptive parents for more than a year

Muon Production in Relativistic Cosmic-Ray Interactions

Cosmic-rays with energies up to $3\times10^{20}$ eV have been observed. The nuclear composition of these cosmic rays is unknown but if the incident nuclei are protons then the corresponding center of mass energy is $\sqrt{s_{nn}} = 700$ TeV. High energy muons can be used to probe the composition of these incident nuclei. The energy spectra of high-energy ($>$ 1 TeV) cosmic ray induced muons have been measured with deep underground or under-ice detectors. These muons come from pion and kaon decays and from charm production in the atmosphere. Terrestrial experiments are most sensitive to far-forward muons so the production rates are sensitive to high-$x$ partons in the incident nucleus and low-$x$ partons in the nitrogen/oxygen targets. Muon measurements can complement the central-particle data collected at colliders. This paper will review muon production data and discuss some non-perturbative (soft) models that have been used to interpret the data. I will show measurements of TeV muon transverse momentum ($p_T$) spectra in cosmic-ray air showers from MACRO, and describe how the IceCube neutrino observatory and the proposed Km3Net detector will extend these measurements to a higher $p_T$ region where perturbative QCD should apply. With a 1 km$^2$ surface area, the full IceCube detector should observe hundreds of muons/year with $p_T$ in the pQCD regime.Comment: 4 pages, 2 figures - To appear in the conference proceedings for Quark Matter 2009, March 30 - April 4, Knoxville, Tennessee. Tweaked formatting at organizers reques

On the homotopy invariance of configuration spaces

For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of Poincare embedding theory and fiberwise algebraic topology.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-35.abs.htm

Market free lunch and large financial markets

The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage are characterized in terms of no asymptotic market free lunch; the difference lies in the set of utilities. In particular, it is shown directly that no asymptotic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free lunch. In principle, the paper can be seen as the large financial market analogue of [Math. Finance 14 (2004) 351--357] and [Math. Finance 16 (2006) 583--588].Comment: Published at http://dx.doi.org/10.1214/105051606000000484 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org