A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy-traffic regime. We derive the scaling limit of the multi-dimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.Comment: Final version accepted for publication in Queueing Systems, Theory and Application

Strong "quantum" chaos in the global ballooning mode spectrum of three-dimensional plasmas

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a $k$-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.Comment: 4 pages typeset, including 2 figures. Paper accepted for publication in Phys. Rev. Letter

Patent Litigation in Europe

We compare patent litigation cases across four European jurisdictions—Germany, the UK (England and Wales), France, The Netherlands—using case-level data gathered from cases filed in the four jurisdictions during the period 2000–2008. Overall, we find substantial differences across jurisdictions in terms of caseloads—notably, courts in Germany hear by far the largest number of cases, not only in absolute terms, but also when taking macro-economic indicators into account—and we further find important cross-country variances in terms of case outcomes. Moreover, we show empirically that a considerable number of patents are litigated across multiple European jurisdictions; and further, that in the majority of these cases divergent case outcomes are reached across the different jurisdictions, suggesting that the long-suspected problem of inconsistency of decision-making in European patent litigation is in fact real. Finally, we note that the coming into force of the Unified Patent Court in Europe may, in the long term, help to alleviate this inconsistency problem

Density-functional theory of inhomogeneous electron systems in thin quantum wires

Motivated by current interest in strongly correlated quasi-one-dimensional (1D) Luttinger liquids subject to axial confinement, we present a novel density-functional study of few-electron systems confined by power-low external potentials inside a short portion of a thin quantum wire. The theory employs the 1D homogeneous Coulomb liquid as the reference system for a Kohn-Sham treatment and transfers the Luttinger ground-state correlations to the inhomogeneous electron system by means of a suitable local-density approximation (LDA) to the exchange-correlation energy functional. We show that such 1D-adapted LDA is appropriate for fluid-like states at weak coupling, but fails to account for the transition to a ``Wigner molecules'' regime of electron localization as observed in thin quantum wires at very strong coupling. A detailed analyzes is given for the two-electron problem under axial harmonic confinement.Comment: 8 pages, 7 figures, submitte

Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

We construct affinization of the algebra $gl_{\lambda}$ of ``complex size'' matrices, that contains the algebras $\hat{gl_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\hat{gl_{\lambda}}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure

Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.Comment: 61 pg

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Lymphoid tumours and breast cancer in ataxia telangiectasia; substantial protective effect of residual ATM kinase activity against childhood tumours

BACKGROUND: Immunodeficiency in ataxia telangiectasia (A-T) is less severe in patients expressing some mutant or normal ATM kinase activity. We, therefore, determined whether expression of residual ATM kinase activity also protected against tumour development in A-T. METHODS: From a total of 296 consecutive genetically confirmed A-T patients from the British Isles and the Netherlands, we identified 66 patients who developed a malignant tumour; 47 lymphoid tumours and 19 non-lymphoid tumours were diagnosed. We determined their ATM mutations, and whether cells from these patients expressed any ATM with residual ATM kinase activity. RESULTS: In childhood, total absence of ATM kinase activity was associated, almost exclusively, with development of lymphoid tumours. There was an overwhelming preponderance of tumours in patients <16 years without kinase activity compared with those with some residual activity, consistent with a substantial protective effect of residual ATM kinase activity against tumour development in childhood. In addition, the presence of eight breast cancers in A-T patients, a 30-fold increased risk, establishes breast cancer as part of the A-T phenotype. CONCLUSION: Overall, a spectrum of tumour types is associated with A-T, consistent with involvement of ATM in different mechanisms of tumour formation. Tumour type was influenced by ATM allelic heterogeneity, residual ATM kinase activity and age