Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Localization in disordered superconducting wires with broken spin-rotation symmetry

Localization and delocalization of non-interacting quasiparticle states in a superconducting wire are reconsidered, for the cases in which spin-rotation symmetry is absent, and time-reversal symmetry is either broken or unbroken; these are referred to as symmetry classes BD and DIII, respectively. We show that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for the transfer matrix, as in some previous work, then when there are more than two scattering channels, all terms that break a certain symmetry are lost. It was already known that the resulting FP equation exhibits critical behavior. The additional symmetry is not required by the definition of the symmetry classes; terms that break it arise from non-Gaussian probability distributions, and may be kept in a generalized FP equation. We show that they lead to localization in a long wire. When the wire has more than two scattering channels, these terms are irrelevant at the short distance (diffusive or ballistic) fixed point, but as they are relevant at the long-distance critical fixed point, they are termed dangerously irrelevant. We confirm the results in a supersymmetry approach for class BD, where the additional terms correspond to jumps between the two components of the sigma model target space. We consider the effect of random $\pi$ fluxes, which prevent the system localizing. We show that in one dimension the transitions in these two symmetry classes, and also those in the three chiral symmetry classes, all lie in the same universality class

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with computer algebra systems. As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified. The submission also contains an example implementation of the algorithms and formulas of the paper as a package for Maple 10, the technical documentation for this implementation, and a worksheet carrying out the computations for the space SU(3)/SO(3) used in the proof of Proposition 6.1 of the paper.Comment: 23 pages, also contains two Maple worksheets and technical documentatio

Connecting geodesics and security of configurations in compact locally symmetric spaces

A pair of points in a riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.Comment: 27 pages, 2 figure

Generalized quantum tomographic maps

Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.Comment: 8 page

Higher-order mesoscopic fluctuations in quantum wires: Conductance and current cumulants

We study conductance cumulants $>$ and current cumulants $C_j$ related to heat and electrical transport in coherent mesoscopic quantum wires near the diffusive regime. We consider the asymptotic behavior in the limit where the number of channels and the length of the wire in the units of the mean free path are large but the bare conductance is fixed. A recursion equation unifying the descriptions of the standard and Bogoliubov--de Gennes (BdG) symmetry classes is presented. We give values and come up with a novel scaling form for the higher-order conductance cumulants. In the BdG wires, in the presence of time-reversal symmetry, for the cumulants higher than the second it is found that there may be only contributions which depend nonanalytically on the wire length. This indicates that diagrammatic or semiclassical pictures do not adequately describe higher-order spectral correlations. Moreover, we obtain the weak-localization corrections to $C_j$ with $j\le 10$.Comment: 7 page

Use of warfarin anticoagulation in patients with atrial fibrillation in Iceland

Neðst á síðunni er hægt að nálgast greinina í heild sinni með því að smella á hlekkinn View/OpenObjective: Despite convincing evidence that warfarin anticoagulation reduces the risk of thromboembolism in patients with atrial fibrillation, recent data suggests that this therapy may be underutilized. Some patients are at higher risk than others and known risk factors for thromboembolism in nonvalvular atrial fibrillation include hypertension, diabetes, a prior history of a cerebrovascular accident or a transient ischemic attack and age over 65 years. Additionally, decreased left ventricular function and an enlarged left atrium increase the risk of emboli. Objective: To study the use of anticoagulation in patients with nonvalvular atrial fibrillation in Iceland we looked at the pattern of warfarin use in two different settings, the emergency room at a University Hospital in Reykjavik and those followed at the Solvangur Health Center, a primary health clinic, in Hafnarfjordur. Methods: Prospective data collection at the University Hospital and retrospective chart review at Solvangur Health Center. Results: A total of 68 patients (39 men, average age 73 years) with known preexisting atrial fibrillation were seen at the University Hospital during the 4 month study period. Thirty six (53%) were taking warfarin. Of the 32 not taking warfarin, 8 (25%) had a contraindication to anticoagulation. A large majority (96%) of the cohort had at least one risk factor for thromboembolism in atrial fibrillation. Fourteen (54%) of those not taking warfarin were on aspirin. At Solvangur Health Center, 40 of 71 patients (56%) (46 men, average age 72 years) with atrial fibrillation were taking warfarin while 4 of the 31 (13%) not on warfarin had a contraindication to the use of the medication. However, 14 (45%) of those not on warfarin were taking aspirin. In all 94% of the patients at Solvangur Health Center had at least one risk factor for thromboembolism. Conclusions: The use of warfarin in patients with atrial fibrillation in Iceland was found to be less than optimal. We speculate that reluctance to use anticoagulants in the elderly and perhaps lack of awareness of the data showing benefit of anticoagulation may contribute to this. Given the relatively easy access of physicians to anticoagulation clinics, the added burden of following an anticoagulated patient is unlikely to be a factor.Inngangur: Sjúklingar með gáttatif eru í aukinni hættu á að fá segarek sem oft hefur slæm áhrif á lífsgæði þeirra og horfur. Nokkrar stórar rannsóknir á undanförnum árum hafa sýnt fram á að blóðþynning með warfaríni getur dregið talsvert úr hættu á segareki hjá þeim sem hafa gáttatif án lokusjúkdóms. Aspirín dregur lítillega úr hættu á segareki en er engan veginn eins öflugt í því skyni og warfarín. Áhættuþættir fyrir segareki hjá sjúklingum með gáttatif án lokusjúkdóms eru: aldur yfir 65 ára, háþrýstingur, sykursýki, fyrri saga um heilaáfall auk stækkaðrar vinstri gáttar og skerts vinstri slegils. Blóðþynningarmeðferð með warfaríni er sérlega gagnleg þeim sjúklingum sem hafa einn eða fleiri ofantalinna áhættuþátta. Efniviður og aðferðir: Rannsókn þessi var tvíþætt. Annars vegar var notkun warfaríns og aspirín hjá sjúklingum með áður greint gáttatif könnuð hjá þeim sem leituðu á bráðamóttöku Landspítala við Hringbraut frá febrúar til júní árið 2000. Hins vegar var notkun blóðþynningarlyfja könnuð hjá sjúklingum sem fylgt hafði verið á heilsugæslustöðinni Sólvangi og höfðu staðfest gáttatif á árunum 1995-2000. Auk lyfjanotkunar voru áhættuþættir fyrir segareki kannaðir hjá þessum hópi svo og frábendingar gegn notkun blóðþynningarlyfja. Niðurstöður: Af 68 sjúklingum (39 karlar, meðalaldur 73 ár) sem leituðu á bráðamóttöku Landspítala við Hringbraut höfðu 65 (96%) sjúklinganna að minnsta kosti einn áhættuþátt fyrir segareki. Þrátt fyrir það voru aðeins 36 (53%) á warfarínmeðferð og af þeim 32 sjúklingum sem voru ekki á warfaríni höfðu aðeins átta skýra frábendingu gegn notkun þess og þrír engan áhættuþátt. Þannig var 21 (31%) sjúklingur ekki á warfarín blóðþynningu þrátt fyrir að vera með áhættuþátt fyrir segareki og enga frábendingu gegn warfaríni. Af 71 sjúklingi (46 karlar, meðalaldur 72 ár) sem voru skjólstæðingar Heilsugæslustöðvarinnar Sólvangs í Hafnarfirði voru 40 (56%) sjúklinganna á warfaríni. Af þeim sem tóku ekki warfarín höfðu aðeins tveir (3%) engan áhættuþátt fyrir segareki og fjórir (6%) höfðu skýra frábendingu gegn lyfinu. Tuttugu og fimm (35%) voru þannig ekki á warfaríni þrátt fyrir að ábending hefði verið fyrir slíku. Ályktanir: Niðurstöður þessarar rannsóknar sýna að aðeins rúmur helmingur sjúklinga með gáttatif var á warfarín blóðþynningu þrátt fyrir að nær allir sjúklinganna hefðu einn eða fleiri áhættuþátt fyrir segareki. Notkun blóðþynningarlyfja sem meðferðarúrræði hjá sjúklingum með gáttatif var því verulega ábótavant þrátt fyrir að fjölmargar stórar rannsóknir hafi ótvírætt sýnt að hún dregur úr hættu á segareki og heilablóðfalli. Ef til vill eru þessar niðurstöður ekki nægilega vel kynntar fyrir læknum hérlendis. Með hliðsjón af góðu aðgengi að blóðþynningarþjónustu á Landspítala bæði í Fossvogi og við Hringbraut er ólíklegt að aukið umstang við að sinna blóðþynntum sjúklingi sé mikilvæg ástæða

Weighted Bergman kernels and virtual Bergman kernels

We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.Comment: 12 pages. One-hour lecture for graduate students, SCV 2004, August 2004, Beijing, P.R. China. V2: typo correcte

Poisson homology of r-matrix type orbits I: example of computation

In this paper we consider the Poisson algebraic structure associated with a classical $r$-matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the $r$-matrix type Poisson orbits. Then we describe the $r$-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or $CP^n$-type orbits of $SL(n,C)$. Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions on $CP^n$-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld--Sklyanin Poisson brackets which belong to the $r$-matrix Poisson family

Exact and semiclassical approach to a class of singular integral operators arising in fluid mechanics and quantum field theory

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated to Riemannian symmetric spaces of rank one with positive, negative or vanishing curvature. For all other cases an accurate semiclassical approximation is derived, based on the identification of the operators with a peculiar Schroedinger-like operator.Comment: 12 pages, 1 figure, amslatex, bibtex (added missing label eq.11