Holomorphic Quantization on the Torus and Finite Quantum Mechanics

We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.Comment: 12 pages LaTeX (needs amssymb.sty). Version as will appear in J. Phys.

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $\chi^{(3)}$ and $\chi^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $n$-particle contributions $\chi^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlev\'e oriented approach.Comment: 21 pages Proceedings of the Counting Complexity conferenc

Quantum Mechanics on the cylinder

A new approach to deformation quantization on the cylinder considered as phase space is presented. The method is based on the standard Moyal formalism for R^2 adapted to (S^1 x R) by the Weil--Brezin--Zak transformation. The results are compared with other solutions of this problem presented by Kasperkovitz and Peev (Ann. Phys. vol. 230, 21 (1994)0 and by Plebanski and collaborators (Acta Phys. Pol. vol. B 31}, 561 (2000)). The equivalence of these three methods is proved.Comment: 21 pages, LaTe

The Real Symplectic Groups in Quantum Mechanics and Optics

text of abstract (We present a utilitarian review of the family of matrix groups $Sp(2n,\Re)$, in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of $Sp(2n,\Re)$. Global decomposition theorems, interesting subgroups and their generators are described. Turning to $n$-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under $Sp(2n,\Re)$ action are delineated.)Comment: Review article 43 pages, revtex, no figures, replaced because somefonts were giving problem in autometic ps generatio

Medium effects in proton-induced K0K^{0} production at 3.5 GeV

We present the analysis of the inclusive $K^{0}$ production in p+p and p+Nb collisions measured with the HADES detector at a beam kinetic energy of 3.5 GeV. Data are compared to the GiBUU transport model. The data suggest the presence of a repulsive momentum-dependent kaon potential as predicted by the Chiral Perturbation Theory (ChPT). For the kaon at rest and at normal nuclear density, the ChPT potential amounts to $\approx 35$ MeV. A detailed tuning of the kaon production cross sections implemented in the model has been carried out to reproduce the experimental data measured in p+p collisions. The uncertainties in the parameters of the model were examined with respect to the sensitivity of the experimental results from p+Nb collisions to the in-medium kaon potential.Comment: 13 pages, 14 figure

p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.Comment: 36 page

eLearning resources to supplement postgraduate neurosurgery training.

BACKGROUND: In an increasingly complex and competitive professional environment, improving methods to educate neurosurgical residents is key to ensure high-quality patient care. Electronic (e)Learning resources promise interactive knowledge acquisition. We set out to give a comprehensive overview on available eLearning resources that aim to improve postgraduate neurosurgical training and review the available literature. MATERIAL AND METHODS: A MEDLINE query was performed, using the search term "electronic AND learning AND neurosurgery". Only peer-reviewed English-language articles on the use of any means of eLearning to improve theoretical knowledge in postgraduate neurosurgical training were included. Reference lists were crosschecked for further relevant articles. Captured parameters were the year, country of origin, method of eLearning reported, and type of article, as well as its conclusion. eLearning resources were additionally searched for using Google. RESULTS: Of n = 301 identified articles by the MEDLINE search, n = 43 articles were analysed in detail. Applying defined criteria, n = 28 articles were excluded and n = 15 included. Most articles were generated within this decade, with groups from the USA, the UK and India having a leadership role. The majority of articles reviewed existing eLearning resources, others reported on the concept, development and use of generated eLearning resources. There was no article that scientifically assessed the effectiveness of eLearning resources (against traditional learning methods) in terms of efficacy or costs. Only one article reported on satisfaction rates with an eLearning tool. All authors of articles dealing with eLearning and the use of new media in neurosurgery uniformly agreed on its great potential and increasing future use, but most also highlighted some weaknesses and possible dangers. CONCLUSION: This review found only a few articles dealing with the modern aspects of eLearning as an adjunct to postgraduate neurosurgery training. Comprehensive eLearning platforms offering didactic modules with clear learning objectives are rare. Two decades after the rise of eLearning in neurosurgery, some promising solutions are readily available, but the potential of eLearning has not yet been sufficiently exploited

Wide-Field Precision Kinematics of the M87 Globular Cluster System

We present the most extensive combined photometric and spectroscopic study to date of the enormous globular cluster (GC) system around M87, the central giant elliptical galaxy in the nearby Virgo cluster. Using observations from DEIMOS and LRIS at Keck, and Hectospec on the MMT, we derive new, precise radial velocities for 451 GCs around M87, with projected radii from ~ 5 to 185 kpc. We combine these measurements with literature data for a total sample of 737 objects, which we use for a re-examination of the kinematics of the GC system of M87. The velocities are analyzed in the context of archival wide-field photometry and a novel Hubble Space Telescope catalog of half-light radii, which includes sizes for 344 spectroscopically confirmed clusters. We use this unique catalog to identify 18 new candidate ultra-compact dwarfs, and to help clarify the relationship between these objects and true GCs. We find much lower values for the outer velocity dispersion and rotation of the GC system than in earlier papers, and also differ from previous work in seeing no evidence for a transition in the inner halo to a potential dominated by the Virgo cluster, nor for a truncation of the stellar halo. We find little kinematical evidence for an intergalactic GC population. Aided by the precision of the new velocity measurements, we see significant evidence for kinematical substructure over a wide range of radii, indicating that M87 is in active assembly. A simple, scale-free analysis finds less dark matter within ~85 kpc than in other recent work, reducing the tension between X-ray and optical results. In general, out to a projected radius of ~ 150 kpc, our data are consistent with the notion that M87 is not dynamically coupled to the Virgo cluster; the core of Virgo may be in the earliest stages of assembly.Comment: 47 pages, ApJS in press. Redacted long data tables available on reques