On a Generalized Oscillator: Invariance Algebra and Interbasis Expansions

This article deals with a quantum-mechanical system which generalizes the ordinary isotropic harmonic oscillator system. We give the coefficients connecting the polar and Cartesian bases for D=2 and the coefficients connecting the Cartesian and cylindrical bases as well as the cylindrical and spherical bases for D=3. These interbasis expansion coefficients are found to be analytic continuations to real values of their arguments of the Clebsch-Gordan coefficients for the group SU(2). For D=2, the superintegrable character for the generalized oscillator system is investigated from the points of view of a quadratic invariance algebra.Comment: 13 pages, Latex file. Submitted for publication to Yadernaya Fizik

Superintegrability on the two-dimensional hyperboloid

In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras

Superintegrability on the two dimensional hyperboloid II

This work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admit separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C.P.Boyer, E.G.Kalnins and P.Winternitz, which haven't yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.Comment: 18 pages, LaTex; 1 figure (eps

Cosmic microwave background snapshots: pre-WMAP and post-WMAP

Abbreviated: We highlight the remarkable evolution in the CMB power spectrum over the past few years, and in the cosmological parameters for minimal inflation models derived from it. Grand unified spectra (GUS) show pre-WMAP optimal bandpowers are in good agreement with each other and with the one-year WMAP results, which now dominate the L < 600 bands. GUS are used to determine calibrations, peak/dip locations and heights, and damping parameters. These CMB experiments significantly increased the case for accelerated expansion in the early universe (the inflationary paradigm) and at the current epoch (dark energy dominance) when they were combined with `prior' probabilities on the parameters. A minimal inflation parameter set is applied in the same way to the evolving data. Grid-based and and Monte Carlo Markov Chain methods are shown to give similar values, highly stable over time and for different prior choices, with the increasing precision best characterized by decreasing errors on uncorrelated parameter eigenmodes. After marginalizing over the other cosmic and experimental variables for a weak+LSS prior, the pre-WMAP data of Jan03 cf. the post-WMAP data of Mar03 give Omega_{tot} =1.03^{+0.05}_{-0.04} cf. 1.02^{+0.04}_{-0.03}. Adding the flat prior, n_s =0.95^{+0.07}_{-0.04} cf. 0.97^{+0.02}_{-0.02}, with < 2\sigma evidence for a log variation of n_s. The densities have concordance values. The dark energy pressure-to-density ratio is not well constrained by our weak+LSS prior, but adding SN1 gives w_Q < -0.7. We find \sigma_8 = 0.89^{+0.06}_{-0.07} cf. 0.86^{+0.04}_{-0.04}, implying a sizable SZ effect; the high L power suggest \sigma_8 \sim 0.94^{+0.08}_{-0.16} is needed to be SZ-compatible.Comment: 36 pages, 5 figures, 5 tables, Jan 2003 Roy Soc Discussion Meeting on `The search for dark matter and dark energy in the Universe', published PDF (Oct 15 2003) is http://www.cita.utoronto.ca/~bond/roysoc03/03TA2435.pd

Quantum Systems with Hidden Symmetry. Interbasis Expansions

This monograph is the English version of the book "Quantum systems with hidden symmetry. Interbasis expansions" published in 2006 by the publishing house FIZMATLIT (Moscow) in Russian. When compiling this version of the book, typos and inaccuracies noted since the release of the Russian edition have been corrected

How Filaments are Woven into the Cosmic Web

Observations indicate galaxies are distributed in a filament-dominated web-like structure. Numerical experiments at high and low redshift of viable structure formation theories also show filament-dominance. We present a simple quantitative explanation of why this is so, showing that the final-state web is actually present in embryonic form in the overdensity pattern of the initial fluctuations, with nonlinear dynamics just sharpening the image. The web is largely defined by the position and primordial tidal fields of rare events in the medium, with the strongest filaments between nearby clusters whose tidal tensors are nearly aligned. Applications of the cosmic web theory to observations include probing cluster-cluster bridges by weak gravitational lensing, X-rays, and the Sunyaev-Zeldovich effect and probing high redshift galaxy-galaxy bridges by low column density Lyman alpha absorption lines.Comment: 9 pages, gzipped uuencoded postscript file, 4 figures in separate files. The text + figures are also available from anonymous ftp site: ftp://ftp.cita.utoronto.ca/ftp/cita/bond/bkp_natur

Low molecular weight blood plasma proteome – a source of differential diagnostic biomarkers of ovarian cancer

At present, there is no screening test for the early detecting of ovarian cancer, one of the most lethal form of gynaecological malignancy in the worldwide. In this study the new methodology for the search of tumor markers of ovarian cancer, involving profiling the low-molecular blood plasma proteomes, is developed, unified and approved. The given approach included three basic components: pre-preparation of samples, matrix-assisted laser desorption / ionization time-of-flight mass spectrometry and bioinformatics software for mass spectral data processing. Opportunities and prospects of the developed approach for the detection of potential ovarian cancer markers were shown. For search of potential tumor markers, screening of 56 blood plasma samples from ovarian cancer patients and 36 benign ovarian neoplasia samples were carried out.As a result of the present research, peptides / polypeptides which can be used in future for detecting this pathology were found out

An infinite family of superintegrable Hamiltonians with reflection in the plane

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly solvable. The angular part of the wave function is expressed in terms of little -1 Jacobi polynomials. The spectra exhibit "accidental" degeneracies. The superintegrability of the model is proved using the recurrence relation approach. The (higher-order) constants of motion are constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page