Laplace equations, conformal superintegrability and B\^ocher contractions

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems and their algebras are related by geometric limits, induced by generalized In\"on\"u-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of B\^ocher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra $so(4,C)$ to itself. Here we announce main findings, with detailed classifications in papers under preparation.Comment: 10 pages, 2 figure

Rotation and scale space random fields and the Gaussian kinematic formula

We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper. This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley-Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strong contraction of the representations of the three dimensional Lie algebras

For any Inonu-Wigner contraction of a three dimensional Lie algebra we construct the corresponding contractions of representations. Our method is quite canonical in the sense that in all cases we deal with realizations of the representations on some spaces of functions; we contract the differential operators on those spaces along with the representation spaces themselves by taking certain pointwise limit of functions. We call such contractions strong contractions. We show that this pointwise limit gives rise to a direct limit space. Many of these contractions are new and in other examples we give a different proof

A Possible Hidden Population of Spherical Planetary Nebulae

We argue that relative to non-spherical planetary nebulae (PNs), spherical PNs are about an order of magnitude less likely to be detected, at distances of several kiloparsecs. Noting the structure similarity of halos around non-spherical PNs to that of observed spherical PNs, we assume that most unobserved spherical PNs are also similar in structure to the spherical halos around non-spherical PNs. The fraction of non-spherical PNs with detected spherical halos around them, taken from a recent study, leads us to the claim of a large (relative to that of non-spherical PNs) hidden population of spherical PNs in the visible band. Building a toy model for the luminosity evolution of PNs, we show that the claimed detection fraction of spherical PNs based on halos around non-spherical PNs, is compatible with observational sensitivities. We use this result to update earlier studies on the different PN shaping routes in the binary model. We estimate that ~30% of all PNs are spherical, namely, their progenitors did not interact with any binary companion. This fraction is to be compared with the ~3% fraction of observed spherical PNs among all observed PNs. From all PNs, ~15% owe their moderate elliptical shape to the interaction of their progenitors with planets, while \~55% of all PNs owe their elliptical or bipolar shapes to the interaction of their progenitors with stellar companions.Comment: AJ, in pres

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index $H \to 0$ (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the position of the maximum $x_m$ through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the $\beta$-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary $\beta >0$ and positive integer $n$ in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix A from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit $n \to 0$ and to negative Dyson index $\beta \to -2$, we obtain the moments of $x_m$ and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.Comment: 64 page, 5 figures, with Appendix A written by Alexei Borodin and Vadim Gorin; The appendix H in the ArXiv version is absent in the published versio

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra so(4,C) to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of e(2,C) and so(3,C) and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems

Geodesic-Controlled Developable Surfaces for Modeling Paper Bending

We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces. © 2007 The Eurographics Association and Blackwell Publishing Ltd.link_to_subscribed_fulltex