15 research outputs found
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
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 (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 through its moments. Using replica, this requires
calculating moments of the density of eigenvalues in the -Jacobi
ensemble. Using Jack polynomials we obtain an exact and explicit expression for
both positive and negative integer moments for arbitrary and
positive integer 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 and to negative Dyson index , we obtain the
moments of 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