150,476 research outputs found
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
Optimal detection of sparse principal components in high dimension
We perform a finite sample analysis of the detection levels for sparse
principal components of a high-dimensional covariance matrix. Our minimax
optimal test is based on a sparse eigenvalue statistic. Alas, computing this
test is known to be NP-complete in general, and we describe a computationally
efficient alternative test using convex relaxations. Our relaxation is also
proved to detect sparse principal components at near optimal detection levels,
and it performs well on simulated datasets. Moreover, using polynomial time
reductions from theoretical computer science, we bring significant evidence
that our results cannot be improved, thus revealing an inherent trade off
between statistical and computational performance.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1127 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Safe Compositional Specification of Networking Systems: A Compositional Analysis Approach
We present a type inference algorithm, in the style of compositional analysis, for the language TRAFFIC—a specification language for flow composition applications proposed in [2]—and prove that this algorithm is correct: the typings it infers are principal typings, and the typings agree with syntax-directed type checking on closed flow specifications. This algorithm is capable of verifying partial flow specifications, which is a significant improvement over syntax-directed type checking algorithm presented in [3]. We also show that this algorithm runs efficiently, i.e., in low-degree polynomial time.National Science Foundation (ITR ANI-0205294, ANI-0095988, ANI-9986397, EIA-0202067
On Point Spread Function modelling: towards optimal interpolation
Point Spread Function (PSF) modeling is a central part of any astronomy data
analysis relying on measuring the shapes of objects. It is especially crucial
for weak gravitational lensing, in order to beat down systematics and allow one
to reach the full potential of weak lensing in measuring dark energy. A PSF
modeling pipeline is made of two main steps: the first one is to assess its
shape on stars, and the second is to interpolate it at any desired position
(usually galaxies). We focus on the second part, and compare different
interpolation schemes, including polynomial interpolation, radial basis
functions, Delaunay triangulation and Kriging. For that purpose, we develop
simulations of PSF fields, in which stars are built from a set of basis
functions defined from a Principal Components Analysis of a real ground-based
image. We find that Kriging gives the most reliable interpolation,
significantly better than the traditionally used polynomial interpolation. We
also note that although a Kriging interpolation on individual images is enough
to control systematics at the level necessary for current weak lensing surveys,
more elaborate techniques will have to be developed to reach future ambitious
surveys' requirements.Comment: Accepted for publication in MNRA
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