52 research outputs found
Spherical Functions Associated With the Three Dimensional Sphere
In this paper, we determine all irreducible spherical functions \Phi of any K
-type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by
associating to \Phi a vector valued function H=H(u) of a real variable u, which
is analytic at u=0 and whose components are solutions of two coupled systems of
ordinary differential equations. By an appropriate conjugation involving Hahn
polynomials we uncouple one of the systems. Then this is taken to an uncoupled
system of hypergeometric equations, leading to a vector valued solution P=P(u)
whose entries are Gegenbauer's polynomials. Afterward, we identify those
simultaneous solutions and use the representation theory of \SO(4) to
characterize all irreducible spherical functions. The functions P=P(u)
corresponding to the irreducible spherical functions of a fixed K-type \pi_\ell
are appropriately packaged into a sequence of matrix valued polynomials
(P_w)_{w\ge0} of size (\ell+1)\times(\ell+1). Finally we proved that \widetilde
P_w={P_0}^{-1}P_w is a sequence of matrix orthogonal polynomials with respect
to a weight matrix W. Moreover we showed that W admits a second order symmetric
hypergeometric operator \widetilde D and a first order symmetric differential
operator \widetilde E.Comment: 49 pages, 2 figure
Some comments on the inverse problem of pure point diffraction
In a recent paper, Lenz and Moody (arXiv:1111.3617) presented a method for
constructing families of real solutions to the inverse problem for a given pure
point diffraction measure. Applying their technique and discussing some
possible extensions, we present, in a non-technical manner, some examples of
homometric structures.Comment: 6 pages, contribution to Aperiodic 201
A new algorithm for computing the Geronimus transformations for large shifts
A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift ? transforms the monic Jacobi matrix associated with a measure d? into the monic Jacobi matrix associated with d?/(x????)?+?C?(x????), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C?=?0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision
A new approach to quantitative propagation of chaos for drift, diffusion and jump processes
This paper is devoted the the study of the mean field limit for many-particle
systems undergoing jump, drift or diffusion processes, as well as combinations
of them. The main results are quantitative estimates on the decay of
fluctuations around the deterministic limit and of correlations between
particles, as the number of particles goes to infinity. To this end we
introduce a general functional framework which reduces this question to the one
of proving a purely functional estimate on some abstract generator operators
(consistency estimate) together with fine stability estimates on the flow of
the limiting nonlinear equation (stability estimates). Then we apply this
method to a Boltzmann collision jump process (for Maxwell molecules), to a
McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision
jump process with (stochastic) thermal bath. To our knowledge, our approach
yields the first such quantitative results for a combination of jump and
diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction
of a few typos, to appear In Probability Theory and Related Field
Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy
Pairs of matrices whose commutator differ from the identity by a
matrix of rank are used to construct bispectral differential operators with
matrix coefficients satisfying the Lax equations of the Matrix KP
hierarchy. Moreover, the bispectral involution on these operators has dynamical
significance for the spin Calogero particles system whose phase space such
pairs represent. In the case , this reproduces well-known results of
Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to
the bispectrality of (scalar) differential operators. This new class of pairs
of bispectral matrix differential operators is different than
those previously studied in that acts from the left, but from the
right on a common eigenmatrix.Comment: 16 page
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and, particularly for applications in the
geosciences, for scalar and vectorial signals defined on the surface of a unit
sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics,
edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be
published by Springer Verlag. This is a slightly modified but expanded
version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the
Handbook, when it was called: Slepian functions and their use in signal
estimation and spectral analysi
A New Euler's Formula for DNA Polyhedra
DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus
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