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Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS Wales Institute of Mathematical and Computational Sciences. Analysis Cluster. Research interests include:. Nodal length of random spherical harmonics.
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Wales Institute of Mathematical and Computational Sciences
Research interests include:
Nodal length of random spherical harmonics
Geometric methods for image processing
Let M be a compact surface, and - Laplacian on M. We are interested in the eigenvalues and eigenfunctions of : .
It is well-known that the spectrum is discrete: , and
The nodal line of is simply its zeros set . We are interested in the geometry of the nodal lines as , most basically, in lj, the nodal length of .
Yau conjectured that is commensurable with in the sense that
. This conjecture was settled by Bruning & Donnelly-Fefferman in real analytic case (still open in the smooth case). Berry proposed
to model using random planewaves (RWM) with wavenumber .
Zelditch (?) conjectured that the nodal lines of generic chaotic manifolds are equidistributed. In particular, . The latter is wrong for completely integrable systems, such as the sphere or the torus.
We study the nodal length distribution of random spherical harmonics. Let Enbe the space of spherical harmonics of degree n (eigenvalue n(n+1)) – its dimension 2n+1, and pick any L2-orthonormal basis . We define the random spherical harmonic
, where ak are standard Gaussian i.i.d. Let
be the nodal length of the random spherical harmonic. We want to study the distribution of for large n. It is easy to compute that its expected value is of order n, consistent to Yau. Also, it follows from Yau that is bounded from below and above. However, does this it really fluctuate between two different numbers?
We may infer further information about the equidistribution conjecture using the following principle: Suppose X holds for generic eigenfunctions on completely integrable M. Then X also holds for all eigenfunctions on generic chaotic M.
Our primary focus is the variance of the nodal length. We proved the following:
Theorem (2008): .
It already answers our first inquiry: since the variance of the normalized length, , vanishes, it does not fluctuate for typical spherical harmonics, so that typically, we have . However, it is also important to evaluate the variance. We proved:
Theorem (2009): . The constant 65/32 in the theorem is different than the one predicted by Berry (1/64), based on the RWM.
The cover of the book "Entdeckungenüber die Theorie des Klanges" by Ernst F.F. Chladni (Discoveries concerning the theory of sound; Leipzig, 1787)
Removal of high-density noise from large images
Left: Lena (1229x1229) with 99.5% salt and pepper noise
Right: recovered image
Identification of multiscale
Left: multiscale medial axis of a Chinese character including fine branches
Right: main branches of medial axis
The nodal structure of a random spherical harmonic. The blue and red connected components are positive and negative nodal domains respectively. The nodal lines are the domain boundaries.
Left: detection of crossing points
Right: detection of end points
Patent application“Image processing” GB 0921863.7, filed December 2009
Joint project Kewei Zhang (Maths, Swansea), Antonio Orlando (Engineering, Swansea), Elaine Crooks (Maths, Swansea)
Past Cluster Workshops
Future Cluster Activities
Poster Presenters: Dr I Wigman, Dr E Crooks