A trillion triangles: New computer methods reveal secrets of ancient math problem

Mathematicians from North America, Europe, Australia, and South America have resolved the first one trillion cases of an ancient mathematics problem. The advance was made possible by a clever technique for multiplying large numbers. The numbers involved are so enormous that if their digits were written out by hand they would stretch to the moon and back. The biggest challenge was that these numbers could not even fit into the main memory of the available computers, so the researchers had to make extensive use of the computers' hard drives.

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Original source : http://www.physorg.com/news172819291.html...

Finite Section Method for a Banach Algebra of Convolution Type Operators on $L^p(mathbb{R})$ with Symbols Generated by $PC$ and $SO$. (arXiv:0909.3821v1 [math.FA])

We prove the applicability of the finite section method to an arbitrary
operator in the Banach algebra generated by the operators of multiplication by
piecewise continuous functions and the convolution operators with symbols in
the algebra generated by piecewise continuous and slowly oscillating Fourier
multipliers on $L^p(mathbb{R})$, $1<p<infty$.

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Original source : http://arxiv.org/abs/0909.3821...

A Spectral Method for the Eigenvalue Problem for Elliptic Equations. (arXiv:0909.3607v1 [math.NA])

Let $Omega$ be an open, simply connected, and bounded region in
$mathbb{R}^{d}$, $dgeq2$, and assume its boundary $partialOmega$ is smooth.
Consider solving the eigenvalue problem $Lu=lambda u$ for an elliptic partial
differential operator $L$ over $Omega$ with zero values for either Dirichlet
or Neumann boundary conditions. We propose, analyze, and illustrate a 'spectral
method' for solving numerically such an eigenvalue problem. This is an
extension of the methods presented earlier in [5],[6].

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Original source : http://arxiv.org/abs/0909.3607...