Interesting Eigenvectors of the Fourier Transform

It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions – two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with four different eigenvalues. That is, the Fourier transformof each of the four components is simply that component multiplied by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular interest find application in coding, communication and imaging. Some of the underlying mathematics goes back to the times of Carl Friedrich Gauss

Freedom and constraints in the K3 landscape

We consider ``magnetized brane'' compactifications of the type I/heterotic string on K3 with U(1) background fluxes. The gauge group and matter content of the resulting six-dimensional vacua are parameterized by a matrix encoding a lattice contained within the even, self-dual lattice Γ[superscript 3,19]. Mathematical results of Nikulin on lattice embeddings make possible a simple classification of all such solutions. We find that every six-dimensional theory parameterized in this way by a negative semi-definite matrix whose trace satisfies a simple tadpole constraint can be realized as a K3 compactification. This approach makes it possible to explicitly and efficiently construct all models in this class with any particular allowed gauge group and matter content, so that one can immediately ``dial-a-model'' with desired properties