We use the automorphism group Aut(H), of holes in the lattice L8=A2⊕A2⊕D4, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, L4=A2⊕A2, enters the construction because a subgroup of Aut(L4) is isomorphic to Aut(H). The lattices L8 and L4, when glued together through this relationship, provide an alternative construction of the laminated lattice in twelve dimensions with kissing number 648.

An information theoretic criterion for the feasibility of reconstructing diffraction signals from noisy tomographs, when the positions of the tomographs within the signal are unknown, is derived. For shot-noise limited data, the number of detected photons per tomograph for successful reconstruction is much smaller than previously believed necessary, growing only logarithmically with the number of contrast elements of the diffracting object.

The proposed experiments to image single molecules with x-ray free-electron lasers present an unprecedented challenge in data processing. We describe for non-experts the computational tasks and some recent progress in solving them. © 2009 OSA.

We introduce the EMC algorithm for reconstructing a particle's three-dimensional (3D) diffraction intensity from very many photon shot-noise limited two-dimensional measurements, when the particle orientation in each measurement is unknown. The algorithm combines a maximization step (M) of the intensity's likelihood function, with expansion (E) and compression (C) steps that map the 3D intensity model to a redundant tomographic representation and back again.

Many difficult computational problems involve the simultaneous satisfaction of multiple constraints that are individually easy to satisfy. These constraints might be derived from measurements (as in tomography or diffractive imaging), interparticle interactions (as in spin glasses), or a combination of sources (as in protein folding). We present a simple geometric framework to express and solve such problems and apply it to two benchmarks.

The problem of reconstructing an object from diffraction data that has been incoherently averaged over a discrete group of symmetries is considered. A necessary condition for such data to uniquely specify the object is derived in terms of the object support and the symmetry group. An algorithm is introduced for reconstructing objects from symmetry-averaged data and its use with simulations is demonstrated.

The most efficient known method for solving certain computational problems is to construct an iterated map whose fixed points are by design the problem's solution. Although the origins of this idea go back at least to Newton, the clearest expression of its logical basis is an example due to Mermin. A contemporary application in image recovery demonstrates the power of the method.

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ2 and its sections ("cuts") by ℂ. More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ2 generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity.

A recent proposal by Maris, 1 that single electron bubbles in helium might fission into separate, particle-like entities, does not properly take into account the failure of the adiabatic approximation when, due to tunneling, there is a long electronic time scale. The point along the fission pathway of a photoexcited p-state bubble, where the adiabatic approximation first breaks down, occurs well before the bubble waist has pinched down forming two cavities.