Diophantine equations are in general undecidable, yet appear readily in string theory. We demonstrate that numerous classes of Diophantine equations arising in string theory are decidable and propose that decidability may propagate through networks of string vacua due to additional structure in the theory. Diophantine equations arising in index computations relevant for D3-instanton corrections to the superpotential exhibit propagation of decidability, with new and existing solutions propagating through networks of geometries related by topological transitions. In the geometries we consider, most divisor classes appear in at least one solution, significantly improving prospects for Kähler moduli stabilization across large ensembles of string compactifications.
In a gas aggregation cluster source, nanoparticles are formed by aggregation of single atoms to particles as well as by agglomeration of smaller particles to form larger particles. Because the small particles are often electrically charged, it is possible that particles are mutually attracted or repelled by electric forces. In this work we report the observation of bound multinanoparticle systems which are the result of the electrical and centrifugal forces acting on nanoparticles in a magnetron sputtering cluster source. Transmission electron microscopy analysis of the deposited 'planetary' nanoparticle systems exhibits characteristic inter-particle distances and sizes, consistent with different charge states of the particles. Simulations confirm the presence of metastable bound nanoparticle systems up to at least 7/8 satellites. The discovery of such 'planetary' nanoparticle systems inside a cluster source opens up possibilities for more research into their properties and how they can be further controlled and explored.
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