A. The filling length of an edge-circuit η in the Cayley 2-complex of a finitely presented group is the least integer L such that there is a combinatorial null-homotopy of η down to a basepoint through loops of length at most L. We introduce similar notions in which the null-homotopy is not required to fix a basepoint, and in which the contracting loop is allowed to bifurcate. We exhibit...

We derive a upper bound on the free energy of a Bose gas system at density ̺ and temperature T . In combination with the lower bound derived previously by Seiringer [15], our result proves that in the low density limit, i.e., when a̺ ≪ 1, where a denotes the scattering length of the pair-interaction potential, the leading term of ∆f the free energy difference per volume between interacting and id...

We identify classes of functions for which a No Free Lunch result does and does not hold, with particular emphasis on the relationship between No Free Lunch and problem description length. We show that a NFL result does not apply to a set of functions when the description length of the functions is sufficiently bounded. We consider sets of functions with non-uniform associated probability distr...