Move-minimizing puzzles, diamond-colored modular and distributive lattices, and poset models for Weyl group symmetric functions

The move-minimizing puzzles presented here are certain types of one-player combinatorial games that are shown to have explicit solutions whenever they can be encoded in a certain way as diamond-colored modular and distributive lattices. Such lattices can also arise naturally as models for certain algebraic objects, namely Weyl group symmetric functions and their companion semisimple Lie algebra representations. The motivation for this paper is therefore both diversional and algebraic: To show how some recreational move-minimizing puzzles can be solved explicitly within an order-theoretic context and also to realize some such puzzles as combinatorial models for symmetric functions associated with certain fundamental representations of the symplectic and odd orthogonal Lie algebras

The logic of causally closed spacetime subsets

The causal structure of space-time offers a natural notion of an opposite or orthogonal in the logical sense, where the opposite of a set is formed by all points non time-like related with it. We show that for a general space-time the algebra of subsets that arises from this negation operation is a complete orthomodular lattice, and thus has several of the properties characterizing the algebra physical propositions in quantum mechanics. We think this fact could be used to investigate causal structure in an algebraic context. As a first step in this direction we show that the causal lattice is in addition atomic, find its atoms, and give necesary and sufficient conditions for ireducibility.Comment: 17 pages, 8 figure

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

Thermodynamic stability of folded proteins against mutations

By balancing the average energy gap with its typical change due to mutations for protein-like heteropolymers with M residues, we show that native states are unstable to mutations on a scale M* ~ (lambda/sigma_mu)^(1/zeta_s), where lambda is the dispersion in the interaction free energies and sigma_mu their typical change. Theoretical bounds and numerical estimates (based on complete enumeration on four lattices) of the instability exponent zeta_s are given. Our analysis suggests that a limiting size of single-domain proteins should exist, and leads to the prediction that small proteins are insensitive to random mutations.Comment: 5 pages, 3 figures, to be published in Physical Review Letter