Role of initial conditions in the Universe formation

A dependence of low energy physics on initial conditions in the framework of multidimensional gravity is discussed. It is shown that the observable symmetries could be a result of specific topologies originated from space-time foam.Comment: 20 page

Snow Leopard Permutations and Their Even and Odd Threads

Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$.Comment: 25 pages, 6 figures. Version 3 is modified to use standard Discrete Mathematics and Theoretical Computer Science but is otherwise unchange