Graph limits of random graphs from a subset of connected k-trees.

For any set Ω of non-negative integers such that , we consider a random Ω-k-tree G that is uniformly selected from all connected k-trees of (n + k) vertices such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to Ω. We prove that G, scaled by where H is the kth harmonic number and σ  > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω-k-tree to an infinite but locally finite random Ω-k-tree G.

RNA secondary structures in a polymer-zeta model how foldings should be shaped for sparsification to establish a linear speedup.

Various tools used to predict the secondary structure for a given RNA sequence are based on dynamic programming used to compute a conformation of minimum free energy. For structures without pseudoknots, a worst-case runtime proportional to n3, with n being the length of the sequence, results since a table of dimension n2 has to be filled in while a single entry gives rise to a linear computational effort. However, it was recently observed that reformulating the corresponding dynamic programming recursion together with the bookkeeping of potential folding alternatives (a technique called sparsification) may reduce the runtime to n2 on average, assuming that nucleotides of distance d form a hydrogen bond (i.

Equality of Shapley value and fair proportion index in phylogenetic trees.

The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (J Math Biol 67:1163-1170, 2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley value) and the fair proportion index. He gave an explanation of this correlation by showing that the contribution of both indices to an edge of the tree becomes identical as the number of taxa tends to infinity.