No abstract uploaded!

No abstract uploaded!

In this paper, a random graph process {$G$($t$)}_{$t$≥1}is studied and its degree sequence is analyzed. Let {$W$_{$t$}}_{$t$≥1}be an i.i.d. sequence. The graph process is defined so that, at each integer time$t$, a new vertex with$W$_{$t$}edges attached to it, is added to the graph. The new edges added at time$t$are then preferentially connected to older vertices, i.e., conditionally on$G$($t$-1), the probability that a given edge of vertex$t$is connected to vertex$i$is proportional to$d$_{$i$}($t$-1)+δ, where$d$_{$i$}($t$-1) is the degree of vertex$i$at time$t$-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_{W},τ_{P}}, where τ_{W}is the power-law exponent of the initial degrees {$W$_{$t$}}_{$t$≥1}and τ_{P}the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

We prove that all maximal-tb positive Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m), (3,3), (3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2,Z) and the mapping class group M0,4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

No abstract uploaded!