Small sets with large power sets

Small vulnerable sets determine large network cascades in power grids.

The understanding of cascading failures in complex systems has been hindered by the lack of realistic large-scale modeling and analysis that can account for variable system conditions. Using the North American power grid, we identified, quantified, and analyzed the set of network components that are vulnerable to cascading failures under any out of multiple conditions. We show that the vulnerab...

Some small large sets of t-designs

Small Sets Satisfying the Central Sets Theorem

The Central Sets Theorem is a powerful theorem, one of whose consequences is that any central set in N contains solutions to any partition regular system of homogeneous linear equations. Since at least one set in any finite partition of N must be central, any of the consequences of the Central Sets Theorem must be valid for any partition of N. It is a result of Beiglböck, Bergelson, Downarowicz...

Sets with large additive energy and symmetric sets

We show that for any set A in a finite Abelian group G that has at least c|A|3 solutions to a1 + a2 = a3 + a4, ai ∈ A there exist sets A ′ ⊆ A and Λ ⊆ G, Λ = {λ1, . . . , λt}, t ≪ c −1 log |A| such that A′ is contained in {∑t j=1 εjλj | εj ∈ {0,−1, 1} } and A′ has ≫ c|A|3 solutions to a1 + a ′ 2 = a ′ 3 + a ′ 4, a ′ i ∈ A ′. We also study so–called symmetric sets or, in other words, sets of lar...

Reasoning with Large Data Sets

Efficient reasoning is a critical factor for successful Semantic Web applications. In this context, applications may require vast volumes of data to be processed in a short time. We develop novel reasoning techniques which will extend current reasoning methods as well as existing database technologies in order to enable large scale reasoning. We propose advances and key design principles primar...