A Unified Convergence Theorem

A Unified Achievability Theorem

We prove a single achievability theorem that recovers many achievability results in network information theory as simple corollaries. Our result is enabled by many novel elements such as a generalized network model that includes multiple sources and channels interacting with each other, a unified typicality constraint that generalizes cost and distortion constraints, a unified coding strategy t...

A metastable dominated convergence theorem

The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( ∫ fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound...

A Convergence Theorem For Controlled . . .

Unified Hanani-Tutte Theorem

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly inD. The theorem is implicit in the proof of the strong Hanani–Tu...

A Unified Theorem on SDP Rank Reduction

We consider the problem of finding a low–rank approximate solution to a system of linear equations in symmetric, positive semidefinite matrices, where the approximation quality of a solution is measured by its maximum relative deviation, both above and below, from the prescribed quantities. We show that a simple randomized polynomial– time procedure produces a low–rank solution that has provabl...