16 September 2017 | 10:07 pm

Suppose we have an matrix that is expressed in block-matrix form as where is an matrix, is an matrix, is an matrix, and is a matrix for some . If is invertible, we can use the technique of Schur complementation to express the inverse of (if it exists) in terms of the inverse of , […]

12 September 2017 | 5:43 pm

Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions. Roth’s theorem is the special case when one considers arithmetic progressions of length three. Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory. However, […]

5 September 2017 | 8:56 pm

Fix a non-negative integer . Define an (weak) integer partition of length to be a tuple of non-increasing non-negative integers . (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the […]