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246A, Notes 1: complex differentiation

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the… 6,298 more words


The two-dimensional case of the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture

In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely… 1,352 more words


Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture

Given any finite collection of elements in some Banach space , the triangle inequality tells us that

However, when the all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. 3,427 more words


275A, Notes 5: Variants of the central limit theorem

In the previous set of notes we established the central limit theorem, which we formulate here as follows:

Theorem 1 (Central limit theorem) Let be iid copies of a real random variable of mean and variance , and write .

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Thanksgiving Latte: Matcha Apple Nut Crisp

Thanksgiving is right around the corner and what better way to celebrate than teatime with family & friends. We have a lovely Matcha latte recipe to share with you, featuring one of our new teas,  182 more words


275A, Notes 4: The central limit theorem

Let be iid copies of an absolutely integrable real scalar random variable , and form the partial sums . As we saw in the last set of notes… 7,774 more words


275A, Notes 2: Product measures and independence

In the previous set of notes, we constructed the measure-theoretic notion of the Lebesgue integral, and used this to set up the probabilistic notion of expectation on a rigorous footing. 5,661 more words