Tableaux Let λ be a partition and let μ = (μ1, . . . , μn) ∈ Z≥0 be a sequence of nonnegative integers. A column strict tableau of shape λ and weight μ is a filling of the boxes of λ with μ1 1s, μ2 2s, . . ., μn ns, such that (a) the rows are weakly increasing from left to right, (b) the columns are strictly increasing from top to bottom. If p is a column strict tableau write shp(p) and wt(p) f...

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1 Maximum flow In the maximum flow problem, we are given a directed graph with a distinguished source node and a distinguished sink node. In addition, each edge has a capacity, which is the largest amount of flow that can go over this edge. Our goal is to route as much flow from the source to the sink as we can, along the edges of the graph. For example, consider the directed graph with capacit...

Professor: Mark van der Laan Email: [email protected] Office: 108 Haviland Hall Office Hours: MW 1-2 Textbook: None required, but material is covered in a handout distributed to the class. Optional reading is The Statistical Analysis of Failure Time Data by Kalbfleisch and Prentice. Grading: Based on attendance, class notes (students write their notes for one lecture as a LaTex document), ...

In theoretical studies, we often develop structures that are dedicated to specific problems. In practice, often it is unrealistic to create many different indexes on the same dataset to support various types of queries, because doing so will incur prohibitive space consumption and update overhead. Therefore, it would be nice to have a single, all-around, structure, which occupies small space, c...

1 Riemann integration 2 1.1 Partitions and Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A criterion for integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Upper and Lower Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The refinement of a partition . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Properties of upper an...

These are the notes for the 2011 Summer Tutorial on Coding Theory. I have not gone through and given citations or references for all of the results given here, but the presentation relies heavily on two sources, van Lint’s Introduction to Coding Theory and the book of Huffman and Pless Fundamentals of Error-Correcting Codes. I also used course notes written by Sebastian Pancratz from a Part II ...

Today we prove the Goldreich-Levin theorem.

We begin this lecture by discussing how to compare running times of functions in an abstract, mathematical way. The same underlying mathematics can be used for other purposes, like comparing memory consumption or the amount of parellism permitted by an algorithm. We then use this to take a first look at sorting algorithms, of which there are many. In this lecture it will be selection sort becau...

Caution! These lecture notes are very rough. They are mainly intended for my own use during lecture. They almost surely contain errors and typos, and tend to be written in a stream-of-consciousness style. In many cases details, precise statements, and proofs are left to Durrett’s text, homework or presentations. But perhaps these notes will be useful as a reminder of what was done in lecture. I...