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Probablog

October 22, 2015

Basic Matrix Inequalities

Sometimes it’s useful to think of real numbers as symmetric 1-by-1 matrices. By increasing the dimension, we may formulate matrix versions of various properties of the real numbers. The analogue of a positive real number is a positive definite matrix, and the analogue of a non-negative number is a positive semi-definite matrix.


October 20, 2015

Erdős Discrepancy Problem

Terry Tao has recently solved the Erdős Discrepancy Problem. The goal is to show that every infinite sequence of ±1 has infinite discrepancy. The discrepancy of a sequence is defined to be the largest magnitude of the sum of an evenly spaced subsequence. That’s the entire problem statement!