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Maps such as $\eta$ and $\xi$ are called measure-preserving and are studied in ergodic theory. In particular ergodic theory views these as dynamical systems, because the maps can be iterated. One th …

I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies: Complete independence (the random variables $\eta(B_1), \ldots, \eta(B_n …

One way to come at this is to try to stretch your intuition even farther, toward the Johnson-Lindenstrauss lemma, which says that while we can only fit $n$ orthogonal vectors into $\mathbb{R}^n$, we c …

One example that is close to, if not exactly of this type, is Veit Elser's demonstration that machine learning techniques can learn how to do fast matrix multiplication from examples of matrix product …

No, the condition on the marginals is not sufficient. Consider for example three 0-1 valued random variables with bivariate marginals $p_i(s,t) = 0.5(1-\delta(s,t))$ for all $i, s, t$. The necessary …

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to in …

Could one of these two be what you're looking for? J. Milnor, Analytic proofs of the “hairy ball theorem” and the Brouwer fixed-point theorem, Amer. Math. Monthly 85 (1978), no. 7, 521–524. MR MR505 …

I enjoyed a series of talks by Bernd Sturmfels on some such interrelationships, which it looks like are written up in a paper by Rostalski and Sturmfels called "Dualities in Convex Algebraic Geometry. …

For the second question (theoretical computer science) I strongly recommend Sipser's Introduction to the Theory of Computation. It is a very easy read for someone with a math background, and requires …

My impression is that most math papers (and almost all of the most important ones) are now published in English. Not long ago (historically) publishing in French, German, Russian, etc. were more comm …

This isn't a direct answer to your question (I don't have a good book recommendation because that's not my field), but if there is a higher level course on differential equations or dynamics of some s …

There are lots of good answers here, so I'm not going to add any additional book recommendations. I just want to warn you of one misconception I had when I was in your position. It is best illustrat …

In some sense this is (part of) the theory of linear programming. If you want a reference for that, check out Bertsimas and Tsitsiklis' Introduction to Linear Optimization.