Matrix Multiplication, the human way!

Having to do copious calculations by hand when preparing for an exam, I came to realize that there was an alternative way of interpreting a matrix multiplication. This new insight would allow me to instantly guess the following product without ever doing any numerical multiplication:
\[\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & -8 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 0 \\
0& 1 & 0 \\
1& 0 & 0
\end{bmatrix}
=
\begin{bmatrix}
3 & 2 & 0 \\
6 & 5 & 0 \\
0 & -8 & 0
\end{bmatrix}\]

Was there a way to have known that the first column of the product would be the third column of the first matrix?

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Optimal Control Theory to Settle Reinhardt’s Conjecture

The 2010’s are a Golden Age for packing problems. In 2014, Hales announced the long-awaited completion of a high-profile machine proof project called FlySpeck, which verified his proof of Kepler’s conjecture. Johannes Kepler, in 1600, conjectured that the densest way to pack spheres is in cannonball stacks. This was Hilbert’s 18th problem. The final proof was quite involved and its correctness needed massive computer checking. For example, verifying the nonlinear inequalities took 5000 hours on the Microsoft Azure cloud. Other optimal packing problems have been solved recently too—with simpler methods. Cohn–Kumar–Miller–Radchenko–Viazovka [3] and Viazovka [8] found optimal sphere packings in dimensions 8 and 24. Viazovka was awarded a 2017 Clay Research Prize for this work.

 

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What is a Manifold? (6/6)

In posts 1-3 we were able to reduce all of the geometry of a curve in 3-space to an interval [a,b]\subset \mathbb{R} along with two or three real-valued functions. We also discussed when two sets of such data give equivalent (overlapping) curves. This enabled us to patch together a collection of such sets of data into one unified spatial curve.

We then studied the specific example of re-defining the metric on the plane so that its geometry is precisely that of a 2-sphere. We saw that for measurements of angles, lengths, and areas, all we need is a dot-product on vectors. Given an open domain in the plane, once we have a dot-product, we will be able to make such measurements. Our goal in this post is to make the following definition of a manifold more tangible.

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Ordered Fields and When You Can’t Order Them

The real numbers \mathbb{R} have an ordering on them–given two numbers x and y, we can tell whether x = y, x > y, or y<x. So as math people, we like to generalize this to other sets–when can we say that a general set is ordered? In this post, we’re going to explain the explicit of idea of what it means for a field to be ordered, and then show that the complex numbers \mathbb{C} can’t be ordered–no matter what ordering you put on it. (If you don’t know what a field is, just think of the real numbers \mathbb{R} or the complex numbers \mathbb{C}.)

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On “Imposter Syndrome”

Here’s how it happens: You’re in graduate school and were one of the best people in your major from your school. Honestly, that’s how you got into graduate school in the first place. You go in the first few weeks, you meet your new peers, and you engage in mathematical discussion. It’s really fun, being with people who are just as excited about math as you are. But then, a horrible thing happens. Someone, in conversation, mentions something you don’t know. And not only that, but the way they talk about it suggests that anyone who knows anything about anything knows what they’re talking about. Or maybe, in an even worse turn of events, this person is a professor. What are you going to do?

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