Real Numbers Base…Factorials! And A By-product

PROPOSITION 1:  For a real number  x  there exists a sequence $ x_1, x_2, x_3,…$ of integers such that

$ \hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots,  \hspace{2cm} (*) $

where $x_1$ can be any integer, but for $ n \geq 2$, $x_n \in \{ 0,1,…,n-1 \}.$ Furthermore, if we require that the partial sums be strictly smaller than  x, then such a representation is unique.

Remark: One cannot help recalling decimal or binary expansion of numbers. Notice that $\frac{n}{n!}=\frac{1}{(n-1)!}$ (drops back to previous digit), so the bound on $x_n$ is logical. Continue reading “Real Numbers Base…Factorials! And A By-product” »

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Daily Quizzes: the Good, the Bad, and the Ugly—Part 2

You may recall that quite some time ago, I tried to convince you that giving your students a one- or two-question quiz every single day had a myriad of good aspects. You can check out why I loved this method in Part 1. As a quick refresher, I taught Calculus I four days a week the semester that I employed this method. Now, we’re going to discuss the bad (easily fixable) and ugly (not so easily fixable) issues which I ran into that semester. To keep this post from being a total downer, we are also going to talk about a new experiment I tried the next semester that I taught.  Continue reading “Daily Quizzes: the Good, the Bad, and the Ugly—Part 2” »

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Shedding light on AI’s black boxes

A recent special issue in Science highlights the increasingly important role that artificial intelligence (AI) plays in science and society. Providing a small but compelling sample of the types of challenges AI is equipped to tackle—from aiding chemical synthesis efforts to detecting strong gravitational lenses—the issue captures the palpable excitement about AI’s potential in a world saturated with data.

But one article in particular, “The AI detectives,” captured my attention. Rather than highlighting a specific application of AI, as the other articles do, this piece draws attention to the lack of transparency in certain machine learning algorithms, particularly neural networks. The inner workings of such algorithms remain almost entirely opaque, and they are accordingly termed “black boxes”: though they may generate accurate results, it’s still unclear how and why they make the decisions they do.

Machine Learning by XKCD is licensed under CC BY 2.5.

Researchers have recently turned their attention to this problem, seeking to understand the way these algorithms operate. “The AI detectives” introduces us to these researchers, and to their approaches to unlocking AI’s black boxes. Continue reading “Shedding light on AI’s black boxes” »

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Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds)

In the last post, I attempted to give an overview of the state of affairs in four-manifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) four-manifolds and their intersection forms afforded by Freedman’s theorem, and briefly touched on the relevance of this relationship to the difference between continuous and smooth topology in dimension four. Today, we will elaborate on this a bit by describing what happens to Freedman’s theorem in the smooth category and will try to give a vague idea (to be expanded upon next time) of why gauge theory might have something to say about smooth manifolds.

Continue reading “Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds)” »

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AMS Notices Spotlight September 2017

Hello and welcome to the September AMS Notices Spotlight. Since the last spotlight, many of you have started a new school year and if you haven’t started yet you are getting ready to start very soon. With that in mind, take a moment before the busyness of the semester sets in and peruse the September AMS Notices. There are many great articles in this month’s notices, including a sampler from the three sectional meetings that are going to occur this fall and several articles about bikes and math. We mentioned in our last spotlight that every issue of the AMS Notices includes a dedicated graduate student section. In the graduate student section, there is usually an interview of someone notable, and at least one article written with graduate students in mind. This month we are highlighting one of these articles. Continue reading “AMS Notices Spotlight September 2017” »

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