Chebyshev and Russian transliteration

The purpose of this post is to expand on the following sentence [1]: Russian letters are created by entering [Ctrl-k followed by] a corresponding Latin letter followed by an equals sign -, or, in a few places, a percent sign %. The Russian alphabet has 33 letters, so there couldn’t be a Latin letter for […] The post Entering Russian characters in...

Let J(x) be the function plotted below. This is the Bessel function J1, but we drop the subscript because it’s the only Bessel function we’re interested in for this post. You can think of J as a sort of damped sine. We can create versions of J with different frequencies by multiplying the argument x […] The post An unexpected triangle first...

Many are voicing concern that the world is running out of data and that this will be a blocker to progress toward smarter AI models. One paper in fact projects timelines for when we will run out. AI researchers are looking for ways to adapt.  Nvidia has trained a specific model to generate synthetic data for […] The post Can AI Models Reason: Is...

Sometimes it’s useful to apply dimensional analysis where it doesn’t belong, to imagine things having physical dimension when they don’t. This post will look at artificially injecting dimensions into equations involving factorials and related functions. Factorials The factorial of n is defined as the product of n terms. If each of these terms had...

Yesterday I wrote about how the right notation could make Newton’s interpolation theorem much easier to remember, revealing it as an analog of Taylor series. This post will do something similar for the binomial theorem. Let’s start with the following identity. It’s not clear that this is true, or how one might generalize it. But […] The post...