John    If you’ve followed this blog for a while, you may know that I do a lot of work with data privacy and that I have an interest in modal logic. Recently...

John    The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function Î (s) in the critical strip, i.e. the region of the...

John    The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of length n in time O(n log n). OpenAI recently...

John    For a real number x, the irrationality index Î (x) is a way of measuring how well x can be approximated by rational numbers. If x is rational, Î (x)...

John    I was skimming through David Well’s book The Penguin Dictionary of Curious and Interesting Geometry and was surprised by this entry: Lissajous...

John    Gödel’s incompleteness theorem illustrated the need to distinguish between what is true and what is provable. There are true statements that cannot...

John    An earlier post presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his...

John    The previous post discussed a superficial connection between modal logic and topology, that both use the terms regular and normal to indicate added...

John    You can’t say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you...

John    Euclidean geometry dates back at least to Euclid (circa 300 BC), and so you might think it’s been pretty well picked over by now. And yet people...

John    Despite the predictions that no one would ever put build data centers in space, Google is starting on Thursday. Google’s prototype satellite will be...

John    In the novel Carry On, Mr. Bowditch, a sailor asked Nathaniel Bowditch to teach him how to do navigational calculations, but the man only knows how...

John    A couple days ago a friend told me about the book Carry On, Mr. Bowditch, a fictional account of the life of Nathaniel Bowditch (1773-1838). I’ve...

John    Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines....

John    Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve....

John    There’s a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this...

John    I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some...

John    I’ve written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation...

John    Yesterday I wrote about the canonical example of how vector embeddings of words add: king ’ man woman queen This should be interpreted as saying that...

John    The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 à 5 à 8 à 13 39 à 40. I ran across...

John    The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly...

John    Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors. For vectors x and y, where Î is...

John    Suppose I give you an n-digit number and ask you what it represents. This seems impossible, and in theory it is impossible. But in practice it’s...

John    The Greek letter Î (beta) and the German letter à (eszett) look similar, especially in some fonts. Now suppose an OCR program sees some character...

John    The idea of using computers to assist with proofs is not new. The first major computer-assisted proof was published in 1976, the proof of the four...

John    A rising tide may lift all boats, but the AI tide lifts some boats much more than others. By all accounts, the best programmers have had the biggest...

John    Yesterday OpenAI announced a proof that settled a long-standing question about the Navier-Stokes equations from fluid dynamics. The announcement has...

John    There are rumors that a long-standing math problem, one of the Millennium Prize problems, has been solved. The problem concerns technical properties...

John    The Online Etymological Dictionary gives the following etymology for grok: grok (v.) understand empathically, 1961, an arbitrary formation by U.S....

John    The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric...