Wayne Joubert    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...

John    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...

John    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...

John    Suppose one company would like to buy another company’s client list, but only if the lists don’t overlap too much. Neither company wants to hand over their list to the other before a sale takes place. What can they do A low tech solution would be for both parties to provide their client lists to a...

John    Newton’s interpolation formula looks awfully complicated until you introduce the right notation. With the right notation, it looks like a Taylor series. Not only is this notation simpler and more memorable, it also suggests extensions. The notation we need comes in two parts. First, we need the...

John    The previous post was an AI generated podcast that I friend made by crawling my web site. I decided to create an actual podcast for posting occasional audio files. I expect to post very sporadically. I’ve posted two audio files, and I have one more in mind to post some day. Maybe that’ll be the end...

John    A friend just sent me an audio file of a podcast about my consulting that he created by asking an AI to crawl my web site. The podcast is a remarkably natural sounding conversation between two synthetic hosts. The only clues that the audio is automatically generated are a couple of mispronounced...

John    A recent article reported on the successful factoring of a bit RSA key. The process took worth of rented computing power. What does that say about the security of RSA encryption The following Python function estimates the computation steps required to factor a number b bits long using the...

Wayne Joubert    We’re awaiting the release of OpenAI’s o model later this month. Its performance is impressive on very hard benchmarks like SWE bench Verified, Frontier Math and the ARC AGI benchmark discussed previously in this blog . And yet at the same time some behaviors of the frontier AI models are very...

John    One difficulty when working at a command line is navigating between directories, particularly between locations with long paths. There are several ways to mitigate this. One of the simplest is using cd to return to the previous directory. Another is to use pushd and popd. Still another is to set...

John    The security of RSA encryption depends on the difficulty of factoring the product of two large primes. If you can factor large numbers efficiently, you can break RSA. But if can break RSA, can you factor large numbers Sorta. It’s conceivable that there is a way to break RSA encryption without...

John    Steganography attempts to prevent messages from being read by unintended recipients by hiding the messages rather than or in addition to encrypting them. Steganography is used when you not only want to keep your communication private, you want to hide the fact that you’ve communicated at all. Fun...

John    A blog carnival is a way to discover new blogs. Writers on a given topic, such as math, take turns hosting the carnival, featuring recent posts from various writers. Blog carnivals were once much more common, but most have faded away. The Carnival of Mathematics, however, is one of the oldest...

John    The previous post showed that there are Abelian groups that have elements. Implicitly that means there are Abelian groups up to isomorphism, i.e. groups that are not in some sense the same even if they look different. Sometimes it is clear what we mean by the same and there’s no need...

John    Every finite Abelian group can be written as the direct sum of cyclic groups of prime power order. To find the number of Abelian groups of order we have to find the number of ways to partition the factors of into prime powers. Now à . We can partition The post Abelian...

John    Here’s a visualization of how the day of the week for New Year’s Day changes. The green diamonds represent leap years and the blue squares represent ordinary years. The day of the week for New Year’s Day advances one day after each ordinary year and two days after each leap year, hence the diagonal...

John    RSA public key cryptography begins by finding a couple large primes. You essentially do this by testing random numbers until you find primes, but not quite. Filippo Valsorda just posted a good article on this. Suppose you’re looking for a bit prime number. You generate random bit numbers...

John    The previous post looked into the common definition of Unix time as the number of seconds since January, GMT and why it’s not exactly true. It was true for a couple years before we started inserting leap seconds. Strictly speaking, Unix time is the number of non leap seconds since January...

John    The time it takes earth to orbit the sun is not a simple multiple of the time it takes earth to rotate on its axis. The ratio isn’t even constant. The time it takes earth to circle the sun wobbles a bit, and the rotation of the earth is slowing down slightly. The ratio is The post Unix Time and...

John    I looked at Hacker News to see which posts on this site were most popular. I didn’t look at my server logs, but generally the posts that get the most traffic are posts that someone submits to Hacker News. Older posts popular this year Two posts written earlier got a lot of traffic this year, ...

John    Stirling’s asymptotic series for the gamma function is Now suppose you’d like to find an asymptotic series for the function Î z . Since the series for Î has the form f z times an infinite sum, it would make sense to look for a series for Î of the form f z times an infinite sum. The hard ...

John    My nephew recently told me about being on a camping trip and seeing a long line of lights in the sky. The lights turned out to be Starlink satellites. It’s fairly common for people report seeing lines of these satellites. Why would the satellites be in a line Wouldn’t it be much more efficient to ...

John    I’ve been playing around with Grok today, logging into some of my X accounts and trying out the prompt Draw an image of me based on my posts. In most cases Grok returned a graphic, but sometimes it would respond with a text description. In the latter case asking for a photorealistic image...

Wayne Joubert    OpenAI has just released its full o model a new kind of model that is more capable of multi step reasoning than previous models. Anthropic, Google and others are no doubt working on similar products. At the same time, it’s hotly debated in many quarters whether AI models actually reason in a way...

John    A couple days ago I analyzed the observation that repeatedly pressing the cosine key on a calculator leads to a fixed point. After about iterations the number no longer changes. This post will analyze the same phenomenon a different way. Interval arithmetic Interval arithmetic is a way to get...

John    The previous post presented an approximation for’ x and said that it was related to a probability function. This post will make the connect explicit. Let X be a normally distributed random variable with mean Î and variance Ï . Then the CDF of X is So if Î and Ï The post Normal...

John    I recently ran across the fact that is a remarkably good approximation for’ x . Since the integral above defines the error function erf x , modulo a constant, this says we have a good approximation for the error function again provided’ x . The error function is closely related to The...

John    No matter what number you start with, if you press the cos key on a calculator repeatedly, the numbers eventually quit changing. This fact has been rediscovered by countless children playing with calculators. If you start with, which is likely the default when you turn on a calculator, you’ll hit...

John    The previous post explained why the Gregorian calendar is the way it is, and that it consists of a whole number of weeks. It follows that the Gregorian calendar repeats itself every years. For example, the calendar for will be exactly the same as the calendar for and . There are...

John    The time it takes for the earth to orbit the sun is not an integer multiple of the time it takes for the earth to rotate on its axis, nor is it a rational number with a small denominator. Why should it be Much of the complexity of our calendar can be explained by rational The post Gregorian...