The visualization is interesting Superposition of 2 gaussians has oscillating cross term. Cancels if you integrate along X or P direction joint x-p direction, parallel to the troughs, cross term won't cancel Instead it cancels with the gaussians to give diffraction pattern for X-P

in quantum physics position X and momentum P can’t be simultaneously measured. But, linear combinations aX+bP 𝘢𝘳𝘦 observable, and their distributions can be used to back out the joint X,P probability density. The Wigner distribution but, it is 𝘸𝘦𝘪𝘳𝘥

• gamma function • Wigner quasiprobabilities • fractional Fourier transforms • Mistakes in probability (historical) • Ito’s lemma • Randomness in number theory • clickbait math, actually surprisingly interesting • probability doesn’t exist! which YT vid you want to see next?

New YouTube video posted “The Joy of Coupling” Happy Valentine’s Day! youtu.be/rpW2ZYtXn6o?...

YouTube video coming up tomorrow, on my channel, especially for Valentine’s Day! “The Joy of Coupling” watch this space…

just invented perpetual motion

You’re probably familiar with Fourier transforms. But, did you know that it can be viewed as a 90° rotation in the time-frequency domain and we can rotate by any angle via fractional Fourier transform. Explains why applying FT twice gives reflection (rotation by 180°).

Inaugural Board of Peace meeting is under way

Hey US-Americans, can you just give Trump Alaska? He won’t know the difference and might keep him quiet for a bit

“do you even lift bro?”

#QOTD “I never said even a fraction of the shit that’s ascribed to me” – 𝐴𝑙𝑏𝑒𝑟𝑡 𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛

Woke up to this sight

hey everyone! I won the knowbell prize!

Kids are really cute when they’re at that age where they haven’t yet worked out that giving you their snack means less for them

hey @johnbull.wtf you can’t just post this and delete it!

A second hand yacht? Really? How poor do they think we are?

so how hard are these Erdös problems anyway?

Land line analogue tape analogue clock real life

You’re probably familiar with Fourier transforms. But, did you know that it can be viewed as a 90° rotation in the time-frequency domain and we can rotate by any angle via fractional Fourier transform. Explains why applying FT twice gives reflection (rotation by 180°).

Ghost fox? So the other night a fox came up to me with a tennis ball in its mouth. Quite unusual so tried taking a picture, but fumbled trying to select camera app and it was already walking away by the time I was ready. But why can you see through it?

“jones dual” to avoid confusion

Anyone had success using AI to help generate Manim animations? I tried Claude, but due to autocorrect (Manim->mania…) it used JavaScript, and produced nice results. But when I asked to do in Manim, was full of bugs, did not do the correct thing when these were fixed, and eventually it gave up

cool 2026 fact: it’s a sum of just two primes!

(Relatively) new YouTube video posted! “Bell’s Theorem, the intuitive way” youtu.be/zSawtSy9Z5I

New YouTube video uploaded on connections between Riemann zeta and Brownian motion! What does Riemann Zeta have to do with Brownian Motion? youtu.be/YTQKbgxbtiw

It's a while since my last YT video upload. Nothing is happening, I am busy working on the next one. Just taking longer than expected (been very busy recent weekends). Will be on connections between Riemann zeta and Brownian motion. ETA a few days to a week.

Helter skelter

New YouTube video: Wild Expectations! This looks at weird properties of conditional expectations, such as two random variables being bigger than each other 'on average' youtu.be/4ZwRXVVepj8?...

Whatever you do, 𝙛𝙤𝙧 𝙩𝙝𝙚 𝙡𝙤𝙫𝙚 𝙤𝙛 𝙂𝙤𝙙, do not ask random variables to be Lebesgue measurable! If you were so stupid to do that, then 𝙮𝙤𝙪 𝙬𝙤𝙪𝙡𝙙 𝙣𝙤𝙩 𝙚𝙫𝙚𝙣 𝙗𝙚 𝙖𝙗𝙡𝙚 𝙩𝙤 𝙖𝙙𝙙 𝙧𝙖𝙣𝙙𝙤𝙢 𝙫𝙖𝙧𝙞𝙖𝙗𝙡𝙚𝙨 𝙩𝙤𝙜𝙚𝙩𝙝𝙚𝙧.

New video released on mixing quantum and classical probabilities. The Algebra of Mixed Quantum States youtu.be/K6h62Gr0nwg

New video released on the Gaussian correlation inequality. This unexpected proof shocked mathematicians! youtu.be/WJGR1oc6Gxo?...

I am not sure if the distribution is symmetric under X(mu,t)->1-X(1-mu,t). It is for individual times, as it matches Dirichlet distribution, but probably not for the entire paths wrt t. Which is disappointing. Maybe it can be modified?

Here's the method of simulation, and also shows that the joint distribution of X(μ,t) is uniquely determined if we impose independent ratios property wrt μ.

Simulating the martingales X(μ,t) which are beta distributed and martingale wrt t. X(μ,t) ~ Beta(μ(1-t)/t, (1-μ)(1-t)/t) The (1-t)/t scaling is so that on range 0<t<1 we cover entire set of Beta distributions. For each t, X(μ_{i+1},t)-X(μ_i,t) have Dirichlet distribution.

Probability fact: A sequence X_0,X_1,X_2,…,X_i,… of Gamma(a_i) rv’s has independent increments X_1-X_0,X_2-X_1,… if and only if it has independent ratios X_0/X_1,X_1/X_2,… in which case a_{i+1}>=a_i and, X_{i+1}-X_i~Gamma(a_{i+1}-a_i) X_i/X_{i+1}~Beta(a_i,a_{i+1})

Probability fact: If X,Y are independent Gamma(a), Gamma(b) random variables then X/(X+Y), X+Y are independent Beta(a,b), Gamma(a+b) rvs. Equivalently: if X,Y are independent Beta(a,b), Gamma(a+b) random variables then XY, (1-X)Y are independent Gamma(a), Gamma(b) rvs.

My extension of Hermite-Hadamard: If c = (pa+qb)/(p+q) for p,q > 0 then f(c)≤M(t)≤(pf(a)+qf(b))/(p+q) where M(t) is the average value of f under Beta(qt,pt) distribution scaled to interval [a,b]. M(t) is ctsly decreasing from M(0)=(pf(a)+qf(b))/(p+q) to M(∞) = f(c) pbs.twimg.com/media/GoSUpd...

Ok, I managed to prove this! So Beta(at,bt) is decreasing in the convex order and can find reverse martingale (continuous Itô diffusion) X(t)=E[X(s) | {X(u),u >=t}] (all s < t) with marginals X(t)~Beta(at,bt)

Question: fixed a, b > 0, are distributions Beta(at,bt) decreasing in convex order over t > 0? Equivalent to existence of a reverse-time martingale X_t ~ Beta(at,bt). Equivalently: -(d/dt)E[(x-X_t)_+] >=0 for all 0 < x < 1. Plots suggest so: parameterised as s=a+b,mean=a/s

New YouTube video posted "Brownian motion - A Beautiful Monster" youtu.be/IgMmsnzye1s?...

New YouTube video posted "Brownian motion - A Beautiful Monster" youtu.be/IgMmsnzye1s?...

“Reciprocal tariffs”

New Youtube short: Infinite complexity of Brownian motion An extreme zoom in to Brownian motion youtube.com/shorts/EvP9G...

Maybe you’ve heard that Henri Poincaré said that Weierstrass' work (on continuous functions without derivatives) is "an outrage against common sense". That’s what Wikipedia, and many other sources say. But, no! Looking up the original source, he's actually being positive about Weierstrass' work!

Fractional Brownian motion, varying the Hurst parameter H between 0 and 1. H=0.5 corresponds with standard Brownian motion, and the path has fractal dimension 2-H

Constructing Weierstrass' function. Continuous, nowhere differentiable, unbounded variation, no intervals of increase or decrease. Fractal dimension 3/2. All things it has in common with mathematical Brownian motion

Happy Feigenbaum approximation day! 14/3 = 4.66…

“They don’t know I’m universal for chaotic maps”

The American Mathematical Society has also started a page to coordinate support for professional mathematics, so far focusing on executive orders impacting the National Science Foundation: www.ams.org/government/g...

New YouTube video posted "Measuring the Earth...from a vacation photo!" (correct link this time: youtu.be/038AkmPvltA)