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Almost Sure @almostsure.bsky.social · 07/08/2026

New YouTube video uploaded! "What is a fractional Fourier transform?" youtu.be/DViyFiwRrwY?...

youtu.be

What is a fractional Fourier transform?

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 08/06/2026

New Youtube video released! An innocent looking meme that's been floating around the internet for a few years, looking like your standard clickbait slop, but actually hides some interesting math using elliptic curves youtu.be/tXTV86cZZ6s

youtu.be

The advanced math behind the meme

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 16/04/2026

On the new proof of Erdös 1196, here's my proof using the (more natural) downwards Markov chain rather than upwards. A bit incomplete since I am out of time for now, but all the main details are there. It is just tidying up really that remains. @teorth.bsky.social you agree this works?

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Almost Sure @almostsure.bsky.social · 31/03/2026

New YT video posted, on Wigner quasiprobability distributions! youtu.be/988P5tIhhak

youtu.be

Wigner's probabilities are...weird

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 24/02/2026

Wavefunction and Wigner phase-space distribution for a quantum harmonic oscillator. It just rotates at constant speed, as with classical mechanics But for non-quadratic potential, such as pendulum type dynamics, the distribution spreads out and generates interference patterns

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Almost Sure @almostsure.bsky.social · 19/02/2026

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

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Almost Sure @almostsure.bsky.social · 19/02/2026

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

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Almost Sure @almostsure.bsky.social · 16/02/2026

• 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?

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Almost Sure @almostsure.bsky.social · 14/02/2026

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

youtu.be

The Joy of Coupling

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 13/02/2026

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

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Almost Sure @almostsure.bsky.social · 27/01/2026

just invented perpetual motion

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Reposted by Almost Sure

Almost Sure @almostsure.bsky.social · 12/01/2026

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°).

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Almost Sure @almostsure.bsky.social · 22/01/2026

Inaugural Board of Peace meeting is under way

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Almost Sure @almostsure.bsky.social · 21/01/2026

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

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Almost Sure @almostsure.bsky.social · 18/01/2026

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

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Almost Sure @almostsure.bsky.social · 16/01/2026

hey everyone! I won the knowbell prize!

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Almost Sure @almostsure.bsky.social · 16/01/2026

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

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Almost Sure @almostsure.bsky.social · 14/01/2026

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

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Almost Sure @almostsure.bsky.social · 14/01/2026

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

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Almost Sure @almostsure.bsky.social · 14/01/2026

so how hard are these Erdös problems anyway?

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Almost Sure @almostsure.bsky.social · 13/01/2026

Land line analogue tape analogue clock real life

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Almost Sure @almostsure.bsky.social · 12/01/2026

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Almost Sure @almostsure.bsky.social · 12/01/2026

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?

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Almost Sure @almostsure.bsky.social · 12/01/2026

“jones dual” to avoid confusion

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Almost Sure @almostsure.bsky.social · 09/01/2026

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

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Almost Sure @almostsure.bsky.social · 02/01/2026

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

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Almost Sure @almostsure.bsky.social · 02/01/2026

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

youtu.be

Bell's theorem, the intuitive way

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 09/11/2025

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

youtu.be

What does Riemann Zeta have to do with Brownian Motion?

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 02/11/2025

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.

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Almost Sure @almostsure.bsky.social · 23/09/2025

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?...

youtu.be

Wild Expectations

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 18/09/2025

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.

Absolutely continuous rvs summing to singular rv

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Almost Sure @almostsure.bsky.social · 23/08/2025

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

youtu.be

The Algebra of Mixed Quantum States

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 03/07/2025

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

youtu.be

This unexpected proof shocked mathematicians

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 17/04/2025

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?

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Almost Sure @almostsure.bsky.social · 16/04/2025

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.

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Almost Sure @almostsure.bsky.social · 15/04/2025

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.

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Almost Sure @almostsure.bsky.social · 14/04/2025

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})

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Almost Sure @almostsure.bsky.social · 12/04/2025

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.

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Almost Sure @almostsure.bsky.social · 12/04/2025

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...

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Almost Sure @almostsure.bsky.social · 12/04/2025

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)

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Almost Sure @almostsure.bsky.social · 12/04/2025

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

Plot of derivatives showing positive, as required for convex order

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Reposted by Almost Sure

Almost Sure @almostsure.bsky.social · 06/04/2025

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

youtu.be

Brownian Motion - A Beautiful Monster

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 06/04/2025

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

youtu.be

Brownian Motion - A Beautiful Monster

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 29/03/2025

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

youtube.com

Infinite complexity of Brownian motion #maths #probability #stochastic

YouTube video by Almost Sure

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Almost Sure @almostsure.bsky.social · 21/03/2025

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!

Translation. “A hundred years ago, such a function *would have been* regarded as an affront…” “Weierstrass has given us a useful warning and taught us to appreciate…” etc

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Almost Sure @almostsure.bsky.social · 21/03/2025

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

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