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)

Part 2 of the collaboration with Terence Tao on the cosmic distance ladder is now out. It covers how we first learned the distances to planets, stars, and galaxies far, far away. youtu.be/hFMaT9oRbs4

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

My next YT video is ready. Just the thumbnail left to do. Will be a bit of a departure this time, no probability involved but is a bit of fun

Visualizing the curvature of the Earth

New YouTube Video uploaded Borwein integrals: expecting the unexpected youtu.be/px1JLasCN2c

So I was chilling at the beach, and happened to measure the size of the Earth! Island is visible, checking Google maps it’s Nevis Island. Looks like it slopes steeply into the sea, but pics show a lot of flatter land. Why? Lower bits are over horizon, due to Earth’s curvature!

calculating 1/e by holding dinner parties and, then, handing people their coats at random

Borwein integrals

Have a great 2025!

New YouTube video posted: Exponentials and exponentiation youtu.be/28LZ_CH1xhs?...

It's always difficult to remember which Bernoulli

Nearly there

Will AI really get better at math than human mathematicians? I’m sceptical but, maybe. However, will they be able to make better math YouTube videos than humans? I think not! btw, I recently started a YT channel Check it out (more videos coming)! YouTube.com/@almostsure

Christmas banner image activated!

Quick "proof" of Taylor expansion using operator exponential

Building a slide rule Commentary will be added, but just posting what I have so far, so leaving you to work out what is going on

mathematicians will tell you that dy/dx is *not* a ratio. Don’t believe them! It is the derivative of y with respect to x divided by 1

New YouTube video: Martingales and the Peacock Problem! youtu.be/sqrgqhl8C-A?...

That time of year again: bottling the blackberry liqueur! Last time I only made two bottles, hopefully this time I have enough

New YouTube video: Martingales and the Peacock Problem! youtu.be/sqrgqhl8C-A?...

Davinci Resolve Fusion for green screen. Node based, so very flexible. DeltaKeyer does green screen removal, with plenty of settings, and can apply separately on different areas. Can combine with other effects, like colour adjustments to the blackboard to make it more visible. Awesome, and free!

Recently (well, last year), a paper was published proving some anti-concentration bounds for Rademacher sums, which I had previously conjectured on mathoverflow! These extend a previously unsolved 1994 conjecture. pbs.twimg.com/media/F5yHsW...

It’s not often that I see my name in the abstract of a maths paper. So, I was stoked to see this Even if was mainly a complaint!

derangement sighted in the wild!