iiinova — where your digital self keeps evolving

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 · 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 · 27/01/2026

just invented perpetual motion

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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 · 16/01/2026

hey everyone! I won the knowbell prize!

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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 · 12/01/2026

Another aspect on fractional Fourier transforms. Its generator is the same as the quantum harmonic oscillator, so they are the same as time evolution of the quantum system!

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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 · 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 · 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 · 05/10/2025

I posted this as a YouTube short, so I’ll refer to my description there

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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 · 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

here's another plot (more 'mu' points, fewer time points. And, gamma(1) process scaled to hit 1 at time 1 (time parameter mu to compare). Corresponds to t=0.5 in the surface plot. You can see its dominated by a few large jumps. gamma(40) process is shown in the 3rd plot, corresponds with t=0.024

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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 · 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

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

This later quote does sound negative, although in context it’s not so bad. You wouldn’t start students off on the most general class of cts functions, that are nowhere differentiable, without first mastering smooth functions, would lose sight of why cts functions were introduced in the first place

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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

It is Hölder continuous with exponent 1/2. Unlike Brownian motion, which is only locally Hölder continuous for exponents 1/2-ε

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

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

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Almost Sure @almostsure.bsky.social · 25/02/2025

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

Feigenbaum constant in the corner, other constants partying

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Almost Sure @almostsure.bsky.social · 22/02/2025

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

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Almost Sure @almostsure.bsky.social · 04/02/2025

Visualizing the curvature of the Earth

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Almost Sure @almostsure.bsky.social · 01/02/2025

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

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Almost Sure @almostsure.bsky.social · 22/01/2025

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!

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Almost Sure @almostsure.bsky.social · 30/12/2024

It's always difficult to remember which Bernoulli

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Almost Sure @almostsure.bsky.social · 23/12/2024

Christmas banner image activated!

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Almost Sure @almostsure.bsky.social · 23/12/2024

Quick "proof" of Taylor expansion using operator exponential

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Almost Sure @almostsure.bsky.social · 17/12/2024

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

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Almost Sure @almostsure.bsky.social · 07/12/2024

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

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Almost Sure @almostsure.bsky.social · 07/12/2024

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

Martingales and the Peacock Problem thumbnail

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Almost Sure @almostsure.bsky.social · 06/12/2024

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!

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Almost Sure @almostsure.bsky.social · 06/12/2024

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...

Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem

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Almost Sure @almostsure.bsky.social · 03/12/2024

You and me both, Maxwell, you and me both

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Almost Sure @almostsure.bsky.social · 03/12/2024

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!

Isometry Group of Gromov-Hausdorff space. Abstract referencing me, George Lowther, and complaining about lack of follow-up to online posts and claims

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Almost Sure @almostsure.bsky.social · 02/12/2024

derangement sighted in the wild!

Commentators by football pitch with oddly sized jackets

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Almost Sure @almostsure.bsky.social · 02/12/2024

Almost 2025 the only square year of my lifetime!

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