My opinions are not my own. They're beamed to me by aliens.
Nov 6, 2022 • 21 tweets • 4 min read
I want to follow up what I said about Feynman's sum over all paths approach to optics (and QM) by giving some actual details. I'm going to try to use no more that high school geometry and some basic complex numbers...
I'm going to assume you know that exp(iθ) = cos(θ)+isin(θ) (and so is essentially the position of a point on the unit circle that has started at 1 and is travelled anti-clockwise around the origin through angle θ)...
Nov 4, 2022 • 6 tweets • 2 min read
The most Lovecraftian thing I know in reality is Feynman's sum over paths formulation of quantum mechanics. Seriously...
As kids we're taught light travels in straight lines. What Feynman realised is that light takes every path. *Every* path. That thing you're looking at, it's the sum of contributions from every possible path any light could have taken to get there...
Aug 19, 2022 • 5 tweets • 2 min read
I don't understand how the concept of expectation has taken such an iron grip on so many people's minds. The button on the right is not equivalent to $25,000,000.
Expectation is a cool concept because often we have concentration theorems. Like the central limit theorem says that when we have many repeated plays, it's quite likely we can get away with replacing a gamble with its expectation. But there are no such theorems about unique plays
Dec 12, 2021 • 14 tweets • 4 min read
Some musings on life, the universe and everything. But first a mathematical preliminary. We often use the exponential distribution to model the time before a “memoryless” random event happens, like nuclear decays...
Imagine a universe with lots of planets and the physics and biology in this universe works like this: a certain (large) N random events have to happen in a row before intelligent life that can contemplate its existence evolves...
Oct 23, 2021 • 6 tweets • 1 min read
Sometimes it's hard to pick out an element of a set even though there is an abundance of elements. Here's one of my favourites...
...every real number can be written as a continued fraction which means you can write it as a+1/(b+1/(c+1/(d+1/(e+1/...)))). For example for the golden ratio we use a=b=c=d=e=...=1 and for sqrt(2) we use a=1 and b=c=d=e=...=2...
Aug 8, 2021 • 11 tweets • 2 min read
My view of tensor products is this: if you know you're going to compute a bilinear product of a and b, but you don't know which one (maybe a dot, or cross, or something else random)...
...then a⊗b is a vector that contains precisely the information you need to do this with a linear map - no more, no less.
Apr 11, 2020 • 12 tweets • 4 min read
Suppose we have 3 job types that run in the same unit time: A, B and C.
A requires just the CPU, C requires just the GPU, but B requires both.
We set up a job queue with a scheduler that works down the queue to find and run the first job available to run as soon as it's ready...
Then in any queue with a C just before an A we can move the A before the C while remaining completely fair (i.e. without changing the order in which jobs get run) because an A never holds up a C or vice versa...
Feb 15, 2020 • 7 tweets • 2 min read
There's a numerical method "design pattern" I've seen many times but I've never seen pointed out explicitly. So here goes. The idea is that you have a forward operation that you want to invert. At first it looks really hard but then it turns out the inverse operation is...
...almost identical to the forward operation apart from (1) you have to pick away at the edges first and (2) you need to keep feeding your output back in as input.
A good example is the infinite impulse response (IIR) filter. It looks just like a FIR filter, ie. ...
Aug 18, 2019 • 33 tweets • 7 min read
Couple of years ago I did some GPS related work taking data from the analogue-to-digital converters in the radio receivers to the binary data stream. I think I'm going to (very slowly) tweet a series of tweets on the large amount of pretty mathematics involved at each stage.
Used box like this ettus.com/product-catego…
Measures varying electric (I think) field E(t) in antenna. Challenge: we want 1.5GHz signal. Nyquist sampling theorem says we need 3GHz samples. How to do that on a PC? en.wikipedia.org/wiki/Nyquist%E…
Mar 17, 2019 • 5 tweets • 2 min read
If all your descendants are made happy by all of their descendants being happy, then all your descendants are happy.
(See RTed thread for partial explanation and connection to Löb & Gödel's theorems)
Small print: descendant = any descendant alive or dead from past or present
which you may recognise as a form of Löb's theorem
Jul 26, 2018 • 4 tweets • 2 min read
@F_Vaggi That code is pretty much how you might write a recursive function for finite case using standard row ops (en.wikipedia.org/wiki/Gaussian_…). Suppose you already did rows 2..n, use standard operations to eliminate top row to the right of leading 1.
@F_Vaggi That suggests method where you recurse and do rows 2..n first and then do top row. The weird thing is that it works for infinite matrices. The code works when you do rows 2..∞ first.
Apr 21, 2018 • 5 tweets • 1 min read
By treating matrices as forming a "latent space" for the configuration of their eigenvalues you can produce curiously purposeful looking interpolations between sets of points
BTW I decide the start and end configuration. But I only specify the *set* of locations. The Matrix decides for itself the actual permutation of the points.
