A few Numberphile videos + Final version of Borwein integrals

Congrats on the new videos! And darn, I didn't get my feedback on your sinc() video to you on time. But most of my feedback was on the convolutions part anyway, so hopefully it's still helpful for your next explainer: * Overall: This got me far more excited about convolutions! I've heard of convolutions before, but only the discrete-case, and for image processing / machine learning. The math facts that "multiplying = convoluting after a Fourier transform", and that "a Fourier transform is its own inverse", are new & intensely surprising to me! * My biggest structure-level feedback is that the explanation for convolutions should *not* have started with that really weird g(x) function; curvy, has negative values, asymmetric even. I was surprised that *only after* that explanation, you then said "think of a moving average" and showed the simple rect() case. It seems like it's more logical, and pedagogically helpful, to anchor it on the simple rect() case first, then move upwards in complexity: 1) Simple rect() – for moving (unweighted) averages 2) [Introducing: continuous fn] A Gaussian curve – for smoothing out functions. Two useful applications: in image processing, you can use this to blur an image. And just last week, I used a Gaussian convolution to turn a noisy, gap-filled histogram into a smooth function. (see also: Kernel Density Estimation. Here's a short, cool interactive explanation: https://mathisonian.github.io/kde/ ) 3) [Introducing: negative values] Some function that goes 0, 0, 0, -1, 2, -1, 0, 0, 0. This lets you do *edge detection*, especially helpful for machine vision. 4) [Introducing: asymmetry] Any arbitrary function, the full general case. I personally don't know of any (simple-to-explain) useful applications for an asymmetric convolution. But I guess a 0, 0, 0, -1, 1, 0, 0, 0 function can do a... crappy first derivative? And in image processing there is the cheesy "embossing" effect: https://muthu.co/basics-of-image-convolution/ * On that note, if it's not too much extra work, I think showing convolutions for image processing could be 1) a cool visual, 2) a good warm-up to continuous functions, since pixels are discrete, and 3) a real-world application. Actually, a cooler application is Convolutional Neural Networks (CNNs) in machine vision, which were what made AI image recognition *finally* practical. I think an even cooler fun fact is that CNNs were directly inspired by how the visual cortex in our brains work. So just like how our ears' spiral-cochlea *physically* do an inverse Fourier Transform, our visual cortices *physically* do convolution! (well, sorta) I just think that's neat. * Unless I blinked & missed it, the "why (t-x) and not (x-t)" part never seemed to be explained? Not a big deal, but still seems weird. I hope my feedback's still helpful and not totally outdated now!

Nicky Case