MOM Alok’s thinking about conic sections again. The Classical Groups The classical groups (that I was thinking about) are: Real Orthogonal Group: \(O(n, \mathbb{R})\). Real-valued matrices \(A\) such that \(A^T A = I\) where \(A^T\) is the transpose of \(A\), and \(I\) is the identity matrix. Complex Orthogonal Group: \(O(n, \mathbb{C})\)....
But this is impossible by definition The title may seem like a contradiction. How can you differentiate something that’s not even continuous? The usual definition of the derivative of a function \(f\) at a point \(a\) is given by the limit: \[f'(a) := \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\] If \(f\) is differentiable at \(a\), then it is...
Nonstandard analysis gives a nice link between continuous and bounded functions, which are small and large-scale notions.1 Let \(X\) and \(Y\) be topological spaces. Identify them with their nonstandard extensions. A function \(f: X \rightarrow Y\) is continuous if \(x \approx x' \implies f(x) \approx f(x')\). Intuitively, infinitely close points...
Let \(H > \mathbb{Nat}\) be unlimited. Then the linear map \(T(x: \mathbb{R}^*): \mathbb{R}^* := Hx\) is discontinuous. Why, its discontinuity is equivalent to it being unbounded. This holds in general, but this example is the germ of generality. See my previous post for definitions of bounded and continuous. The map \(T\) is unbounded since it...
(written long time ago, publish or languish) These are some notes I made for Davide Radaelli for the first section of Schuller’s lectures on physics. Let’s turn Boolean algebra into something we know better: arithmetic. First we’ll set False to 0 and True to 1. To handle overflow, any arithmetic is mod 2. So even numbers are \(0\) and odd numbers...
The 2 Aspects There’s 2 Aspects to things in general. I will call them Mapping Out and Mapping In, in titlecase so you know they’re distinct concepts. warmup: 0 -> 1 Here, 0 is an initial object and 1 is a terminal object. 0 is Mapped Out of because it’s 0 -> and not -> 0. 1 is Mapped Into because it’s -> 1 and not 1 ->. The defining property of...
GPT-4 did this in 1 shot. The prompt is at https://chat.openai.com/share/b2af7f22-2953-4b25-af69-bd16077a6770 Code ran first time, always a rush when that happens. For a periodic minimal hypersurface, one would typically expect the surface to repeat itself in a regular pattern across the space. This periodicity could be along one or more axes,...
Been going through Mosteller’s 50 challenging problems in probability. There are 100 coins in a box. In each box there’s 1 fake. 100 boxes are tested. What is the chance of going undetected? What if it’s n instead of 100? Same as above but now each box has m fakes. Same sampling procedure. What is the chance of getting exactly r ...
I’ve been learning about the neural tangent kernel and some things confused me. In the NTK paper, the network layers have the form \(\frac{1}{\sqrt{n_A}}A\) where \(n_A\) is the number of neurons in \(A\). Why the square root? So I worked it out. Setup Input: \(x: \mathbb{^*R_{\lim}}\), where \(\mathbb{^*R_{\lim}}\) means a limited (hyper)real...
You replied to my dissection post and I decided then that I wished to be your friend. You didn’t leave your email, but asked how old I was in earth years. My email is alokbeniwal@gmail.com, please reach out if you see this. With love, Alok