discontinuous linear functions

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...

(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...