The argument describes an algorithm that can be translated into code.
1/(1-x)^(2) at 0 is 1
(1/(1-x)^(2) - 1)/x = (1 - 1 + 2x - x^(2))/x = 2 - x at 0 is 2
(1/(1-x)^(2) - 1 - 2x)/x^(2) = ((1 - 1 + 2x - x^(2) - 2x + 4x^(2) - 2x(3))/x(2) = 3 - 2x at 0 is 3
and so on
Let f(x) = 1/((x-1)^(2)). Given an integer n, compute the nth derivative of f as f^((n))(x) = (-1)(n)(n+1)!/((x-1)(n+2)), which lets us write f as the Taylor series about x=0 whose nth coefficient is f^((n))(0)/n! = (-1)^(-2)(n+1)!/n! = n+1. We now compute the nth coefficient with a simple recursion. To show this process works, we make an inductive argument: the 0th coefficient is f(0) = 1, and the nth coefficient is (f(x) - (1 + 2x + 3x^(2) + … + nx(n-1)))/x(n) evaluated at x=0. Note that each coefficient appearing in the previous expression is an integer between 0 and n, so by inductive hypothesis we can represent it by incrementing 0 repeatedly. Unfortunately, the expression we’ve written isn’t well-defined at x=0 since we can’t divide by 0, but as we’d expect, the limit as x->0 is defined and equal to n+1 (exercise: prove this). To compute the limit, we can evaluate at a sufficiently small value of x and argue by monotonicity or squeezing that n+1 is the nearest integer. (exercise: determine an upper bound for |x| that makes this argument work and fill in the details). Finally, evaluate our expression at the appropriate value of x for each k from 1 to n, using each result to compute the next, until we are able to write each coefficient. Evaluate one more time and conclude by rounding to the value of n+1. This increments n.
I don’t think you need permission to send someone an email directly addressed to and written for them. I don’t have context for the claims about Kagi being disputed, but I’d be frustrated if someone posted a misinformed rant about my work and then refused to talk to me about it. I might even write an email. Doesn’t sound crazy. If there’s more to the “harassment” that I don’t know about, obviously I’m not in favor.
No. sqrt(2) is an irrational number characterized as the positive solution to x^2 - 2 = 0. It’s described by a very small amount of data. Even its decimal expansion can be determined up to any precision by a simple algorithm.
But something has to be written on the birth certificate and social security card, and that’s what everything else will expect you to use. I think just due to technical limitations (e.g. of the printer/template for those things) it wouldn’t be allowed, but I dunno about legally
Java is a fine choice. Much prefer it over pseudocode.
I have read programs a lot shorter than 500 lines which I don’t have the expertise to write.
These things are specifically not defined by the protocol. They could be. They’re not, by design.
It doesn’t, it just delegates the responsibility to something else, namely xdg-desktop-portal and/or your compositor. The main issue with global hotkeys is that applications can’t usually set them, e.g. Discord push-to-talk, rather the compositor has to set them and the application needs to communicate with the compositor. This is fundamentally different from how it worked with X11 so naturally adoption is slow.
Okay, but this makes more sense as an instance method rather than a static one
Instance properties are PascalCase.
Yeah, properties (like a field but with a getter and/or setter method, may or may not be backed by a field) are PascalCase
That’s an instance property
Yes, with Iosevka font
Stokes’ theorem. Almost the same thing as the high school one. It generalizes the fundamental theorem of calculus to arbitrary smooth manifolds. In the case that M is the interval [a, x] and ω is the differential 1-form f(t)dt on M, one has dω = f’(t)dt and ∂M is the oriented tuple {+x, -a}. Integrating f(t)dt over a finite set of oriented points is the same as evaluating at each point and summing, with negatively-oriented points getting a negative sign. Then Stokes’ theorem as written says that f(x) - f(a) = integral from a to x of f’(t) dt.
Hom functors exist for locally small categories, which is just to say that the hom classes are sets. The distinction can be ignored often because local smallness is a trivial consequence of how the category is defined, but it’s not generally true
Docker is lighter and easier to manage than a VM. I run a collection of services as docker compose services inside a NixOS host VM. It’s easy to start, stop, monitor, update etc. even from a different computer (via ssh or docker contexts). It’s great.
deleted by creator
Not the same issue