How Sridhar Thinks

The internal logic of effective regular categories (aka, "exact categories", though this term is overloaded, or "Barr-exact categories") can be given in many different ways, but is perhaps most easily spelt out in terms of thinking of relations, rather than functions as such. Any…

A result sometimes observed as surprising, shocking, etc, is that $$\int_{0}^{\infty} \mathrm{sinc}(x) dx = \pi/2$$, and indeed $$\int_{0}^{\infty} \mathrm{sinc}(x) \mathrm{sinc}(x/3) dx = \pi/2$$ and $$\int_{0}^{\infty} \mathrm{sinc}(x) \mathrm{sinc}(x/3) \mathrm{sinc}(x/5) dx =…

Let $$G$$ be a finite subgroup of the multiplicative group of a field. We shall show that it is cyclic. In fact, we will show something far stronger: if $$G$$ is a group with at most $$n$$ solutions to $$x^n = 1$$ (i.e., at most $$n$$ elements of order dividing $$n$$) for each $$…

Let $$F$$ be an endofunctor and let $$A : FX \to X$$ be an algebra for that endofunctor. $$A$$ gives rise to another F-algebra $$FA : FFX \to FX$$, and this has a "tautological" F-algebra homomorphism back into $$A$$ with underlying map $$A$$ itself (the relevant commutative squa…

By the Chinese Remainder Theorem, the ring of integers modulo N is the product of the rings of integers modulo p^n, for each prime power p^n in the factorization of N. To study the integers modulo p^n, it is helpful to consider the p-adics. By Fermat's little theorem and Hensel's…

Let's consider the general problem of determining the volume $$v(K)$$ of the region where $$\sum f_i(x_i) < K$$; that is, of determining $$\int_{\sum f_i(x_i) < K} \prod dx_i$$. Let us define new variables $$y_i = f_i(x_i)$$; the problem is now expressed as $$\int_{\sum y_i…

Let $$\leq$$ be an arbitrary binary relation (not necessarily transitive or reflexive, despite the notation). We will impose one condition: $${y \mid y \leq x}$$ should be finite for every $$x$$ (this finiteness condition can maybe be relaxed in some form for what we're doing her…

Suppose given a difference relation $$F(x + 1) - F(x) = f(x)$$, where $$f$$ is known but $$F$$ is not. What should $$F$$ be? Of course, there are many choices of $$F$$; on each equivalence class of integer-separated inputs, we can choose one starting value arbitrarily at one inpu…

A) Say relation → is "confluent against" ↓ if, given a→b and a↓c, ∃d with b↓d and c→d (ie, given the top and left of a square, you can fill out the rest). Claim: If → is confluent against ↓, then →* is confluent against ↓*, where * denotes reflexive transitive closure Proof: Give…

Under what circumstances can a regular n-gon be made such that all its points are lattice points, for some lattice? Here, I do not mean to restrict only to square lattices, but to any discrete (i.e., points cannot be found arbitrarily close to each other) subset of the plane clos…

Suppose you have a set of men $$M$$ and women $$W$$, and every man comes with a well-order on the women (where we take the smallest woman in a set to be the man's favorite among that set; a well-order amounts to the same thing as a way of assigning to each inhabited set a favorit…

Here's a simple fact from order theory: Suppose $$X$$ is a least element of partial order $$C$$. Then $$X$$ is the meet of all of $$C$$. In fact, for any order-preserving function $$F$$, we have that $$F(X)$$ is the meet of all of $$F(C)$$. In this post, we will strengthen this f…

Laypeople seem to spend an inordinate amount of mental energy on something called "Order of Operations" that they imagine to be deeply important and fundamental in mathematics. It's not their fault. This is how their teachers present it to them. This is what they are trained to t…

When asked what the difference between sequent calculus and natural deduction logical systems is, everyone (e.g., Wikipedia, but also everyone you meet in the world too) says a bunch of stuff that makes no sense. For example, as to whether sequents are involved, or whether sequen…

How I think about the so-called "Fundamental Theorems of Calculus" is a little different from how others think about them. I don't even think of these as having to do with derivatives and integrals, as such. Part of them is an idea from elementary school: subtraction and addition…

In this post, I will accumulate (and perhaps relate) several different proofs of the sine product formula $$\sin(\pi x) = \pi x \prod_{n \geq 1} \left(1 + \frac{x}{n}\right) \left(1 + \frac{x}{-n}\right)$$, or the essentially equivalent (by logarithmic differentiation) cotangent …

An often observed fact in type theory/functional programming/etc is that that, given a type constructor T(X) [a type T parametrized by an input type X], the corresponding type with one "hole" acts like the derivative of T(X). (See "The Derivative of a Regular Type is its Type of …

I have a job now where I do stuff with supply chains and inventory control/optimization. I'd never thought about or been aware of any of this stuff before, but it turns out there's a surprisingly large amount of interesting math in "supply chain theory". So, inbetween my explorat…

The Basel problem can be solved by simple integration! Recall that the Basel problem is to determine the value of $$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots$$. Consider $$f(x) = \ldots + e^{-3x} + e^{-2x} + e^{-1x} + e^{0x} + e^{1x} + e^{2x} + e^{3x} + \ldots$$. By …

Suppose you wanted to extend the factorial function to arbitrary arguments. How might you do it? Well, of course, there are a million ways to do it. (Where "a million" = "infinitely many"). You could say the factorial function is the normal thing at natural numbers, and $$\sqrt{7…

There are many “different” proofs of the irrationality of $$\pi$$ which are all based on the same underlying idea. The proof I describe in this post is also based on the same underlying idea as all the other ones you'll find in the wild, but in a different presentation, one I per…

[There's a better version of this post coming when I copy over https://howsridharthinks.wordpress.com/2019/11/19/difference-equations-infinite-sums-generalized-factorial-zeta-functions-etc/, but it's not fully ready yet. This is an archive of an old Quora answer, in the meanwhile…

Consider the question of how many primes are $$\leq n$$; let us call this quantity $$\pi(n)$$, as is traditional. Understanding $$\pi(n)$$ is not just a random curiosity, but in fact ubiquitously useful for answering other questions in number theory, as the primes entirely determ…

In a past life, I worked for 3blue1brown, and I discovered and made a video for them on a simple new proof of the Wallis product and the sine product more generally. Alas, I no longer work for 3blue1brown. But I had a post on a number of supplements to that video, which I will ke…

The $$n$$-dimensional unit sphere (in the indexing in which the Earth is a 3-dimensional sphere) has an inner volume which is its surface area divided by $$n$$ (by considering each tiny patch of its surface as the base of a figure tapering down to its center), and at the same tim…