Top new questions this week:

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...

(induction) (big-list) (philosophy)

Has anybody ever considered "full derivative"?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which ...

(finite-differences) (nonstandard-analysis) (infinitesimals)

Why is anti-symmetry a desirable quality in determinants?

I hear the determinant of matrix can be defined using 3 facts. 1. It is multilinear. 2. It is anti-symmetric. 3. It is scaled so the determinant of the identity is 1. But, I don't understand why ...

(linear-algebra)

How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?

So while doing all sorts of proving and disproving statements regarding irrational numbers, I ran into this one and it quite stumped me: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational. I tried ...

(number-theory) (irrational-numbers)

What skill do I lack to factor multivariate polynomials?

Ok so I can factor easily regular quadratic polynomials, i.e. $5x^2+7x+9$ (I'm not sure whether that's prime, just made it up), and I was working on solving $y^2+(x^2+2x−2)y+(x^3−x^2−2x)$ by ...

(algebra-precalculus) (polynomials)

Given $2015$ points, show that it is possible to separate them such that $1007$ of them lie inside the circle

Suppose there are $2015$ points on a plane, no three collinear. Show that you can draw a circle such that exactly $1007$ lie inside the circle. Here is my solution: First draw a line that ...

(combinatorics) (proof-verification)

Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...

(category-theory) (big-list) (applications)

Greatest hits from previous weeks:

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...

(probability) (random) (philosophy)

Multiple-choice question about the probability of a random answer to itself being correct

I found this math "problem" on the internet, and I'm wondering if it has an answer: Question: If you choose an answer to this question at random, what is the probability that you will be correct? ...

(probability)

Does the vector space of compactly-supported continuous functions $X \rightarrow \mathbb{R}$ satisfy an interesting universal property?

Let $S$ denote a set. Then the vector space $FS$ freely generated by $S$ can be identified with the set of all finitely-supported functions $S \rightarrow \mathbb{R}$. This gave me the following idea; ...

(general-topology) (vector-spaces) (category-theory) (topological-vector-spaces)

Reference for Cavalieri's principle

Does someone know of a reference where I can see Cavalieri's principle (basically the principle that generalized areas can be obtained by multiplying "base times height" -- for constant ...

(linear-algebra) (geometry) (reference-request)
Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...