Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

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)  
asked by Asvin 27 votes
answered by user28111 43 votes

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)  
asked by Anixx 15 votes
answered by Furihr 2 votes

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)  
asked by Rioghasarig 13 votes
answered by pjs36 24 votes

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)  
asked by Elad Avron 13 votes
answered by kobe 40 votes

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)  
asked by Abba 13 votes
answered by pjs36 16 votes

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)  
asked by rah4927 13 votes
answered by Oscar Cunningham 5 votes

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)  
asked by Martin Brandenburg 12 votes
answered by Maciej Piróg 3 votes

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)  
asked by Andrew Cheong 135 votes
answered by josinalvo 59 votes

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)  
asked by Christofian 186 votes
answered by Henning Makholm 193 votes

Can you answer these?

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)  
asked by goblin 5 votes

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)  
asked by user225369 6 votes

When does Sheafification commute with direct image?

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 ...

(algebraic-geometry) (category-theory) (sheaf-theory)  
asked by Dorebell 5 votes
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