Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

A way to calculate e?

Define three sequences: The first sequence is $$n^n: 1,\ 4,\ 27,\ 256,\ 3125,\ 46656, \ldots$$ The second sequence is that of the ratios between adjacent members of the first series, or ...

(sequences-and-series)  
asked by Thomas Pogge 26 votes
answered by Ron Gordon 19 votes

Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: ...

(quaternions)  
asked by The Turtle 21 votes
answered by Danikar 52 votes

Counting matchings, the modern way

A hundred years ago, if you had $k$ men and $k$ women and wanted to marry them all off in pairs, it was easy to see that there are exactly $k!$ ways to do that. Today, however, societal standards ...

(combinatorics)  
asked by Henning Makholm 21 votes
answered by joriki 19 votes

We all use mathematical induction to prove results, but is there a proof of mathematical induction itself?

I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like $$1+2+3+\cdots+n = ...

(proof-strategy) (induction)  
asked by bodacydo 19 votes
answered by Peter Smith 29 votes

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...

(abstract-algebra) (vector-spaces) (definition) (axioms)  
asked by David Zhang 18 votes
answered by user7530 13 votes

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...

(inequality) (exponential-function) (exponentiation) (pi)  
asked by mathlove 16 votes

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...

(algebra-precalculus) (exponentiation) (quadratics)  
asked by Veo 15 votes
answered by John 25 votes

Greatest hits from previous weeks:

Definite Integral of square root of polynomial

I need to learn how to find the definite integral of the square root of a polynomial such as: $$\sqrt{36x + 1}$$ or $$\sqrt{2x^2 + 3x + 7} $$ EDIT: It's not guaranteed to be of the same form. ...

(calculus) (integration)  
asked by notbad.jpeg 5 votes
answered by user17762 5 votes

What is the intuitive relationship between SVD and PCA

Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important ...

(linear-algebra) (statistics)  
asked by wickedchicken 151 votes
answered by Guess who it is. 141 votes

Can you answer these?

Maximal possible dimension of an abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$.

Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...

(lie-algebras)  
asked by user265435 4 votes

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...

(linear-algebra) (abstract-algebra) (vector-spaces) (representation-theory) (multilinear-algebra)  
asked by Thomas Banks 5 votes

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...

(calculus) (real-analysis) (integration) (definite-integrals) (special-functions)  
asked by Chris's sis the artist 4 votes
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