## 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)

### 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)

### 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)

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)
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 ...