Mathematics Weekly Newsletter - Friday, February 27, 2015

Top new questions this week: Can I recover a group by its homomorphisms? There is finitely generated group $G$ which I don't know. For every finite group $H$ I can think of, I know the number of homomorphisms $G \to H$ up to conjugation. (By this I mean that two ... (group-theory) (finite-groups) (finitely-generated) (group-homomorphism)   asked by Turion 29 votes answered by anomaly 27 votes The series $\sum_{n=1}^\infty\frac1n$ diverges!! We all know that the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ diverges and grows very slowly!! I have seen many proofs of the result but recently found ... (calculus) (sequences-and-series) (proof-verification) (divergent-series)   asked by tone 25 votes answered by robjohn 17 votes Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$? My question is on the existence (or not) of a function $f:\mathbb{R}\to\mathbb{R}$ which satisfy the equation: $$f(f(x))=x^2-x+1 \text{ for every }x\in\mathbb{R}$$ Supposing that such a map do exist ... (algebra-precalculus) (functional-equations)   asked by aly 23 votes answered by Nilan 7 votes Elegant Proof of a simple inequality I'm looking for an elegant proof of the following identity: for $w_1,w_2,z_1,z_2\ge 0$, $w_1w_2+z_1z_2\le \max\{z_1,w_1\}\max\{z_2,w_2\}+\min\{z_1,w_1\}\min\{z_2,w_2\}$ The proof I currently have ... (combinatorics) (algebra-precalculus) (inequality)   asked by user218296 22 votes answered by Joffan 17 votes Inequality from Chapter 5 of the book *How to Think Like a Mathematician* This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained ... (algebra-precalculus) (inequality)   asked by Gwydyon 16 votes answered by Crostul 41 votes Exotic bijection from $\mathbb R$ to $\mathbb R$ Clearly there is no continuous bijections $f,g~:~\mathbb R \to \mathbb R$ such that $fg$ is a bijection from $\mathbb R$ to $\mathbb R$. If we omit the continuity assumption, is there such an example ... (real-analysis)   asked by anonynous 15 votes answered by echinodermata 14 votes Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles? The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ... (sequences-and-series) (visualization) (egyptian-fractions)   asked by alex.jordan 14 votes Greatest hits from previous weeks: 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 I use my powers for good? I hesitate to ask this question, but I read a lot of the career advice from mathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ... (soft-question) (career-development)   asked by Flounderer 726 votes answered by Gerry Myerson 239 votes Can you answer these? On $1/7$ in base $12$ Remember something from seventh grade: \begin{align} & 142857 \\ {}+ {}& 142857 \\ \\ & 285714 \\ {}+{} & 142857 \\ \\ & 428571 \\ {}+{} & 142857 \\ \\ & 571428 \\ ... (arithmetic)   asked by Michael Hardy 4 votes Weakening Goldbach hypothesis by allowing finitely many composites and 1 Is it an open question whether there is a finite set $N$ of positive integers such that for every positive even integer $n$ there are $n_1,n_2\in\mathbb P\cup N$ such that $n=n_1+n_2$? ($\mathbb P$ ... (elementary-number-theory) (open-problem) (goldbachs-conjecture)   asked by curious 4 votes Identifying the cotangent bundle of the flag variety Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ... (algebraic-geometry) (differential-geometry) (representation-theory) (lie-groups) (algebraic-groups)   asked by user219145 4 votes

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