Mathematics Weekly Newsletter - Friday, May 29, 2015

Top new questions this week: If I know the order of every element in a group, do I know the group? Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ... (abstract-algebra) (group-theory) (finite-groups) (examples-counterexamples)   asked by Walter 33 votes answered by Dietrich Burde 30 votes Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence? So, $$1,1,2,3,5,8,13,21...$$ Any connection to primes?...it appears not. However, in between the Fibonacci numbers are how much primes? Let's see: 1 and 1 ZERO 1 and 2 NADA 2 and 3 ZILCH 3 and 5 ZIP ... (prime-numbers) (fibonacci-numbers) (pattern-recognition)   asked by HyperLuminal 29 votes answered by nullUser 29 votes I think I see mysterious lines inside triangles—how to prove their existence? Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ... (geometry) (triangle)   asked by Silenttiffy 28 votes answered by achille hui 5 votes Computing irrational numbers I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy? (algebra-precalculus) (number-theory) (algorithms) (irrational-numbers)   asked by user242891 27 votes answered by Ethan Bolker 19 votes Tough question taken from a quite easy maths puzzle book I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ... (elementary-number-theory) (fractions)   asked by Silenttiffy 24 votes answered by Peter Woolfitt 28 votes Axiomatic definition of sin and cos? I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ... (trigonometry) (axioms)   asked by gavenkoa 21 votes answered by Steven Taschuk 18 votes Can two perfect squares average to a third perfect square? My question is does there exist a triple of integers, $a<b<c$ such that $b^2 = \frac{a^2+c^2}{2}$ I suspect that the answer to this is no but I have not been able to prove it yet. I realize ... (elementary-number-theory) (pythagorean-triples)   asked by Elliot 17 votes answered by Will Jagy 22 votes Greatest hits from previous weeks: Finding an equation for a circle given its center and a point through which it passes No idea how to do this, I used to have these conic shapes committed to memory but I forget them already. I am supposed to find an equation for the circle that has center $(-1, 4)$ and passes through ... (algebra-precalculus) (circle)   asked by user138246 5 votes answered by David Mitra 10 votes What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors? (linear-algebra) (soft-question) (eigenvalues-eigenvectors)   asked by Ryan 79 votes answered by Arturo Magidin 110 votes Can you answer these? Importance of compactness in Rudin problem. Okay the problem goes like this: Suppose X, Y, Z are metric spaces, and Y is compact. Let $f$ map X into Y, let g be a continuous one-to-one mapping of Y into Z, and put $h(x)=g(f(x))$ for $x\in ... (real-analysis)   asked by user160110 5 votes Diffeomorphism-invariant spaces of smooth functions Let's start with an interesting story. In his celebrated Partial Differential Relations (p. 146), the great Misha Gromov gives a nice exercise of which the following is a (strict) part. Exercise. ... (real-analysis) (invariant-theory)   asked by PseudoNeo 6 votes Aut(G) is abelian I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ... (group-theory) (finite-groups)   asked by Ivan 5 votes

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