Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)

Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including ...

(calculus) (real-analysis) (derivatives)  
asked by Matt24 24 votes
answered by Thomas Andrews 50 votes

Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers?

For instance, we can certainly represent π in this fashion. $$ \frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\! $$ $\ln(2)$ is also irrational. And even that can be represented as an ...

asked by shreedhar 22 votes
answered by Brevan Ellefsen 41 votes

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...

(probability) (recreational-mathematics) (puzzle) (average) (geometric-probability)  
asked by John Smith 18 votes
answered by martin 10 votes

Have there been (successful) attempts to use something other than spheres for homotopy groups?

Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties: For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian. $\pi_n$ defines ...

(algebraic-topology) (homotopy-theory)  
asked by Najib Idrissi 18 votes
answered by Qiaochu Yuan 12 votes

why $ \rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$

Let $n\ge 7$ be positive integers,show that $$f(n)=\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$$ Anyone know this problem background?or maybe have best proof or best result?

(number-theory) (inequality) (prime-numbers) (approximation) (least-common-multiple)  
asked by Kelantan Kota 16 votes
answered by Will Jagy 5 votes

Can we recover a space from its continuous functions?

Let $X$ be a topological space and let $\mathcal{F}(X,\Bbb R)$ be the set of continuous function from $X$ to $\Bbb R$. Can we recover the topology of $X$ by only the knowledge of ...

asked by user275913 14 votes
answered by David C. Ullrich 13 votes

Group with two generators of order 3 is finite

A group $G$ is generated by two elements $a$ and $b$ such that for any $g\in G:$ $g^3=e$. Show that $G$ is finite. I don't understand how this is so. I don't know if $G$ is abelian, so I can ...

(abstract-algebra) (finite-groups)  
asked by user20467 13 votes
answered by Hagen von Eitzen 16 votes

Greatest hits from previous weeks:

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...

(linear-algebra) (reference-request) (eigenvalues-eigenvectors)  
asked by Phonon 31 votes
answered by Arturo Magidin 49 votes

Real-world applications of prime numbers?

I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently. The problems are interesting per se, but I ...

(prime-numbers) (applications)  
asked by Sylverdrag 30 votes
answered by Gadi A 24 votes

Can you answer these?

Evaluating an intriguing expression.

Let $x_1,x_2...x_{2014} \in R, \not= 1$ such that $$x_1+x_2...+\ x_{2014}=1$$ and $$\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}...+\frac{x_{2014}}{1-x_{2014}}=1$$ ...

(algebra-precalculus) (systems-of-equations)  
asked by Kugelblitz 4 votes

Every vector bundle has a metric connection?

Let $(E,g)$ be a vector bundle with a metric over a manifold $M$. Does $(E,g)$ always admit a compatible (metric) connection? If so, are there examples where there exists only one such metric ...

(differential-geometry) (vector-bundles) (connections)  
asked by Asaf Shachar 5 votes

Can this approximate closed form of Apery's constant $\zeta(3)$ be improved?

I know that an approximate closed form is not really a solution. However, I would like to present a method that gives a closed form of $\zeta(3)$ that is accurate to the 5th decimal, hoping that it ...

asked by Majid Fekri 5 votes
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