MathOverflow Weekly Newsletter
MathOverflow Weekly Newsletter

Top new questions this week:

On an example of an eventually oscillating function

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

ca.analysis-and-odes real-analysis  
asked by Twi 19 votes
answered by Noam D. Elkies 22 votes

Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ ...

asked by user64494 18 votes
answered by Robert Israel 13 votes

Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

There is a 50 point bounty on this question. Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define $$\pi_k(x)=\#\{n\leq x: ...

nt.number-theory cv.complex-variables  
asked by Kevin Smith 16 votes
answered by Lucia 5 votes

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?

asked by Nima 15 votes
answered by Włodzimierz Holsztyński 10 votes

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ... geometric-group-theory big-picture  
asked by user68579 15 votes
answered by Francesco Polizzi 23 votes

Free Loop-Space Recognition Principle

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...

at.algebraic-topology homotopy-theory operads loop-spaces  
asked by Nerses Aramian 14 votes
answered by Qiaochu Yuan 6 votes

Number of solutions to equations in finite groups

Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$. Is it always true that the number ... finite-groups character-theory  
asked by pbabcdefp 14 votes
answered by Benjamin Steinberg 15 votes

Greatest hits from previous weeks:

Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...

ho.history-overview soft-question big-list  
asked by romkyns 132 votes
answered by Beren Sanders 150 votes

Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...

ag.algebraic-geometry books big-list textbook-recommendation  
asked by sanokun 91 votes
answered by Javier Álvarez 109 votes

Can you answer these?

How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

rt.representation-theory lie-groups lie-algebras flag-varieties geometric-rep-theory  
asked by Zhaoting Wei 5 votes

An inequality concerning non-negative integer matrices with constant row and column sums

[I posted this question on math.stackexchange a few weeks back, but no luck there so far: ...

inequalities binomial-coefficients nonnegative-matrices  
asked by Navin K. 5 votes

Prime zeta zeros - reference

Is there an online repository for zeros of the prime zeta function? I looked at the Yahoo group Prime numbers and primality testing listed on the MathWorld notebook for the prime zeta function, but ...

asked by martin 3 votes
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