## 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

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 $... mp.mathematical-physics  asked by user64494 18 votes  answered by Robert Israel 13 votes ### Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts? There is a 50 point bounty on this question. Specifically, I find it appealing to count only squarefree numbers having$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? gn.general-topology  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 ... gr.group-theory 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 ... gr.group-theory 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

### 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