 ## Top new questions this week:

### Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...

algorithms integration computer-algebra differential-algebra asked by Timothy Chow 32 votes answered by Sam Blake 16 votes

### Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$

Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it in two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$ Show that ...

co.combinatorics graph-theory ra.rings-and-algebras asked by Daniel Li 22 votes answered by Tony Huynh 22 votes

### Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory ?)

The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...

at.algebraic-topology gt.geometric-topology soft-question stable-homotopy asked by buck 21 votes

### Is being simply connected very rare?

Essentially, my question is how strong a restriction it is to be simply connected. Here is a way of making this precise: Let's say we want to count simplicial complexes (of dimension 2, though that ...

co.combinatorics at.algebraic-topology gt.geometric-topology computational-topology asked by Karim Adiprasito 19 votes answered by Tim Campion 7 votes

### Are the Morse inequalities sharp for 5-manifolds

Given a compact smooth manifold $M$ denote by $b_i(M)$ the $i$-th Betti number and denote by $q_i(M)$ the minimal number of generators for $H_i(M)$. Let $f$ be a Morse function on $M$. The Morse ...

gt.geometric-topology asked by Stefan Friedl 18 votes answered by Danny Ruberman 8 votes

### Is the Alexander horned sphere a cofibration?

The Alexander horned sphere is a closed embedding of $S^2$ into $S^3$ which is not flat because otherwise the Schoenflies Theorem would be true for it. However, not being flat is not the same as not ...

at.algebraic-topology gt.geometric-topology asked by daniel 15 votes answered by Tyrone 17 votes

### Pontrjagin-Thom model for units of the sphere spectrum?

Is there a framed bordism model for the units of the sphere spectrum, $gl_1(S)$? At the level of individual homotopy groups, the Pontrjagin-Thom construction identifies the group of bordism classes of ...

at.algebraic-topology asked by pupshaw 14 votes

## Greatest hits from previous weeks:

### Proofs where higher dimension or cardinality actually enabled much simpler proof?

I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...

ag.algebraic-geometry dg.differential-geometry graph-theory gt.geometric-topology big-list asked by Claus Dollinger 67 votes answered by Gabe K 55 votes

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

nt.number-theory real-analysis soft-question big-list asked by I. J. Kennedy 199 votes answered by Gjergji Zaimi 242 votes

### The enigmatic complexity of number theory

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...

nt.number-theory mathematical-philosophy metamathematics asked by David G. Stork 77 votes answered by Scott Aaronson 105 votes

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...

reference-request big-list examples intuition slick-proof asked by Mariano Suárez-Álvarez 350 votes answered by Mariano Suárez-Álvarez 506 votes

### Coin pusher game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins). Essentially, one has a distribution of ...

pr.probability reference-request st.statistics stochastic-processes asked by Alex R. 11 votes answered by Diamond Jim 5 votes

### John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends. Maybe this is an appropriate time to ask a ...

riemannian-geometry big-list game-theory asked by Paul Siegel 221 votes answered by Deane Yang 61 votes

### Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...

soft-question big-list gm.general-mathematics asked by Ilya Nikokoshev 193 votes answered by Daniel Moskovich 202 votes

## Can you answer these questions?

### Fourier Transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...

reference-request fourier-analysis fourier-transform symmetry critical-point-theory asked by John Clever 1 vote

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\... ag.algebraic-geometry vector-bundles moduli-spaces asked by Aoki 1 vote ### A bilinear estimates involving critical Sobolev norms Given$q>1$, consider the critical Sobolev space$W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in$L^{\infty}(\mathbb{R}^n)\$. I'm wondering if we can recover some critical estimate by considering ...

fa.functional-analysis real-analysis sobolev-spaces asked by Capublanca 1 vote
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