# All Questions

7k views

### Direct proof of irrationality?

There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form: Suppose $a/b=\sqrt{2}$ ...
578 views

### Applications of modular forms outside Number Theory? [closed]

Are there applications of modular forms to areas other than Number Theory (and Galois Theory) such as Combinatorics, Algebraic Topology, Algebraic Geometry, Theoretical Physics,...?
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### Minimum of Milnor number for the curve singularities of fixed multiplicity

An element $F\in \mathbb{C}[[x,y]]$ defines a germ of plane curve. We assume $F(0,0)=0$. The multiplicity $mult$ of the germ is defined to be a minimal number $i$ such that $F\in m^i$ where $m=(x,y)$ ...
917 views

### Unprovable statements S where the only way to prove S is to assume S

Motivation: Incompleteness (and various independence statements) is about unprovable statements. One natural way to make an unprovable statement provable is to assume it as a new axiom. But this feels ...
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### Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...
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801 views

### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...
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### notation for vector product in the space

The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$. However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...
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### Solutions of an nonlinear evolution problem

We consider the following continuous-time nonlinear evolution problem $$\begin{cases} \dot{y}(t)=Ay(t)+F(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}$$ where ...
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### Continuous extensions reals and to p-adic numbers

Assume $f\colon \mathbb Q\to \mathbb Q$ is a function which admits continuous extensions $f_0\colon\mathbb R\to \mathbb R$ and $f_p\colon \mathbb Q_p\to \mathbb Q_p$ for each prime $p$. Is ...
432 views

### RO(G) grading of Mackey functors

If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational ...
8k views

### Research statement in PhD applications--how much is too much?

I intend, in the somewhat near future, to engage preparing my graduate school applications for next year. I have worked hard to secure a solid application as far as coursework, grades, ...
973 views

### Is $x^p-x+1$ always irreducible in $F_p[x]$?

It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
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### Defining Global Choice in terms of strong limit cardinals over $ZF$

In his answer to user33038's mathoverflow question "What axioms are stronger than the Axiom of choice?", Prof. Hamkins writes: "What's more, the axiom of choice is equivalent over $ZF$ to the ...
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### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
32k views

### Math puzzles for dinner [closed]

You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ...
1k views

### A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
1k views

### Heuristic argument for the prime number theorem?

Here is a bad heuristic argument for the prime number theorem. Let n be a positive integer and assume that PNT holds up to n. Then n itself is prime if and only if for each prime p<n the event p|n ...
1k views

### Topple height of randomly stacked bricks

What is the expected height of a stack of unit-length bricks, each one stacked on the previous with a uniformly random shift within $\pm \delta$? The stack topples if the center of gravity of the top ...
524 views

### Determining conjugacy class of a subgroup from intersection with conjugacy classes

Is a subgroup of a finite group uniquely determined, up to conjugation, by the subset of conjugacy classes of the larger group that it intersects?
4k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please). 2) There are proofs ...
422 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$. For $m$ large, let ...
3k views

### Has decidability got something to do with primes?

Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this. Motivation: ...
435 views

### What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum?

The following was posted to math.stackexchange to no avail: http://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e The question ...
Let $g \geq 2$, and consider the moduli space $\bar M_{g,n}$ of stable n-pointed curves of genus g. There is a natural forgetful map to $\bar M_g$, which forgets the markings and contracts any ...