# All Questions

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### Have there been attemps to manage the pool of worldwide mathematics students?

(Foreword: I am well aware that MO is not a blog, and not for argumentative questions. Therefore I have phrased my question in a rather specifically answerable form. If moderators still feel it is ...
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### prove that the sequence a_n = [n((2)^(1/2))] + [n((3)^(1/2))] contains infinitely many even and odd integers

im not sure how to proceed.i have done a few problems similar to this but i can't solve this one. thanks for any help.( by the way this is a problem from croatia team selection test )
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### Proof of the measure representation lemma, used by P. L. Lions in the proof of the concentration-compactness

Please can someone tel me where i can find the proof of this : Thank you
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### Caledonian college level2 [on hold]

assume we have a stick of one meter length. we put 999 ants on the stick, at arbitrary positions and arbitrarily facing either left or right. At a certain time, all ants start moving with the same ...
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### A tensor equation related to an invariant of a diffeomorphism

Let $M$ be an $n$-dimensional differentiable manifold, $f : U \rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$ of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be ...
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Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode $$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$ Where $b$ is a bounded continuous function (not ...
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### Permutation-invariant matrix representation

The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
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### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
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### Something about iterated logarithm [on hold]

that's my first question there. So, can you explain, why iterated log well-defined with base more than e^(1/e). I considered a f = w^(1/w), and prove that f(max) = e^(1/e), so if I prove that log* ...
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### Why are algebraic cycles rational?

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
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### How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
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### For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
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### Complexity :: Integer Programming :: Non-Poly Example

When learning about computational complexity I find that when discussing the NP-Complete problems authors always give examples of such problems that can in fact be solved in poly time. I understand ...
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### Non-discrete modularity measure in graph analysis [on hold]

I work in neuroimaging, and right now graph theory is all the rage. Most graph analyses that parcel brain regions into modules do so in a discrete fashion. This might ignore the idea that one brain ...
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### limit exercise which requires ingenuity1 [on hold]

Hello im an eleventh grader in the best mathematics high school in my country. I wanted to see if any of you guys can help me solve a limit without integrals and L'Hospital, as i havent learnt them ...
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### Is there a nonzero sheaf with all cohomologies vanish?

Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?
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### Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
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### Proof for existence of isoperimetric minimizer by compactness theorem

Let $X$ be an $n$-dim closed Riemannian manifold, then given a number $0 <v < vol(X)$, there exists a Borel subset $A\subset X$ attaining $I(v)$, $I$ is the isoperimetric profile. The existence ...
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### Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces. Let $B_1, \ldots, B_k \in B(\mathcal{H}).$ Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via ...
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### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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### first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
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### Can anybody help me for this counting question? [on hold]

Peter has 12 pairs of socks and 6 pairs of gloves in different colors. His socks are in green, yellow, black, and grey (3 pairs each). Peter's gloves are either blue, black, or red (2 pairs each). ...
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### Lower bound on class number of binary quadratic forms of discriminant of the form n^2+4

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
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### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...