# All Questions

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### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction. Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
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### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
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### I'm PHD student (finance) [on hold]

I want to apply CIR model for a chive asset price. can I do this if Feller condition is considering? please help me
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### When does $R [x]/I$ have infinitely many idempotents in special case?

At < When does $R [x]/I$ have infinitely many idempotents? >, Er_Ro asked the following question. Let $R$ be a commutative ring with identity and $R[x]$ its polynomial ring. I am looking ...
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### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
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### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$. (1). What is the cohomology ring $$H^*(A_4;\mathbb{Z}/3)$$ and its Steenrod operation $P^i$'s? (2). Are there general results about the ...
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### Similarity estimation [on hold]

http://www.diku.dk/summer-school-2014/course-material/mikkel-thorup/bottomk-exercise.pdf Can somebody help with exercise 4 in chapter 2.2? Any hints would be highly appreciated.
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### the first chern class of complex vector bundles

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$). Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class $c_1(\xi^\mathbb{C})$ is zero ...
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### Find an example of functions f, g such that limx→0 f(x) and limx→0 g(x) both do not exist, but limx→0 f(x) + g(x) = 1 [on hold]

Find an example of functions f, g such that limx→0 f(x) and limx→0 g(x) both do not exist, but limx→0 f(x) + g(x) = 1.
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### Modulus of continuity of analytic functions [on hold]

Let $H$ be the class of all analytic functions of the unit disk onto itself. For $r\in (0,2)$ let $$h(r)=\sup\{|f(z)-f(w)|: f\in H, |z-w|\le r\}.$$ How to determine $h$ explicitly? Schwarz lemma ...
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### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and $$\mathrm{Ricc}(g)=\lambda g,$$ $h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
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### Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...
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### Generalized density functions on the natural numbers

If $a_1,a_2,\dots$ are IID random bits (correction as per Anthony Quas: these "bits" are $+1$ and $-1$ with equal probability), then with probability 1, the set of natural numbers $n$ such that ...
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### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...
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### Is the restricted root system of a simple real Lie group irreducible?

As the title asks, is the restricted root system of a simple real Lie group irreducible? I believe this is true but I need a reference to cite.
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### k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)). I realized that ...
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### What function is a Gaussian integral

Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$. Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse ...
Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories? That is, for each spectrum $E$, do we have a lift in the ...
Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...