# All Questions

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### Computer power in plane geometry

I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
2k views

### Probability of random permutation having certain cycles

Are there any good references (either books or on-line) on the subject of the distribution of various cycle properties amongst permutations, particularly ones containing exact, closed-forms? For ...
640 views

can you give me a good paper (in the sense of a simple introduction) about Sasaki-Einstein manifolds? Thank you and best regards Florian M.
1 vote
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### Extension of the projective norm to a cross norm

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
1 vote
18 views

### On an integral equation of Volterra type

Consider the following integral equation $$f(t) = A(t) + \int_0^t K(s,t)f'(s)ds,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$ where $A: \mathbb R_+\to [0,1]$ and $K: \{(s,t): 0\le s\le t\}\to[0,1]$ ...
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### Enlargement of filtration

Let $M_t$ be a continuous time real valued martingale, and $\mathcal F_t$ its natural filtration. Suppose that $\mathcal F_t \setminus \mathcal F_s$ is nonempty for all $t > s$. Let $\mathcal G$ be ...
41 views

### Complete proof about Penrose tilings

It is well known that equivalence classes of Penrose tilings (say, by semikites and semidarts) are in bijection with binary sequences not containing 11 and modulo tail equivalence. However, I couldn't ...
139 views

### Vanishing cycles exact sequence for degeneration of curves

Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$. Let $\eta\in D - \{0\}$ be a general point, and let ...
1 vote
35 views

### Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$

Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
1k views

### Stable homology of arithmetic groups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field. Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL_n(O_F)$, and as a result, ...
1 vote
72 views

### Infinitely many primes that split completely in an arithmetic progression

Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$. ...
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### Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
182 views

### Surprisingly only real points on intersection of certains quadrics

Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by \begin{align} X_e &= 0\\ X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\ X_g &...
231 views

### Quadratic extensions of cyclotomic numbers by absolute values of elements

Summary I was wondering whether there is an explicit description of the extension $\mathbb{Q}(\zeta_n, |z|)/\mathbb{Q}$ obtained from a cyclotomic field $\mathbb{Q}(\zeta_n)$ by adjoining any finite ...
137 views

1 vote
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### Ramsey-theoretic properties of Erdős cardinals

The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous ...
I posted this same question on MSE with no answer. Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...