# All Questions

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### Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...
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### Hodge metric on pair

I am looking for the definition of Hodge metric, like definition 2.2 here https://hal.archives-ouvertes.fr/hal-00322845/document But instead of Vector bundle if we have divisor $D$ with conic ...
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### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...
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### General formula for composite numbers [on hold]

I propose general formula of composite numbers, except divisible by 2 and 3: Positive integers contained in two 2-dimensional arrays:$P1(i,j)=6i^2-1+(6i-1)(j-1)$ and $P2(i,j)=6i^2-1+(6i+1)(j-1)$ are ...
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### Conic sections in high dimensions

Can every $n$-dimensional ellipsoid be obtained as a (spherical) conic section? This is false for generic quadrics but seems true for ellipsoid. Does anybody have any references?
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### Some examples of non trivial principal bundles

1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection ...
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### concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...
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### Category of equicontinuous sets of mappings

Does this category have a name? Does it have any literature? Objects are topological vector spaces. A morphism from A to B is any equicontinuous set of linear mappings from A to B.
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### How to determine whether a power of eta function is a eigenform? [on hold]

I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases ...
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### The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
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### Euler's constant: irrationality and proof theory [on hold]

Let γ represent Euler's constant. Is there a real number x such that there is a proof within Zermelo-Fraenkel set theory (ZF) that x is irrational and there is also a proof within ZF that γ + x is ...
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### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
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### Examples and Counterexamples in Commutative Algebra

There are Counterexamples in Analysis and Counterexamples in Topology. Is there any similar book for commutative algebra? I want to see some more (counter)examples for Atiyah and MacDonald's book. Let ...
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### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
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### Existence of nonlinear equation

How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation: ...
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### Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [on hold]

Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative ...
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### Construction of Stein's exchangeable pair for certain dependent random variables

Given a sequence of exchangeable random variables $X_1,\ldots,X_n$, and a measurable function $g: \mathbb{R}^n \to \mathbb{R}$. Let $S_n=g(X_1,\ldots,X_n)$. Then what is a natural construction of a ...
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### Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note). Let $n$ be the length of ...
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### Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
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### Probability questionss [on hold]

In a population brain volume is distributed according to the normal distribution with a mean value of 1400 cm3 and an SD of 125 cm3. What is the probability that a randomly chosen individual will have ...
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### Probability questions [on hold]

The probability that a new drug prevents infection by a certain flu strain is 40%. What is the probability that the drug will be effective in one out of 5 exposed persons? ( can someone please answer ...
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### Is the complement of the ends of a manifold bounded?

Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...
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### Which fields have no extensions of degree divisble by a fixed prime?

Let $p$ be a prime. What are the most general examples of a field $K$ such that for any finite extension $L/K$ the degree $[L:K]$ is prime to $p$? Certainly, there are algebraically closed examples ...