# All Questions

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### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...
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### A construction with homotopy colimits and homotopy pullbacks for descent

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
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### Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic

It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already ...
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### Is my proof of the Schoenfeld's inequality correct? [on hold]

Full preprint here. Theorem 4.1. For any $x\ge 2$ we have $$$$\theta(x)-x<\frac{1}{8\pi}\sqrt x\log^2 x. \;\;\;\;\;\;\;\;\;\;\;(4.1)$$$$ Proof: It's known that ...
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### Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function ...
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### Quardic Equation [on hold]

We know that a quardic equation have two roots.After solving this equation:8x^2-33x-35=0 we get two roots.The first one is:5 and the second one is:-7/8.But the root -7/8 doesn't satisfy the given ...
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### Generalize Gauss-Bonnet Formula to non-simple closed curves

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense: For a domain $\Omega$ enclosed by an non-simple closed curve ...
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### A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any ...
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### What is the Turing degree of $\mathbb{C}_{exp}$?

Let $\mathbb{C}_{exp}$ be the theory of the complex numbers in the language of exponential rings. I am interested in the Turing degree of $\mathbb{C}_{exp}$. As the natural numbers are definable in ...
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### Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a complex submanifold (closed) of codimension(complex) r. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$. 1) we have the ...
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### Roots of 3 polynomials in two variables in terms of the coefficients [on hold]

Let p,q,h\in\mathbb{C}[x,y] If {p_{ij}}, {q_{ij}} and {h_{ij}} are the coefficients of p,q and h respectively. (...
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### A stronger version of Fermat's last theorem

Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not. The equation $x_1^m+\cdots+x_n^m=1$ has nonzero rational solutions iff $n\geq m$. Here a nonzero rational ...
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### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$. I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...
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### Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies. Do we have the following ...
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### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...
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### On a permutation module for GL(n,q)

Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ...
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### “Epicycles” (Ptolemy style) in math theory?

By analogy: The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell ...
When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos. When $C$ is ...
Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...