All Questions

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L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$\|q\|_1 \leq O(n) \|p\|_1$$ where we define $\|f\|_p := (\int_{-1}^1 |f(x)|^pdx)^{1/p}$? ...
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Prove relations between hypotenuse and cathetus in geometry problem [on hold]

Given a triangle ABC with cathetus 'a' and 'b' and hypotenuse 'c', prove that for every odd 'a' >= 3 (3, 5, 7...), there is an integer 'b' and integer 'c' -> c - b = 1
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Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$

Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008) observed that if $(x_{n})_{n}$ is a sequence in $L_{p}$ that is ...
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Properties of a subring of a 'completion' of k(X_1, X_2, …, X_n)

I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents". I don't even know the name of this ...
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Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
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Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate. Throughout we work in ZF+AD. My question is: If $A$ is an uncountable OD set of reals,...
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Divisor on variety determined by its restriction to curves

Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?
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Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
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Question on a paper by Benjamini/Kozma/Wormald about a “well known fact”

In "The mixing time of the giant component of a random graph" by the aforementioned authors, in the last proof on page 19 it says something along the lines of "It is well known and easy to verify ...
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'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
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Coherence theorem for symmetric lax monoidal functors

Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements: 1) ...
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Is a polarization on an abelian scheme an open condition?

Let $A/S$ be an abelian scheme such that the dual abelian scheme $A^{\vee}/S$ exists and let $\lambda : A \to A^{\vee}$ be a morphism of abelian schemes. Is the locus of points in $S$ where $\lambda$ ...
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Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. To ...
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Covering map of classifying space [on hold]

We know that for any $m \in \mathbb{N}$ the map $p_m: S^1 \to S^1$ is an $m$-sheeted covering of $S^1$. Suppose that $BG$ is the classifying space of an arbitrary group $G$. Does there exist such a ...
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Methods for defining/calculating homotopy limits of quasicategories

I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the ...
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Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
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Meaningful formalization of a continuum of Bernoulli random variables [on hold]

I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...
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A Global Restriction Estimate from Local Estimate

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose ...
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Does the Ruzsa-Szemeredi Theorem also capture graphs decomposable into *nearly* induced matchings?

The well-known Ruzsa-Szemeredi Theorem states that a graph whose edges can be partitioned into $n$ induced matchings has at most $\frac{n^2}{RS(n)}$ edges, for some slow-growing function $RS(n)$. Now,...
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When two non-equivalent binary forms primitively represent the same infinite subset of the integers

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ and $y$ ...
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What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
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How to figure a complentary set of a Diophantine equation [on hold]

Given a subset of the real numbers defined as 2xy-x-y+1 for x >1, y > 1 how can I determine the complementary set?
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Subspaces of $L_{p}(2<p<\infty)$

Let $p>2$ and $X$ a subspace of $L_{p}$. Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. Question: if $X$ is ...
I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is ...
An arrangement of $s$ chords are drawn over a circle so that no three chords intersect at a common point and no two chords are parallel. Denote the arrangement by $\mathcal{H}_{s}$. Does \$\...