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Bound the value of function by integration of derivatives

Given $f(x,y) \in C^2([0,1]^2)$ (by which I mean $C^2$ in some open neighborhood), with $f_x, f_y, f_{xy} \in L^1([0,1]^2, dxdy)$ (which is sure case since they are continuous), does the following hold? $$ \sup |f| \le \iint |f|+|f_x|+|f_y|+|f_{xy}|\, dxdy $$ I think it is safe to discretise this function under the assumptions, divide $f$ into […]

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What is this linear combination of elementary symmetric polynomials?

It is known that elementary symmetric polynomials $e_k$ appear when we expand a linear factorization of a monic polynomial: $${\displaystyle \prod _{j=1}^{n}(\lambda +X_{j})=\lambda ^{n}+e_{1}(X_{1},\ldots ,X_{n})\lambda ^{n-1}+e_{2}(X_{1},\ldots ,X_{n})\lambda ^{n-2}+\cdots +e_{n}(X_{1},\ldots ,X_{n}).}$$ This is, $${\displaystyle \prod _{j=1}^{n}(\lambda +X_{j})= \sum_{k=0}^n\lambda ^{n-k}e_{k}(X_{1},\ldots ,X_{n}).}$$ https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial Now, suppose that $C_0,\dots C_n$ are positive constants. What can I say about $${\displaystyle \sum_{k=0}^nC_k […]

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How to make algebraic dependence explicit

Here is a possibly standard question, from someone who is in no way an expert. The scenario is taken from Shafarevich, Basic Algebraic Geometry 1, ch. 1, §6.3, proof of Thm. 1.25(ii). Let us have a regular map $f:V\rightarrow W$ ($V,W$ open in $\mathbb{A}^{N},\mathbb{A}^{M}$ respectively, and $f(V)$ dense in $W$); passing to the pullback $f^{*}$, […]

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Tightness of Hilbert-space-valued arrays

I posted this on Stackexchange already here. Since I got no answer, I wanted to give it a try here (I hope this question is advanced enough). Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}}^2 < […]

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In $Q_p$: $f(x)=0$ iff $p(f(x))^2$ is a square

I am reading the paper by Jan Denef: the rationality of the Poincaré series associated to the p-adic points on a variety. In the proof of his first lemma he makes the claim that For $f\in\mathbb{Z}_p[x_1,\ldots,x_m]$ a polynomial in $m$ variables over the $p$-adic integers and $x=(x_1,\ldots,x_m)\in Q_p^m$ we get $f(x)=0$ if and only if […]

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Choosing Sets- A coherent solution?

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that: $\bullet$ for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and $\bullet$ $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. I have […]

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Statements about the correlation between invertibility of an operator and the corresponding matrix

Let $A\in M_n$. Prove the following: $(a)\;T\in L\left(M_{n\times 1}\right),\;T(X)=AX$ is invertible $\iff\; A$ is invertible. $(b)\;S\in L\left(M_n\right),\;S(X)=AX$ is invertible $\iff\; A$ is invertible. My attempt: $(a)$ Since $T\in L\left(M_{n\times 1}\right)$, $X$ is a column matrix, i.e. $$X=\begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}$$ If $A$ is regular, then $\operatorname{rank}{A}=n\;\implies\;$ all the columns of $A$ are linearly independent. Let $c_j$ denote columns […]

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De Rham and Koszul complexes

Consider the algebraic de Rham complex of the $n$-dimensional plane: this is merely $$\ldots\rightarrow Sym(V^*)\otimes\bigwedge^{k}V^*\rightarrow Sym(V^*)\otimes\bigwedge^{k+1}V^*\rightarrow\ldots $$ with the standard de Rham differential $fdx_I\rightarrow\sum_{i=1}^n\frac{df}{dx_i}dx_i\wedge dx_I$. On the other hand, there is a Koszul resolution of the ideal $(x_1,\ldots,x_n)\subset Sym(V^*)$ is defined on a chain complex which is termwise the same as above, but with differential […]

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Let $A$ be a finite set in a metric space. Prove that $H_p(A)=0$ for all $p>0$.

Suppose that $(X,d)$ is a metric space and let $p\geq 0, \delta >0$. Let $\mathcal{E}$ be the collection of those subsets of $X$ with diameter $\leq \delta$ together with the set X, and define $d(A) = (\text{diam }A)^p$. We define: \begin{equation*} H_p^{\delta}(A)=\inf\,\{\sum_{j=1}^\infty d(E_j) : E_j \in \mathcal{E} \text { and } A \subset \bigcup_{j=1}^{\infty} E_j […]

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Tightness of Hilbert-space-valued arrays

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}}^2 < \infty$ that is strictly stationary (which means that $W_{n,1},\ldots , W_{n,n}$ is strictly stationary for every $ n$). After Remark 3.3 in “Central Limit and Functional Central […]