Questions tagged [symmetric-polynomials]

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

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Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions

I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is \begin{align} a^{\...
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How to show that $X_m$ is a zero of this polynomial in $R[X_1,…,X_m][X]$?

I posted this question a few days ago but the images didn’t work so nobody knew what I was talking about. I'm self-studying through Amann/Escher Analysis 1 and I'm stuck on a problem in I.8. Here's ...
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A proof involving Vieta's formulas

If $x,y,z$ are complex numbers satisfying $x+y+z = 1, xy+ xz + yz = 1, x yz = 1,$ then must they be the roots of the polynomial $t^3 - t^2 + t - 1$? Also, can this formula be generalized: if $x_1,\...
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1 vote
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Zonal quaternionic polynomials

I've done a R package which computes the Jack polynomials (I started it a couple of years ago). It also computes the zonal polynomials (Jack with $\alpha=2$ up to a normalizing factor), the Schur ...
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2 votes
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What are the Littlewood-Richardson coefficients for hook-shaped diagrams?

I am trying to compute Littlewood-Richardson coefficients involving hook-shaped diagrams. In particular, if $\lambda$ is a hook-shaped Young diagram, $\rho$ is any diagram, and we consider, $$ s_\...
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proving that $p$ divides $f(1,2,\ldots, p -1)$

Let $p$ be a prime and let $f(x_1,\ldots, x_{p-1})$ be a symmetric polynomial in $p-1$ variables. Suppose $f$ is homogeneous of degree $d$ with $(p - 1) \not | d$. Prove that $p$ divides $f(1,2,\ldots,...
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Numerical methods for solving algebraic equations of symmetric linear forms

Question: Given a set of polynomial algebraic equations for $x_i\in\mathbb{R}^n$ of the form \begin{equation}\label{eqs} x_i = f(x_1,...,x_{i-1},x_{i+1},...,x_{d}), \quad \mathrm{for} \quad i=1,...,d,...
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monomial symmetric polynomials

It's necessary to prove $m_\lambda=\sum\limits_{\nu}^{}s_\nu$ where $\lambda=(\lambda_1,...,\lambda_n)$ is partition of n, $\nu$=$\sigma(\lambda)$, $\sigma\in S_n$, $s_\nu$ is Schur polynomial. I ...
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Dimension of the symmetric\alternating k-tensor over an $n$-dimensional vector space.

I want to solve this question: Suppose $V$ is a vector of dimension $n$ over a field $F$ of characteristic not equal to 2. Calculate dim $Sym^{k}(V)$(the symmetric k tensor ). I know that $(Sym^k(V))^*...
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proving a claim about symmetric polynomials

Define an $S_n$-invariant polynomial $f$ of $\mathbb{Z}[x_1,\cdots, x_n]$ so that $f(x_1,\cdots, x_n) = f(x_{\sigma(1)},\cdots, x_{\sigma(n)})$ for any permutation $\sigma\in S_n$, the set of ...
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$\sigma_1\sigma_2 =\Sigma_{i<j}{~~(z_i^2z_j+z_iz_j^2})+3\Sigma_{i<j<k}{~~z_iz_jz_k}$

Let $P\in\mathbb{R}[X]$ of degree $n$ and $z_1,...,z_n $ the complex roots of $P$. We consider that $\sigma_1(z_1,...,z_n)=\Sigma_{i}z_i$ and $\sigma_2(z_1,...,z_n)=\Sigma_{i<j}z_iz_j$. I read in a ...
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Alternating polynomials under the diagonal action

Consider the (usual) action of the symmetric group $\mathcal{S}_n$ on the polynomials $\mathbb{R}[x_1,\ldots ,x_n]$ given by $$ \sigma\colon p(x_1,\ldots ,x_n)\mapsto p(x_{\sigma_1},\ldots ,x_{\...
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Implementation of symmetric tensor decomposition algorithm

Context Any symmetric tensor F of rank $d$ and dimension 2 ($F \in S^d\mathbb{C^2}$ for our purpose) can be associated with a homogeneous polynomial $P(F)\in k[x_0,x_1]_d$ in 2 variables of degree $d$....
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Kind of elementary symmetric product

I have the following polynomial in the variables $A_1, A_2, ..., A_n;B_1, ...,B_n:$ $ f(A_1, A_2, ..., A_n;B_1, ...,B_n) = A_1B_2B_3...B_n + B_1A_2B_3...B_n + ...+B_1...B_{n-1}A_n$ Without the $A_j$ I ...
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How to prove $\operatorname{LM}\left(\prod_{1\le k\le n}\sigma_{n;k}\right)=\prod_{1\le k\le n}\operatorname{LM}(\sigma_{n;k})$?

The "elementary symmetric polynomials" $\sigma_{n;k}$ are: $$\sigma_{n;k}(x_1,\ldots,x_n)=\sum_{1\le i_1\lt\cdots\lt i_k\le n}x_{i_1}\cdots x_{i_k}$$ where $k=1,\ldots,n$. (...) Every ...
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Efficient way to derive coefficient from (1+t) to t

Suppose I have the coefficient $b_i$ of degree $d$ polynomial defined on variable $1+t$. Can i get coefficient $c_i$ that defined on variable $t$ in linear time ? Namely, $$ \sum_{i=0}^d b_i(1+t)^i = \...
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Relation between two polynomial subrings

Let $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ be ring in $x_1,\ldots, x_n$ with coefficients from $\mathbb{Z}$. Let $e_1=x_1+x_2+\cdots + x_n$, $e_2=\sum_{i<j} x_ix_j$, $\cdots$, $e_n=x_1x_2\cdots x_n$ : ...
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2 votes
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How to prove this expression is a polynomial

How can we prove that the expression $$\gamma_{m k}(\lambda)=\sum_{i=1}^{m}\left(\lambda_{m i}+m-1\right)^{k} \prod_{j \neq i}\left(1-\frac{1}{\lambda_{m i}-\lambda_{m j}}\right)$$ is a polynomial in ...
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2 votes
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Fractional exponent elementary symmetric polynomials.

I am wondering if there is any literature on relations between fractional power symmetric polynomials. For a particular example, with the variables $\textbf{x} = (x_1,x_2,\dots x_n),$, can we ...
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Proof and meanings of Lagrange's theorem for polynomials

I'm trying to understand Lagrange's theorem which I found in Harold Edward's book ''Galois Theory''. I was trying to find this theory in internet but all Lagrange's theorems which I found are not ...
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4 votes
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Terminology of "algebraically closed rings"

There are various approaches how to generalize the notion of an algebraically closed field to the context of commutative rings. A good survey is R. Raphael, On algebraic closures. I am interested in ...
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1 vote
1 answer
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Computing certain polynomials related to symmetric functions and $\lambda$-rings in Sage

The definition of a $\lambda$-ring (https://en.wikipedia.org/wiki/%CE%9B-ring) makes use of certain "universal" polynomials $P_n$ and $P_{n,m}$, which basically give you the formulas for ...
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Prove for general form of function at -x containing derivatives of order n

I have stumbled across multiple casses of functions (explicitly Hermit and Legendre polynomials) for which I wanted to prove the symmetry. While doing so I always ended up with the following equations:...
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1 answer
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Factoring the sum of polynomial coefficient "division"

Define the coefficient $B = (b_0, \cdots b_n)$ of elementarty polynomial product of $A=(a_0 \cdots, a_n)$. $B = \sum_{i=0}^N b_ix^i = \prod_{i=0}^N(1+a_ix) $. Use $f(A) = B$ to get it's coefficient. ...
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1 vote
0 answers
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Invariant polynomial function on Lie algebras.

Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \...
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Sum of powers of variables

Let $n$ be a fixed positive integer, and $$ \begin{cases} y_1+y_2+\ldots+y_n=x_1+x_2+\ldots+x_n,\\ y_1^2+y_2^2+\ldots+y_n^2=x_1^2+x_2^2+\ldots+x_n^2,\\ y_1^3+y_2^3+\ldots+y_n^3=x_1^3+x_2^3+\ldots+x_n^...
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If $A$ is a real symmetric matrix with full rank and only simple roots, what can be said about $|B\otimes I+I\otimes B|$, for $B = \sum_k a_k A^k$?

I'm considering a situation in which $A, B > 0$ are real $(p\times p)$ symmetric matrices where $A$ has $p$ distinct eigenvalues $\{\lambda_i\}_{i=1}^p$, and $AB=BA$. I'm interested in the level of ...
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How to derive the mentioned relations (in part a-b-c of example 9) from the Weyl denominator formula?

I want to understand Example 9 (a-b-c) of section 1.5 of the book "Symmetric functions and Hall polynomials (https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)" (pages 78-79): In ...
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1 vote
0 answers
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Solving $\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$ with symmetric polynomials

I need to solve the following with symmetric polynomials, but I do not understand what to do here. $$\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$$
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-1 votes
1 answer
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How to solve this equation with symmetric polynomials? [closed]

Here is what I did $$x + \sqrt {17 - x^2} + x\times\sqrt{17 - x^2} = 9$$ I can`t undestand how to solve it, any help would be appreciated!
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11 votes
1 answer
157 views

What does Heron's formula naturally deform?

Fixing three real numbers $a,b,c>0$ determines a triangle with side-lengths $a,b,c$ (if admissible). Therefore, the area of a triangle is a function in $a,b,c$. Due to the geometry of a triangle, ...
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2 votes
1 answer
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What is the proper math notation for a particular sum of products?

Suppose we have a sequence of distinct $a_i$'s: $\left\{ a_1, a_2, \ldots, a_n \right\}$. We also have a sequence of not necessarily distinct $b_i$'s: $\left\{ b_1, b_2, \ldots, b_n \right\}$. Each $...
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1 vote
0 answers
37 views

Efficient way of simplify sum of product of multiple polynomials

Let $A \in \mathbb{R}^{n\times m}, B \in \mathbb{R}^m$. I'm trying to compute the coefficients of $n$ polynomials $C_i = (c_0^i, c_1^i, \cdots, c_{n-1}^i)$. where $\displaystyle \sum_{j=0}^{n-1} c_j^i ...
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1 vote
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Determine if a function is monotonic

Let $a, b, c, d \in \mathbb{R}$ and $a, b, c, d\gt 1$. For any polynomial $A_n = \sum_{i=0}^n a_ix^i$, i'm interested in a quantity $g(A_n) = \frac{1}{n}\langle (a_0, \cdots, a_n), (1/{n \choose 0}, \...
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2 votes
1 answer
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Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$

Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}$ , $b=\sqrt{3}-\sqrt{5}+\sqrt{7}$, $c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate: $$\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$$ What I have tried so far is writing the ...
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  • 359
0 votes
1 answer
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Root transformation to make normalized coefficient match

Let $f(x) = \prod_{i=0}^n(1+a_ix), a_i \neq 0$, and $C(f(x)): = (c_0, c_i, \cdots, c_n) $, where $f(x) = \sum_{i=0}^n c_i x^i$. For a given monomial $g(x) = (1+mx)(1+nx)$, i'm interested in a quantity ...
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3 votes
1 answer
73 views

Reciprocal binomial coefficient polynomial evaluation

The conventional binomial coefficient can be obtained via $$ f(x, n) = (1+x)^n = \sum_{i=0}^n { n \choose i} x^i $$ And the function $f$ can be every efficiently performed on evaluation. I'm ...
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0 votes
0 answers
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Is it possible to obtain roots from the weighted sum of other polynomials in their root form?

Given $n$ polynomials $A_1, A_2, .. A_n$with the same degree $M$. $A_i = \prod_{j=0}^M(1+Q_{ij}x)$. In their root form $Q$, $Q_{ij} \in \mathbb{R}$. And the function, I'm interested in, $B$ is a ...
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0 votes
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How does the fundamental theorem of symmetric polynomials imply that this number is rational?

In the Problems from the Book by Titu Andreescu, there is a proof of Example 9 on page 494 with the following: Example 9. Let $f$ be a monic polynomial with integer coefficients and let $p$ be a ...
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1 vote
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General formulation of polynomial invariants.

Reading this work https://arxiv.org/pdf/2012.06452.pdf, I'm wondering if it's true that any fundamental invariant polynomial $\{f_i\}_{i=1}^{N_{inv}}$ can be written as $$f_i(x) = \sum_{g \in G}\psi(g ...
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  • 1,702
0 votes
2 answers
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Schur polynomial of partition (1,0)

Schur polynomials $s_\lambda$(or Schur functions) for a partition $\lambda$ is given by the bialternant formula of Jacobi, which is a ratio of two Vandermonde determinants: $$s_\lambda (x_1,x_2,\cdots,...
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  • 433
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0 answers
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Galois group of general equation of degree n

Let $K$ be a field and $\{t_1,t_2,\ldots,t_n\}$ be an algebraically independent set over $K$. For every permutation $\sigma \in S_n$ we have an automorphism of $K(t_1,t_2,\ldots,t_n)$ given by mapping ...
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  • 243
1 vote
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Definition of Macdonald polynomial $P_\lambda^{\mathfrak{g}}$ associated to a Lie algebra $\mathfrak{g}$ (unlike $P_\lambda$)

I want to find the definition for the Macdonald polynomial associated to a Lie algebra $\mathfrak{c}_n$, i.e. $P_\lambda^{\mathfrak{c}_n}(x,t,q)$. This appears in the physics paper https://arxiv.org/...
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8 votes
0 answers
198 views

Local max of ratio of elementary symmetric polynomials

Let $e_k(x_1,\cdots, x_n) := \sum_{1\leq j_1 < \cdots < j_k \leq n}x_{j_1}\cdots x_{j_k}$ be elementary symmetric polynomials (see e.g. https://en.wikipedia.org/wiki/...
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$x_i>0, \forall\ 1\leq i\leq n$ is equivalent to $\sigma_1,\cdots,\sigma_n$ are all $>0$, where $\sigma_i$ are the elementary symmetric polynomials. [duplicate]

$x_i>0, \forall\ 1\leq i\leq n$ is equivalent to $\sigma_1,\cdots,\sigma_n$ are all $>0$, where $\sigma_i$ are the elementary symmetric polynomials. Here $\sigma_1=x_1+\cdots+x_n, \sigma_2=...
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  • 2,031
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$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$

Problem: Prove that in any triangle ABC, the following inequality is true: $$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$$ Anyone can help me? I tried AM-GM for right ...
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  • 409
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0 answers
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Mathematica package for computing Macdonald polynomials

0 I want to implement computation of Macdonald polynomials in mathematica. A similar question was raised in another question 5 years ago (Macdonald-Koornwinder polynomials?), but received no clear ...
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  • 433
3 votes
1 answer
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Prove that: $2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$

Problem: For $a,b,c\ge0: ab+bc+ca>0.$ Prove that: $$2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$$ Recently, i have seen a post on AoPS link My approach: ...
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  • 409
2 votes
2 answers
134 views

In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Problem: Given a,b,c are length of triangle. Prove that: $$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$$ Happy Vietnamese Women's ...
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