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Divisibility of cyclic sums

Lately I have been studying the divisibility of some cyclic sums, and I was wondering about the following Conjecture Let it be a set of distinct integers $S=\{x_1,x_2,…,x_n\}$. Then, $$\prod_{n=1}^{n}x_{n}\nmid\left(\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)+\sum_{cyc}\left(\prod_{n=1}^{n-2}x_{n}\right)+…+\sum_{n=1}^nx_n\right)$$ I would appreciate any help regarding this conjecture proof or refutation. I already noted that there exist sets of distinct integers $S=\{x_1,x_2,…,x_n\}$ such that $$\prod_{n=1}^{n}x_{n}\mid\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)$$ […]

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Differential equations, sectors and inequalities

I am working with the following equation: $$ \lambda f – a \frac{d^2f}{dx^2}=g, $$ $$ \lim_{x\to +\infty}f=\lim_{x\to -\infty}f = f(x_1)=f(x_2)=0, $$ where $a>0$, $f\in H^2(\mathbb{R}\setminus (x_1,x_2))$, $g\in H^2(\mathbb{R}\setminus (x_1,x_2))$, and $$ \lambda\in \{ \lambda\in \mathbb{C}\setminus \{0\}\ : \ \mid \arg\ \lambda \mid\leq \pi -\beta,\ \mid \lambda \mid >\gamma \ \} \quad \mbox{with }\beta\in (0,\pi/ 2),\ […]

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More formulas for joint entropy and for trace form entropies

Linked to some applications of entropy to combinatorics I’m looking for formulas expressing the joint entropy of two r. v. as a function of the conditional entropy . For example For BWS extensive entropy H : $ H(X,Y) = H(X) + H(Y[X) $ For q-Tsallis non extensive entropy H : $ H(X,Y) = H(X) + […]

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Product of quotient space is quotient of product for compact spaces?

Let $X_1, X_2,…, X_n$ be compact (not necessarily Hausdorff) spaces, each with equivalence relation $\sim_k$. Is it possible to find an equivalence relation $\sim$ on $\prod\limits^n_{i=1}X_i$ such that $(\prod\limits^n_{i=1}X_i)/\sim$ is homeomorphic to $\prod\limits^n_{i=1}(X_i/\sim_k)$? I think I have found a counterexample to this problem, but I am not sure about it. Both my ‘proof’ and disproof […]

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Completion of product space

I’m trying to prove the statement: “Suppose that $(Y, \rho, f)$ is a completion of the metric space $X$ and $(Y’,\rho’,f’)$ is a completion of the metric space $X’$. Show that $(Y \times Y’, \rho”, f \times f’)$ is a completion of the product space $X \times X’$ where $\rho”$ is the product metric for […]

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Using summation to calculate the remainder of order 3 of a multivariable function

$R_{p+1}(x) = \sum_{i_1+i_2+…+i_n = p+1} \frac{1}{i_1!i_2!…i_n!} [\frac{\partial^{p+1}f}{\partial x_1^{i_1} x_2^{i_2}…x_n^{i_n}} (c) × (x_1 – x_{0,1})^{i_1} …(x_n – x_{0,n})^{i_n}]$ where $c \in (x_0,x)$ and $x = (x_1,x_2,…,x_n)$, $x_0 = (x_{0,1},x_{0,2},…x_{0,n})$ $p=2$ in this case and $x_0 = (0,0,0)$ $R_3(x) = \frac{1}{3!}[\frac{\partial^3f}{\partial x_3^3}(c) × x_3^3] + \frac{1}{3!}[\frac{\partial^3f}{\partial x_2^3}(c) × x_2^3] +\frac{1}{3!}[\frac{\partial^3f}{\partial x_1^3}(c) × x_1^3] + \frac{1}{2!} [\frac{\partial^3f}{\partial x_2 […]

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Verifying that $F^{\infty}$ is a vector space

I am trying to verify that $F^{\infty}$, which Axler defines as $$F^{\infty} = \{(x_1, x_2, \ldots) : x_j \in F\}$$ is a vector space. I have been able to verify that $F^n$ is a vector space. Is the verification for $F^{\infty}$ the same? Am I able to rigorously say that if I can establish something […]

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Let $H_i$ be a subgroup of $G_i$ for $i=1,2,\dots,n.$ Prove that $H_1×\dots × H_n$ is a subgroup of $G_1 ×\dots × G_n.$

First I suuuuuck at proofs. I think I am on the right track but I need some fine tuning. Or if I am totally off let me know. First we show that $H_i$ is nonempty. Note that since $H_i$ is a subgroup of $G_i, H_i$ contains the identity element. So $e_G \in H_1, e_G \in […]

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Example of bounded linear map $c\to \ell^\infty$ with $Ae_i=0$

I am looking for an example of a non-zero bounded linear map $c\to \ell^\infty$ for which $Ae_i=0$ (the sequence $(e_i)_j=\delta_{ij}$). The only thing I came up with so far is $$(x_1,x_2,x_3,\ldots)\mapsto (x_1 x_2,x_2x_3,x_3x_4,\ldots),$$ but this is not linear..

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Variance of empirical mean squared error of binomial distribution

Given $m$ binomially distributed samples $X_1,X_2,\ldots,X_m\sim\text{Bin}(n,p)$. Let $\hat{p}_i=X_i/n$ be the corresponding estimates for $p$. The empirical mean squared error is given by $Z= \frac{1}{m}\sum_{i=1}^m (\hat{p}_i-p)^2$. I am looking for the expected value and the variance of $Z$. $E(Z) = p(1-p)/n$ follows directly from the variance of the binomial distribution. Is there a simple way to […]