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How can I calculate weighted standard errors and plot them in a bar plot?

I have a data frame of counts. I would like to calculate weighted proportions, plot the proportions, and also plot standard error bars for these weighted proportions. Sample of my data frame: head(df[1:4,]) badge year total b_1 b_2 b_3 b_4 b_5 b_6 b_7 b_8 b_9 b_10 1 15 2014 14 3 2 1 1 1 […]

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Covering number of $L^2$ Ball

What is the covering number $N(\epsilon, B_2, ||\cdot||_2)$ of a ball $B_2$ of radius $r$ under the $L^2$ norm?

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Prove $\|A\times B\| = \|A\|\|B\||\sin(\theta)|$, given $A = (a_1,a_2,a_3)$ and $B = (b_1,b_2, b_3)$, and $\theta$ is the angle between A and B.

I have two vectors in $\textbf{R}^{3}$, $A = (a_1,a_2,a_3)$ and $B = (b_1,b_2, b_3)$ and $\theta$ is the angle between $A$ and $B$. And I need to proof that: $\|A\times B\| = \|A\|\|B\||\sin(\theta)|$.

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Can I prove the Baire Category Theorem in this bizarre way?

I’m trying to prove the Baire Category Theorem, which states that in a complete metric space $X$, the countable union of nowhere dense sets has empty interior. In a metric space, $A$ being nowhere dense (having empty interior) implies that $\sup\limits_{p\in \overline{A}} d(p,(\overline{A})^c)=0$, where $A^c$ is the complement of $A$ in $X$. However, more weakly, […]

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Complex cobordism and Chern numbers

Let $L$ be the Lazard’s universal ring, and $R=\mathbb{Z}[b_1,b_2,\cdots,b_n,\cdots]$, regarded as a graded ring with the degree of $b_i$ equal to $2i$. Let $\theta: L\rightarrow R$ be the homomorphism carrying the universal formal group law $\mu^L$ to the formal group law $$\mu^R(x_1,x_2)=\exp(\log(x_1)+\log(x_2)),$$ where the power series $$\exp(x)=x+\sum_{i\geq 1}b_ix^{i+1},$$ and $\log(x)$ its inverse, denoted as $$\log(x)=x+\sum_{i\geq […]

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Circular permutations with derangement [duplicate]

$5$ boys $B_1,B_2,\ldots,B_5$ and $5$ girls $G_1,G_2,\ldots, G_5$ are to be seated around a round table such that all boys and girls sit alternatively and $B_i$ does not sit beside $G_i$ $ \forall i=1,2\ldots,5$. Find the total number of arrangements possible. My attempt: First the boys can be permuted in $4!$ ways. Now here comes […]

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Expectation and variance for the number of edges joining equal vertices on a simple cycle graph with $k+1$ Bernoulli(1/2) vertices

Let $B_1,B_2,\dots,B_{k+1}\sim \text{Bernoulli}(1/2)$ be i.i.d. for some fixed integer $k \geq 2$. Assign these random variables to the $k+1$ vertices of a simple cycle graph of length $k+1$, and let $N$ be the number of edges that connect vertices that are equal (i.e. $(1,1)$ or $(0,0)$). What is $E[N]$ and $\text{Var}(N)$ ?

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count spanning trees (up to isomorphism) in K3,3

I look through the solution posted on stackexchange about counting spanning trees in K3,3 (the link is here Number of spanning trees of $K_{3,3}$.). But I am still confused about how to count it by adding condition “isomorphism”, since in that post, based on my understanding the answer seems to consider two isomorphic spanning trees […]

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count spanning trees (up to isomorphism) in K3,3 [closed]

I look through the solution posted on stackexchange about counting spanning trees in K3,3 (the link is here Number of spanning trees of $K_{3,3}$.). But I am still confused about how to count it by adding condition “isomorphism”, since in that post, based on my understanding the answer seems to consider two isomorphic spanning trees […]

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evaluating a limit of a function

Let $a, b, c> 0$ and consider the differential equation $af”+bf’+cf = 0$ which satisfyies $f(0) = k$ and $f'(0) = 0.$ Let $t = \dfrac{b}{2\sqrt{ac}}.$ Define $f_t(x)$ to be the solution to the differential equation when $t > 1$ and $f_1(x)$ to be the solution when $t = 1.$ Determine the range of values […]