Tagged Questions

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Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \pmb t(\pmb X) = \frac 1 n ...
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Maximum and minimum of weighted sum

For $w_i\ge 0$ and some constants $\alpha_i , i=1,...,n$, what is the maximum and minimum of $\sum_{i=1}^{n}\alpha_i w_i$ subjected to $\sum_{i=1}^{n}w_i=1$? Intuitively, I put all weight on the ...
Suppose we have two sets of weights $w_i$ and $u_i$; i.e., $\sum w_i =\sum u_i = 1$, and $0 \le w_i \le 1$, $0 \le u_i \le 1$. Consider the weighted average ratio R of these weights: $R \equiv \sum ... 1answer 62 views Is it always true that “max$\ge$average + sigma”? Assume that$i$from$1,\ldots,N$,$x_i \ge 0$and: $$\mathrm{avg} = \frac{\sum_i x_i}{N}$$ $$\sigma = \sqrt\frac{\sum_i{(x_i-\mathrm{avg})^2}}{N}$$ Is that true that: $$\max_i x_i \ge ... 2answers 171 views Average limit superior [duplicate] Let \mathcal{l}_\mathbb{R}^\infty be the space of bounded sequences in \mathbb{R}. We define a map p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R} by$$p(\underline x)=\limsup_{n\to\infty} ... 4answers 177 views If$\sum\limits_{k=1}^n y_k\geq n$and$\sum\limits_{k=1}^n \frac{1}{y_k}\geq n$, then$\prod\limits_{k=1}^n y_k\geq 1$? Let$y_1,\ldots y_n$be positive real numbers satisfying$y_1+\cdots+y_n\geq n$and$\displaystyle{\frac{1}{y_1}+\cdots+\frac{1}{y_n}\geq n}$. Is it true that$y_1y_2\cdots y_n\geq 1\$?
It is well known that, if HM denotes the harmonic mean and AM the arithmetic mean, we have $$AM(x) \ge HM(x)$$ Now I am dealing with the expression $$\frac{1}{HM(x)} - \frac{1}{AM(x)}$$ A ...