User:Hal Canary/scratch

$$\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}$$

$$\bar{y} = \frac{y_1 + y_2 + \ldots + y_n}{n}$$

define $$x_i' = x_i - \bar{x}$$

define $$y_i' = y_i - \bar{y}$$

$$\frac{(x_1 - \bar{x})(y_1 - \bar{y}) + (x_2 - \bar{x})(y_2 - \bar{y}) + \ldots + (x_n - \bar{x})(y_n - \bar{y})}{\sqrt{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}\sqrt{(y_1 - \bar{y})^2 + (y_2 - \bar{y})^2 + \ldots + (y_n - \bar{y})^2}}$$

$$\frac{x_1' y_1' +  x_2' y_2' + \ldots + x_n' y_n' }{\sqrt{(x_1')^2 + (x_2')^2 + \ldots + (x_n')^2}\sqrt{(y_1')^2 + (y_2')^2 + \ldots + (y_n')^2}}$$

$$\underset{x\in{}X}{\operatorname{arg\,max}} \, f(x) := f^{-1}(\underset{x\in{}X}{\max} \, f(x) ) \subset{} X$$