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Diffstat (limited to 'analysis/analysis.tex')
| -rw-r--r-- | analysis/analysis.tex | 16 |
1 files changed, 15 insertions, 1 deletions
diff --git a/analysis/analysis.tex b/analysis/analysis.tex index d3aa0e6..516f59d 100644 --- a/analysis/analysis.tex +++ b/analysis/analysis.tex @@ -1,3 +1,4 @@ + \documentclass{article} \usepackage{amssymb} \usepackage{amsmath} @@ -145,6 +146,19 @@ and $f_i:\bb R\to\bb R$ is $K$-Lipschitz for all $i$ then $f$ is $K$-Lipschitz. \end{proof} \end{theorem} +\begin{theorem} +\label{multiplex} +If $f:\bb R^n\to\bb R^n$, $f(\vec x) = (f_1(\vec x) , f_2(\vec x), \dots, f_n(\vec x))$ +and $f_i:\bb R^n\to\bb R$ is $K$-Lipschitz for all $i$ then $f$ is $K\sqrt{n}$-Lipschitz. +\begin{proof} +\begin{align*} +||f(\vec x) - f(\vec y)||^2 &= \sum_{i=1}^n (f_i(\vec x) - f_i(\vec y))^2\\ +&\leq \sum_{i=1}^n (K||\vec x-\vec y||)^2\\ +&= nK^2||\vec x-\vec y||^2 +\end{align*} +\end{proof} +\end{theorem} + \begin{theorem} \label{dim-reduction} @@ -263,7 +277,7 @@ Given $f,g:\bb R^n \to \bb R$ 1-Lipschitz and differentiable, define $$h(\vec x) = \sin (f(\vec x))\cos(g(\vec x))$$ Clearly $h$ is also differentiable, we will show that it is 1-Lipschitz. $$\pp h{x_i} = \cos(f(\vec x))\cos(g(\vec x)) \pp{f(\vec x)}{x_i}- \sin(f(\vec x))\sin(g(\vec x)) \pp{g(\vec x)}{x_i}$$ -so letting $f(\vec x) = a,~g(\vec x)= b,~p = \cos a\cos b,~q = \sin a\sin b,~\pp{f(\vec x)}{x_i} = s_i,~\pp{g(\vec x)}{x_i} = t_i$, we have +so letting $a=f(\vec x),~b=g(\vec x),~p = \cos a\cos b,~q = \sin a\sin b,~\pp{f(\vec x)}{x_i} = s_i,~\pp{g(\vec x)}{x_i} = t_i$, we have \begin{align*} ||Dh(\vec x)||^2 &= \sum_{i=1}^n \left[ps_i - qt_i\right]^2\\ &=p^2\l(\sum_{i=1}^n s_i^2\r) + q^2\l(\sum_{i=1}^n t_i^2\r)- 2pq\l(\sum_{i=1}^n s_it_i\r)\\ |