\documentclass{article}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{newunicodechar}
  \newunicodechar{⁻}{${}^{-}$}% Superscript minus
  \newunicodechar{²}{${}^{2}$}% Superscript two
  \newunicodechar{³}{${}^{3}$}% Superscript three

\pagestyle{empty}
\begin{document}
$f(u,v) \rightarrow (x,y,x)$
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$Pt = (x, y, z), D_u\vec{f} = (dx/du, dy/du, dz/du), D_v\vec{f} = (dx/dv, dy/dv, dz/dv)$
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$N = D_u\vec{f} \times D_v\vec{f}$
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$Pt = (x, y, z), Du = (dx/du, dy/du, dz/du), Dv = (dx/dv, dy/dv, dz/dv)$
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$N = Du X Dv$
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$F:R^2 \rightarrow R^4$
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$f(u,v) = ((r*cos(v)+a)*cos(u),(r*cos(v)+a)*sin(u),r*sin(v)*cos(u/2),r*sin(v)*sin(u/2))$
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$R^3$
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$ N >= \frac{5}{P}\mathrm{ln}\left(\frac{1}{PU}\right) $
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$_mC_n$
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$\left(\begin{array}{c}m \\ n\end{array}\right)$
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$ A* = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$
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$ L = \left( \begin{array}{cc} 1 & 0 \\ c & 1 \end{array} \right)$
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$ U = \left( \begin{array}{cc} a & b \\ 0 & d \end{array} \right)$
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\[ P[0] X^d + ... + P[d-1] X + P[d] \]
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$K(\text{vertex v}) = 2*\pi - \sum_{\text{facet neighbs f of v}} (\text{angle_f at v})$
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$Area(facet)/3$
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$[1/m^2]$
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$H(vertex v) = \text{average over edges neighbs e of H(e)}$
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$H(edge e) = length(e) * dihedral\_angle(e)$
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$k_\max$
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$k_\min$
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$k_\max = H + \sqrt{H^2 - K}$
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$k_\min = H - \sqrt{H^2 - K}$
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$k_max = H + sqrt(H^2 -K)$
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$k_min = H - sqrt(H^2 -K)$
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$L_1$
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$L_2$
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$L_{\infty}$
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$\infty$
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$-\infty$
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\[ \frac{\sum_{i=1}^{N} \lambda_i}{\sum_{i=1}^{D} \lambda_i} < T \]
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$t$
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$\frac{|t|_\infty}{|t|_0}$
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$|t|_\infty$
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$|t|_0$
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$\frac{|t|_\infty}{2\sqrt{3}r}$
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$r$
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$\frac{R}{2r}$
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$R$
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$\frac{|t|^2_2}{2\sqrt{3}{\cal A}}$
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$|t|^2_2$
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$\cal A$
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$q$
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$\frac{|q|_\infty}{|q|_0}$
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$|q|_\infty$
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$|q|_0$
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$\frac{|q|_1|q|_\infty}{4{\cal A}}$
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$|q|_1$
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${\cal A}$
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$\frac{|q|_2h_{\max}}{\min_i{\cal A}_i}$
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$|q|_2$
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$h_{\max}$
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$\min{\cal A}_i$
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$V$
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$\frac{f^2+g^2}{4{\cal A}}$
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$f^2+g^2$
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$K$
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$\frac{|K|_\infty}{|K|_0}$
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$|K|_\infty$
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$|K|_0$
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$\frac{|K|_\infty}{2\sqrt{6}r}$
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$\frac{R}{3r}$
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$\frac{\frac{3}{2}(l_{11}+l_{22}+l_{33}) - (l_{12}+l_{13}+l_{23})} {3(\sqrt{2}\det{T})^\frac{2}{3}}$
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$T$
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$l_{ij}$
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$L=T^t\,T$
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$H$
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$\frac{|H|_\infty}{|H|_0}$
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$|H|_\infty$
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$|H|_0$
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$ \mathrm{DisplacementMagnitude} cos( 2\pi \mathrm{ModeShapeTime} ) $
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$n$
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$N$
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$\frac{N}{|T|} = n$
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$|T|$
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$|T| = 1 + 2^d + \left(2^d\right)^2 + \cdots + \left(2^d\right)^k$
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$d$
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$k$
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$|T|\approx 2 \left(2^d\right)^k$
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\[ k = \frac{\log{\frac{N}{2n}}}{\log{2^d}} \]
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$1, 2, \ldots, k-1$
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$2^d$
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\end{document}
