\message{ !name(formulas.tex)}\documentclass[12pt]{article} \newcommand{\vs}{\vspace{.1in}} \newcommand{\ds}{\displaystyle} \newcommand{\p}{\partial} \newcommand{\Tr}{\rm Tr} \usepackage[T1]{fontenc} \usepackage{yfonts} \usepackage{graphicx} \usepackage{latexsym} \usepackage{amsmath} \usepackage{amssymb} \def\Dirac{\not\hspace{-1.25mm}\partial} \def\DDirac{\not\hspace{-1.25mm}D} \usepackage{amsfonts} \usepackage{rawfonts} \usepackage{amsxtra} \usepackage{amscd} \usepackage{amsthm} \usepackage{eucal} %\usepackage{tikz} %\usetikzlibrary{arrows,decorations.pathmorphing,decorations.markings,trees,backgrounds,fit,calc,through} \usepackage{slashed} \input{prepictex} \input{pictex} \input{postpictex} \begin{document} \message{ !name(formulas.tex) !offset(-3) } \noindent{1. \it Primes and the zeta function:} $$ \prod_p \frac{1}{1-\frac{1}{p^s}} = \sum_{n=1}^{\infty}\frac{1}{n^s}$$ \vs \noindent{2. \it Archimedes. Volume of Sphere:} \begin{figure}[htp] \centering \includegraphics[width=1.5in]{archimedes.eps} \end{figure} $$ v = \frac{2}{3}V$$ \vs \noindent{3. \it Yang-Baxter Equation} $$R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23}$$ \begin{figure}[htp] \centering \includegraphics[width=2in]{braid.eps} \end{figure} \vs \noindent{4. \it Gauss. Quadratic reciprocity:} $$ \left (~\frac{p}{q}~\right )\left ( ~\frac{q}{p}~\right ) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$ \vs \noindent{5. \it Completing the square:} $$ax^2 + bx + c = a\left (x +\frac{\ds b}{\ds 2a}\right )^2 - \frac{\ds b^2-4ac}{\ds 4a}$$ \vs \noindent{6. \it Newton's Law: } $${\bf F} = m{\bf a}$$ \vs \noindent{7. \it Newton's Law of Gravitation:} $$ F = \frac{\ds G m_1m_2}{\ds r^2}$$ \vs %{\it diagram with three 2x2 tangent circles inscribed in a circle} \noindent{8. \it Pythagoras' Theorem with no-word proof:} $$c^2 = a^2 + b^2 $$ \begin{figure}[ht] \centering \includegraphics[width=3in]{pythagoras.eps} \end{figure} \vs \noindent{9. \it Pascal's Hexagon Theorem:} \begin{figure}[ht] \centering \includegraphics[width=3in]{pascal.eps} \end{figure} \vs \noindent{10. \it Euler's Equation:} $$ e^{\ds i\pi} + 1 = 0$$ \vs \noindent{11. \it Schr\"{o}dinger's Equation:} $$ i\hbar \frac{\ds \partial\psi}{\ds \partial t} = -\frac{\ds \hbar^2}{\ds 2m} \nabla^2\psi + V\psi$$ \vs \noindent{12. \it Maxwell's Equations in Vacuum imply Wave Equation:} \begin{figure}[htp] \centering \includegraphics[width=4in]{light.eps} \end{figure} $$\begin{array}{l} \nabla \cdot {\bf B} = 0 \\ \nabla \times {\bf B} = \frac{\ds 1}{\ds c}\frac {\ds \partial {\bf E}} {\ds \partial t}\\ \nabla \cdot {\bf E} = 0 \\ \nabla \times {\bf E} = -\frac{\ds 1}{\ds c}\frac{\ds \partial {\bf B}} {\ds \partial t}\\ \end{array} ~~~~ \Rightarrow ~~~~\nabla^2 {\bf E} = \frac{\ds 1}{\ds c^2}\frac {\ds \partial^2 {\bf E}} {\ds \partial t^2}$$ \vs \noindent{13. \it Einstein's Equation:} $$E = mc^2$$ \vs \noindent{14. \it Einstein's General Relativity Equation: } $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = 8 \pi T_{\mu\nu}$$ \vs \noindent{15. \it Einstein quotation from Fine Hall Common Room Fireplace:} \vs \centerline {\large \frakfamily Raffiniert ist der Herr Gott, aber boshaft ist er nicht.} \vs \noindent{16. \it Heisenberg Uncertainty Principle:} $$\Delta x \Delta p \geq \frac{\ds \hbar}{\ds 2}$$ \vs \noindent{17. \it Dynkin diagram of $E_8$:} \begin{figure}[htp] \centering \includegraphics[width=2in]{dynkin.eps} \end{figure} \vs \noindent{18. \it Quotation from Harish-Chandra:} \vs \centerline{Why has God made the exceptional groups?} \vs \newpage \noindent{19. \it Limiting ratio of Fibonacci numbers = golden mean = partial fraction expansion:} $$ \lim_{n\rightarrow\infty}\frac{\ds F_{n+1}}{\ds F_n} = \frac{\ds 1 + \sqrt{5}}{\ds 2} = {1}+ \frac{\ds 1}{1+\frac{1}{1+ \frac{1}{1+ \dots } } }$$ \begin{figure}[htp] \centering \includegraphics[width=2in]{spiral.eps} \end{figure} \vs \noindent{20. \it Wilson loop average w.r.t Chern-Simons action gives knot invariant:} \begin{figure}[htp] \centering \includegraphics[width=2in]{trefoil.eps} \end{figure} $$ V_K(q) \,=\, q + q^3 - q^4 $$ \vs $$ V_K\!\left( q = {\rm e}^{\!{{2\pi i}\over{k+2}}} \right) \,=\, {1\over Z} \int_{\mathcal A} \Tr_{\bf 2} P\exp{\!\left(-\oint_K A \right)} \, {\rm e}^{\!{{i k}\over{4 \pi}} \, {\rm CS}(A)} {\mathcal D} A$$ \vs \newpage \noindent{21. \it Gauss-Bonnet for spherical triangle:}\vs \begin{figure}[htp] \centering \includegraphics[width=2in]{bonnet-2x.eps} \end{figure} $$\alpha + \beta + \gamma = \pi + \frac{\ds A}{\ds R^2}$$ \vs \noindent{22. \it Stokes' Theorem} $$\int_{M}d\omega=\int_{\partial M}\omega$$ \newpage \noindent{23. \it Uniformization of a surface of genus $2$:} \begin{center} \mbox{ \beginpicture \setplotarea x from -100 to 100, y from -100 to 100 \circulararc 360 degrees from 0 -100 center at 0 0 \circulararc 45 degrees from 0 -59 center at 0 -100 \circulararc -45 degrees from 0 59 center at 0 100 \circulararc 45 degrees from 0 59 center at 0 100 \circulararc -45 degrees from 0 -59 center at 0 -100 \circulararc 45 degrees from -59 0 center at -100 0 \circulararc -45 degrees from -59 0 center at -100 0 \circulararc 45 degrees from 59 0 center at 100 0 \circulararc -45 degrees from 59 0 center at 100 0 \circulararc 45 degrees from 42 -42 center at 71 -71 \circulararc -45 degrees from 42 -42 center at 71 -71 \circulararc 45 degrees from 42 42 center at 71 71 \circulararc -45 degrees from 42 42 center at 71 71 \circulararc 45 degrees from -42 -42 center at -71 -71 \circulararc -45 degrees from -42 -42 center at -71 -71 \circulararc 45 degrees from -42 42 center at -71 71 \circulararc -45 degrees from -42 42 center at -71 71 \arrow <3pt> [1,3] from 59 2 to 59 -2 \arrow <3pt> [1,3] from -2 -59 to 2 -59 \arrow <3pt> [1,3] from -2 -59 to 2 -59 \arrow <3pt> [1,3] from 40 -44 to 42 -42 \arrow <3pt> [1,3] from 42 -42 to 44 -40 \arrow <3pt> [1,3] from 40 44 to 42 42 \arrow <3pt> [1,3] from 42 42 to 44 40 \arrow <3pt> [1,3] from -40 44 to -42 42 \arrow <3pt> [1,3] from -42 42 to -44 40 \arrow <3pt> [1,3] from -44 40 to -46 38 \arrow <3pt> [1,3] from 42 42 to 44 40 \arrow <3pt> [1,3] from -40 -44 to -42 -42 \arrow <3pt> [1,3] from -42 -42 to -44 -40 \arrow <3pt> [1,3] from -44 -40 to -46 -38 \arrow <3pt> [1,3] from -2 59 to 0 59 \arrow <3pt> [1,3] from 0 59 to 2 59 \arrow <3pt> [1,3] from -4 59 to -2 59 \arrow <3pt> [1,3] from 2 59 to 4 59 \arrow <3pt> [1,3] from -59 2 to -59 0 \arrow <3pt> [1,3] from -59 0 to -59 -2 \arrow <3pt> [1,3] from -59 -2 to -59 -4 \arrow <3pt> [1,3] from -59 4 to -59 2 \endpicture } \mbox{ \beginpicture \setplotarea x from 0 to 290, y from 0 to 100 \arrow <6pt> [1,2] from 145 70 to 145 10 \endpicture } %\begin{center} \mbox{ \beginpicture \setplotarea x from 20 to 290, y from 0 to 60 \ellipticalarc axes ratio 3:1 270 degrees from 150 40 center at 120 30 \ellipticalarc axes ratio 3:1 -270 degrees from 175 40 center at 205 30 \ellipticalarc axes ratio 4:1 -180 degrees from 135 33 center at 120 33 \ellipticalarc axes ratio 4:1 145 degrees from 130 30 center at 120 29 \ellipticalarc axes ratio 4:1 180 degrees from 190 33 center at 205 33 \ellipticalarc axes ratio 4:1 -145 degrees from 195 30 center at 205 29 {\setquadratic \plot 150 40 163 37 175 40 / \plot 150 20 163 23 175 20 / } \endpicture } \end{center} \vs \noindent{24. \it Atiyah-Singer Theorem for Twisted Dirac Operator:} %$$\Dirac : \Gamma ({\mathbb S}_+\otimes E) \to \Gamma ({\mathbb S}_-\otimes E)$$ $$\dim \ker (\Dirac) - \dim \ker (\Dirac^*) = \int_{M^{4k}} \hat{A}(M) \smile ch (E)$$ \vs \noindent{25. \it Generalized Gauss-Bonnet Formula:} $$ \chi (M^{2n})= \frac{1}{(8\pi)^n n!} \int_{M}\underbrace{R^{ij}_{ab}\cdots R^{k\ell}_{cd}}_n \varepsilon^{ab\cdots cd} \varepsilon_{ij\cdots k\ell} ~d\mu $$ \vs \noindent{26. \it Classical Gauss-Bonnet Theorem:} \bigskip $$ \chi (M^{2})= \frac{1}{2\pi} \int_{M}K ~dA $$ \begin{center} \mbox{ \beginpicture \setplotarea x from 0 to 350, y from -5 to 70 \put {$\chi (M)= -4$} [B1] at 40 20 \ellipticalarc axes ratio 5:2 280 degrees from 150 40 center at 120 30 \ellipticalarc axes ratio 5:2 -100 degrees from 175 40 center at 205 30 \ellipticalarc axes ratio 5:2 100 degrees from 175 20 center at 205 30 \ellipticalarc axes ratio 5:2 -280 degrees from 260 40 center at 290 30 \ellipticalarc axes ratio 4:1 -180 degrees from 135 33 center at 120 33 \ellipticalarc axes ratio 4:1 145 degrees from 130 30 center at 120 29 \ellipticalarc axes ratio 4:1 180 degrees from 190 33 center at 205 33 \ellipticalarc axes ratio 4:1 -145 degrees from 195 30 center at 205 29 \ellipticalarc axes ratio 4:1 180 degrees from 275 33 center at 290 33 \ellipticalarc axes ratio 4:1 -145 degrees from 280 30 center at 290 29 {\setquadratic \plot 150 40 163 38 175 40 / \plot 150 20 163 22 175 20 / \plot 235 40 248 38 260 40 / \plot 235 20 248 22 260 20 / } \endpicture } \end{center} \vs \noindent{27. \it Chern-Simons Action:} $$ S_{CS}=\frac{k}{4\pi}\int_M{\rm Tr}\left(A\wedge d A+\frac{2}{3}A\wedge A\wedge A\right)$$ \vs \noindent{28. \it Witten's AdS/CFT prescription:} $$\left\langle e^{\int_{S^d}\varphi_0 {\cal O}}\right\rangle_{CFT_d}=Z_{AdS_{d+1}}[\varphi_0]$$ \vs \noindent{29. \it Snell's (Pascal's) Law:} \begin{figure}[htp] \centering \includegraphics[width=2in]{snell.eps} \end{figure} $$\sin i = n \sin r$$ \vs \noindent{30. \it Veneziano amplitude:} $$S(k_1;k_2;k_3;k_4)=\frac{2ig_o^2}{\alpha'}(2\pi\delta)^{26}\left (\sum_i k_i\right )[B(-\alpha(s),-\alpha(t))+ B(-\alpha(s),-\alpha(u))+B((-\alpha(t),-\alpha(u))]$$\vs \noindent{31. \it Liouville equation:} $$\partial_z\partial_{\bar z}\varphi=\frac{1}{2} e^{\varphi}$$ \vs \noindent{32. \it Virasoro algebra:} $$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n} $$ \vs \noindent{33. \it Poincar\'{e}'s epigraph to ``Sur le probl\`{e}me des trois corps ...'' }\vs \centerline{Nunquam praescriptos transibunt sidera fines.} \centerline{(Never will the stars cross their prescribed limits.)} \vs \noindent{34. \it Yang-Mills equations in two notations:} $$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + f_{abc}A^b_\mu A^c_\nu$$ $$\nabla^{\mu}F_{\mu\nu}^a = 0$$ (or) $$\partial^\mu F_{\mu\nu} -ig \left[ A^{\mu} , F_{\mu\nu} \right] = 0$$ \vs $$F = dA + A \wedge A$$ $$\nabla \wedge \star F =0$$ \vs \noindent{35. \it Yang-Mills Lagrangian} $$\mathcal{L} = -\frac{1}{4 g^2} \textrm{Tr} \, F^2$$ \noindent{36. \it Hodge Theorem:} $$H^p(M) = \{ \varphi \in \Omega^p(M)~|~ d\varphi = 0, ~d\star \varphi = 0\}$$ \noindent{37. \it Dirac Equation:} $$ \begin{array}{rcl} \left(-i\hbar \gamma^\mu \nabla_\mu +mc\right) \psi&=&0\\ \gamma^\mu\gamma^\nu+ \gamma^\nu\gamma^\mu&=&2g^{\mu\nu} \end{array} $$ \noindent{38. \it Dirac Equation (another):} $$\begin{array}{rcl} %(\slash ?\! \! \! \! {D}+im)\psi& =& 0\\ (\DDirac + im)\psi& =& 0\\ D&\equiv&\partial+ieA \end{array} $$ \vs \noindent{39. \it Cauchy's Integral Formula:} $$f(z) = \frac{\ds 1}{\ds 2\pi i}\oint_C \frac{\ds f(\zeta)}{\ds \zeta - z}~d\zeta$$ \vs \noindent{40. \it Hodge Decomposition Theorem:} $$H^n(X,\mathbb C)=\oplus_{p+q=n} H^{p,q}(X)$$ \vs \noindent{42. \it Navier-Stokes equation:} $$ \p_t v_i + v_j \p_j v_i = -\p_i p + \nu \p_j \p_j v_i $$ \vs \noindent{43. \it Kolmogorov Law:} $$ E(k) \sim \varepsilon^{2/3} k^{-5/3} $$ \vs \noindent{44. \it Hubbard model:} $$ H=-t \sum_{\langle i,j\rangle,\sigma}\left( c^\dagger_{i,\sigma}c_{j,\sigma} + h.c.\right) + U \sum_i n_{i\uparrow} n_{i\downarrow} $$ \vs \noindent{41. \it Pentagon identity of dilogarithm:} \begin{figure}[h] \centering \includegraphics[width=3in]{pentagon.eps} \end{figure} $$Li_2(x) = \int_{0}^{1} \frac{dt}{t} \log( 1 - x t )$$ \vs \noindent{45. \it Renormalization:} $$ \beta(g) = \mu \frac{\partial g}{\partial \mu} $$ \vs \noindent{46. \it "no comment"} $$ SU(3) \times SU(2) \times U(1) $$ \vs \noindent{47. \it The entropy formula:} $$ S = -k_B \sum_i p_i \log p_i $$ \vs \noindent{48. \it CKM matrix:} $$\left [\begin{array}{ccc} |V_{ud}|&|V_{us}|&|V_{ub}|\\ |V_{cd}|&|V_{cs}|&|V_{cb}|\\ |V_{td}|&|V_{ts}|&|V_{tb}| \end{array} \right ] = \left [ \begin{array}{lll} 0.9742 &0.2257 &0.0036 \\ 0.2256 &0.9733 & 0.0415\\ 0.0087 &0.0407 &0.9991 \end{array} \right ] $$ \vs \noindent{49. \it Exact homotopy sequence of Stiefel fibration:} $$\cdots \rightarrow \pi_{i+1}S^n \rightarrow \pi_iV_{k-1}{\mathbb R}^n \rightarrow \pi_iV_k{\mathbb R}^{n+1} \rightarrow \pi_iS^n \rightarrow \cdots $$ \noindent{50. \it Ricci flow:} $$\frac{dg_{t}}{dt} = -2Ric(g_{t})$$ \vs \noindent{51. \it Apollonian fractal:} \begin{figure}[htp] \centering \includegraphics[width=5in]{scott_apollonian.eps} \end{figure} \newpage \noindent{52. \it Sierpinski gasket:} \begin{figure}[htp] \centering \includegraphics[width=5in]{scott_sierpinski.eps} \end{figure} \newpage \noindent{53. \it Pascal's triangle, Fibonacci numbers:} \begin{figure}[htp] \centering \includegraphics[width=3in]{triangle.eps} \end{figure} \vs \noindent{54. \it Heat Kernel Version of the Index Theorem:} \[ \operatorname{index}(\slashed{D}) = \operatorname{Trace}[(-1)^Fe^{-\beta H}] \] \noindent{55. \it Aharonov-Bohm Effect:} {\Large $$e^{iq\oint {\vec A}\cdot d{\vec\ell}}=e^{ i2\pi F}$$} \vs \newpage %\noindent{. \it Ising Model:} \noindent{56. \it Group law on cubic:} \begin{figure}[htp] \centering \includegraphics[width=3in]{cubic.eps} \end{figure} $${\bf\tt a}\cdot {\bf\tt b} \cdot {\bf\tt c} = 1$$ \noindent{57. \it Riemann-Roch-Hirzebruch:} $$ \sum_{k=0}^n (-1)^k{\rm dim} \, H^k(X,E)\ \ =\ \ \int_X ch\,E \cup Todd(X) $$ \noindent{58. \it Platonic solids, Euler characteristic:} \begin{figure}[htp] \centering \includegraphics[width=1in]{tetrahedron.eps} \centering \includegraphics[width=1in]{cube.eps} \centering \includegraphics[width=1in]{octahedron.eps} \centering \includegraphics[width=1in]{dodecahedron.eps} \centering \includegraphics[width=1in]{icosahedron.eps} \end{figure} $$ V - E + F = 2$$ \vs \newpage \noindent{59. \it Lorenz attractor:} \begin{figure}[htp] \centering \includegraphics[width=4in]{scott-lorenz.eps} \end{figure} \vs \noindent{60. \it Black hole entropy} $$S_{BH}=\frac{A}{4}$$ \newpage \noindent{61. \it Mandelbrot Set:} \begin{figure}[htp] \centering \includegraphics[width=4in]{mandelbrot-fake-sm.eps} \end{figure} \vs \noindent{64. \it Prime number Theorem:} $$ \pi(n)~\sim~\frac{n}{\ln(n)}$$ \vs \noindent{62. \it Sonnet by Edna St. Vincent Millay} Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone. \vs \noindent{63. \it Eratosthenes' mesurement of rdius of Earth:} \begin{figure}[htp] \centering \includegraphics[width=4in]{milnor-eratos.ps} \end{figure} \vs \noindent{65. \it Supergravity:} $${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$ \vs \noindent{66. \it Fourier transform:} $$\hat{f}(\xi) = \int f(x) e^{2\pi i x\cdot\xi} dx \hspace{1in} f(x) = \int \hat{f}(\xi) e^{-2\pi i x\cdot\xi} d\xi$$ \vs \noindent {67. \it Euler's summation for $\zeta(2)$:} $$ 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\ds \pi^2}{\ds 6}$$ \vs \newpage \noindent{68. \it Kepler's laws:} \begin{figure}[htp] \centering \includegraphics[width=4in]{milnor-keppic.ps} \end{figure} %\centerline{Kepler's Laws:\qquad $r\;d\theta/dt={\rm constant}\,, %\qquad \oint dt \sim s^{3/2}$} \vs \noindent{69. \it Prime number theorem:} $$ \pi(n)~\sim~{n\over \ln(n)} $$ \vs \end{document} $$W(K) = \frac{\ds 1}{\ds Z}\int_{\mathcal A} \mbox{tr} P\int_K A\cdot dt ~e^{i\int_M CS(A)} dA$$ $$W_R(C) \,=\, \Tr_R P\exp{\!\left(-\oint_C A\right)} \langle W_R(C) \rangle = V(q) q = \exp{\!\left(2\pi i/(k+2)\right)}. $$ S_{\rm CS}(A) \,=\, {k \over {4\pi}} \, \int_M \Tr\!\left(A \wedge d A \,+\, {2 \over 3} A \wedge A \wedge A\right) $$ \message{ !name(formulas.tex) !offset(-677) }