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Presentations / Kstmumu / Face2Face_meeting / Method_of_moments / MM_v2.tex
@mchrzasz mchrzasz on 13 Aug 2014 25 KB update
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  1. \documentclass[xcolor=svgnames]{beamer}
  2. \usepackage[utf8]{inputenc}
  3. \usepackage[english]{babel}
  4. \usepackage{polski}
  5. %\usepackage{amssymb,amsmath}
  6. %\usepackage[latin1]{inputenc}
  7. %\usepackage{amsmath}
  8. %\newcommand\abs[1]{\left|#1\right|}
  9. \usepackage{amsmath}
  10. \newcommand\abs[1]{\left|#1\right|}
  11. \usepackage{hepnicenames}
  12. \usepackage{hepunits}
  13.  
  14.  
  15. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
  16. \definecolor{mygreen}{cmyk}{0.82,0.11,1,0.25}
  17.  
  18.  
  19. \usetheme{Sybila} 
  20.  
  21. \title[Method of moments for $\PBzero \to \PKstar \mu \mu$]{Method of moments for $\PBzero \to  \PKstar \mu \mu$}
  22. \author{Marcin Chrz\k{a}szcz$^{1,2}$, Nicola Serra$^{1}$}
  23. \institute{$^1$~University of Zurich,\\ $^2$~Institute of Nuclear Physics, Krakow}
  24. \date{\today}
  25.  
  26. \begin{document}
  27. % --------------------------- SLIDE --------------------------------------------
  28. \frame[plain]{\titlepage}
  29. \author{Marcin Chrz\k{a}szcz}
  30. % ------------------------------------------------------------------------------
  31. % --------------------------- SLIDE --------------------------------------------
  32.  
  33. \begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
  34.  
  35. \center \includegraphics[width=0.8\paperwidth]{diagram.png}\\
  36.  
  37. \end{frame}
  38. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  39. \begin{frame}\frametitle{Quo vadis $\PBzero \to \PKstar \mu \mu$?}
  40.  
  41. \center \includegraphics[width=0.8\paperwidth]{diagram_mm.png}\\
  42.  
  43. \end{frame}
  44.  
  45.  
  46. \section{Introduction}
  47. \begin{frame}\frametitle{Introduction}
  48. Why method of moments:
  49. \begin{enumerate}
  50. \item Complementary approach to LL fits.
  51. \item Allows to extract info measuring quantities in event basis depending on the angular distribution.
  52. \item Used in $\PB \to \rho \Plepton \nu$(SLAC-386 UC-414),\\ $\PJpsi \to \PK \PK \gamma$(PRD 71, 032005 (2005) ), etc.
  53. \end{enumerate}
  54.  
  55.  
  56. \end{frame}
  57. \section{Method of Moments - Theory}
  58. \begin{frame}\frametitle{Method of moments}
  59. {~}
  60. Let's assume we have our pdf with $k$ unknown parameters:~$PDF(x_i, \alpha)$, $dim(\alpha)=k$. One can calculate $k$ moments, which are the functions of $\alpha_i$:
  61. \begin{equation}
  62. \mu_i=f(\alpha_1,..., \alpha_k) = E[W_i]
  63. \end{equation}
  64. For $n$ events, we can estimate:
  65. \begin{equation}
  66. \widehat{\mu}_i=\dfrac{1}{n}\sum_{j=0}^{j=n-1} w_j
  67. \end{equation}
  68. , where $w_j=g(x_i)$
  69.  
  70. \end{frame}
  71.  
  72. \begin{frame}\frametitle{Trivial example}
  73. {~}
  74. Lets see how this works in practice:
  75. \begin{equation}
  76. f(x)=\dfrac{x^{a-1}e^{-x/b}} {b^a \Gamma(a)}
  77. \end{equation}
  78. we measure the moments:\\
  79. \begin{center} $m_1=\dfrac{X_1+X_2+...+X_n}{n}$,\\ $m_2=\dfrac{X_1^2+X_2^2+...+X_n^2}{n}$.\\\end{center}
  80. and calculate them analytically:
  81. \begin{center} $m_1=ab$, $m_2=b^2a(a+1)$\end{center}
  82. So one just needs to solve this and get the answer:
  83. \center $a=\dfrac{m_1^2}{m_2-m_1^2}$, $b=\dfrac{m_2-m_1^2}{m_1}$
  84. \end{frame}
  85.  
  86. \section{Moments of Ss}
  87.  
  88.  
  89. \begin{frame}\frametitle{Our PDF}
  90. {~}
  91. The angular terms:
  92. \begin{small}
  93. \begin{multline}
  94. PDF(\cos \theta_k ,\cos \theta_l, \phi) =\dfrac{9}{32\pi}( \dfrac{3}{4}(1-F_l)\sin^2 \theta_k + F_l \cos^2 \theta_k + (\dfrac{1}{4}(1-F_l)\sin^2 \theta_k \\ -F_l\cos^2) \cos 2\theta_l  + S_3 \sin^2 \theta_k \sin^2 \theta_l cos2\phi + S_4 \sin2 \theta_k \sin \theta_l \cos\phi +\\ S_5 \sin 2 \theta_k \sin \theta_l \cos \phi +  (S_{6s} \sin^2 \theta_k + S_{6c} \cos^2 \theta_k) \cos \theta_l + \\ S_7 \sin  2\theta_k \sin \theta_l \sin \phi +  S_8 \sin 2 \theta_k \sin 2 \theta_l \sin \phi + S_9 \sin^2 \theta_k  \sin^2 \theta_l \sin 2 \phi)
  95. \end{multline}
  96. \end{small}
  97. \only<1>{
  98. Since we are fitting a PDF we need to ensure it is normalized:
  99. \begin{equation}
  100. \int_{-\pi}^{\pi} d\phi \int_{-1}^{1} d cos\theta_l \int_{-1}^{1} d cos\theta_k \dfrac{d^4\Gamma}{dq^2 dcos\theta_k dcos\theta_l d\phi}=1
  101. \end{equation}
  102. }
  103. \only<2>
  104. {
  105. \begin{small}
  106. For further use let's introduce a notation:
  107. \begin{multline}
  108. PDF(\cos \theta_k ,\cos \theta_l, \phi) =\dfrac{9}{32\pi}( \dfrac{3}{4}(1-F_l)\sin^2 \theta_k + F_l \cos^2 \theta_k + \\(\dfrac{1}{4}(1-F_l)\sin^2 \theta_k -F_l\cos^2) \cos 2\theta_l  + \sum_{x=3}^{9} S_x f_x(\cos \theta_k ,\cos \theta_l, \phi)
  109. \end{multline}
  110. \end{small}
  111. }
  112.  
  113. \end{frame}
  114.  
  115. \begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$ 1/2}
  116. {~}
  117. \begin{footnotesize}
  118.  
  119.  
  120.  
  121. Let's calculate the moments(means of the given distribution):
  122. \begin{equation}
  123. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k =\frac{2}{5} (2-F_l)
  124. \end{equation}
  125. \begin{equation}
  126. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \cos^2 \theta_k =\frac{1}{5} (2F_l+1)
  127. \end{equation}
  128. \begin{equation}
  129. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \cos^2 \theta_k  \cos 2\theta_l =-\dfrac{2}{25}(2 + F_l)
  130. \end{equation}
  131. \begin{equation}
  132. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k  \cos 2\theta_l=-\dfrac{1}{25}(1+8F_l)
  133. \end{equation}
  134. \end{footnotesize}
  135.  
  136. \end{frame}
  137.  
  138.  
  139. \begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$ 2/2}
  140. {~}
  141.  
  142.  
  143. \begin{small}
  144.  
  145. Let's calculate the moments(means of the given distribution):
  146. \begin{equation}
  147. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{8}{25}S_x,
  148. \end{equation}
  149. for $x=3,4,8,9$, and:
  150.  
  151. %%
  152. \begin{equation}
  153. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{2}{5}S_x,
  154. \end{equation}
  155. for $x=5,6,7$.\\
  156. New physics apparently as we like orthogonal world:
  157. \begin{equation}
  158. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi~( f_{S_x} \times f_{S_y})		= \alpha_{xy} \delta_{x~y}
  159. \end{equation}
  160.  
  161. \end{small}
  162. \end{frame}
  163.  
  164.  
  165.  
  166. \begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
  167. {~}
  168. \begin{itemize}
  169. \item We are abusing the fact that the basis is orthogonal and moments do not mix.
  170. \item Makes live easier and reduces the systematics.
  171. \item Each of the S does not know about other.
  172. \item In case of full PDF, $S_{1s}$, $S_{2s}$, $S_{1c}$, $S_{2c}$ $S_{6s}$, $S_{6c}$ are not orthogonal.
  173. \item Still we can get them solving equation system:
  174. \end{itemize}
  175. \begin{equation}
  176.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin^2 \theta_k cos \theta_l = 0.1(S_6c+4S_6s)
  177. \end{equation}
  178. \begin{equation}
  179.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} cos \theta_l = 0.25(S_{6}c+2S_{6s})
  180. \end{equation}
  181.  
  182. \small solution: $S_{6c}=2 (4 M_{S_{6c}} - 5 M_{S_{6s}})$, $S_{6s}= -2 M_{S_{6c}} + 5 M_{S_{6s}}$
  183.  
  184.  
  185.  
  186. \end{frame}
  187.  
  188.  
  189.  
  190. \section{Toy MC study}
  191.  
  192. \begin{frame}\frametitle{Moments for $\PB \to \PKstar \mu \mu$}
  193. {~}
  194. Lets see if this method actually works. Let's take some random parameters for the PDF and make a toy.
  195. \begin{columns}
  196. \column{2.5in}
  197. \includegraphics[scale=0.25]{images/J3.png}
  198.  
  199. \column{2.5in}
  200. \includegraphics[scale=0.25]{images/J9.png}
  201.  
  202.  
  203. \end{columns}
  204.  
  205. \end{frame}
  206.  
  207.  
  208.  
  209.  
  210.  
  211. \begin{frame}\frametitle{Error Estimation}
  212. {~}
  213. \begin{itemize}
  214. \item Since moment is the mean of a given distribution the error can be estimated as $mean/RMS$
  215. \item use TOY MC to check this assumption
  216. \item Do not worry, detail description an numbers will come in other presentation.
  217. \end{itemize}
  218.  
  219. \includegraphics[scale=0.3]{plots/conw.png}\\
  220.  
  221.  
  222. \end{frame}
  223. %%%%%%%%%%%%%%%%%%%%%%%%5
  224. \begin{frame}\frametitle{Correlation check}
  225. {~}
  226. \begin{itemize}
  227. \item In theory $S_i$ shouldn't be correlated to $S_j$ in the moment calculation.
  228. \item Lets put this to a test.
  229. \end{itemize}
  230.  
  231. \begin{columns}
  232. \column{2.5in}
  233. \includegraphics[scale=0.2]{plots/J9J4.png}\\
  234.  
  235.  
  236. \column{2.5in}
  237. \includegraphics[scale=0.2]{plots/J8J5.png}\\
  238.  
  239.  
  240. \end{columns}
  241.  
  242.  
  243. \end{frame}
  244.  
  245.  
  246. %%%%%%%%%%%%%%%%%%%%%%%%5
  247. \begin{frame}\frametitle{Correlation check 2}
  248. {~}
  249. \begin{itemize}
  250. \item Let's now FIX $J_x$ and simulate different $J_y$
  251. \item Again theory would suggest that one J shouldn't know about the other, so $J_x$ shouldn't change with scanning $J_y$ parameter
  252. \end{itemize}
  253.  
  254. \begin{columns}
  255. \column{2.5in}
  256. \includegraphics[scale=0.24]{plots/J5_vs_J9_300.png}\\
  257.  
  258.  
  259. \column{2.5in}
  260. \includegraphics[scale=0.24]{plots/J3_vs_J9_300.png}\\
  261.  
  262.  
  263. \end{columns}
  264.  
  265.  
  266. \end{frame}
  267.  
  268.  
  269.  
  270. \section{S-wave}
  271.  
  272.  
  273.  
  274. \begin{frame}\frametitle{S-wave pollution}
  275. {~}
  276. \begin{columns}
  277. \column{3in}
  278. \begin{itemize}
  279. \item Unfortunately in our perfect orthogonal world lives an imposter.
  280. \item This imposter is $\PBzero \to (\PK \Ppi)_{S-wave}~\mu \mu$
  281. \item This "ghost" dilutes our NP! Like dark matter the universe.
  282. \item We need something to bust this ghost away
  283. \end{itemize}
  284.  
  285. \column{2in}
  286. \includegraphics[width=0.95\textwidth]{P5.png}\\
  287.  
  288. \includegraphics[width=0.5\textwidth]{gb.jpg}
  289. \end{columns}
  290.  
  291.  
  292. \end{frame}
  293.  
  294.  
  295.  
  296.  
  297.  
  298. %%%%%%%%%%%%%%%%%%%5
  299. \begin{frame}\frametitle{S-wave hunting}
  300. {~}
  301. Our PDF with the S-wave will look as follows:
  302. \begin{multline}
  303. PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) =\dfrac{9}{32\pi}(  \textcolor{red}{(1-F_s)}(\dfrac{3}{4}(1-F_l)\sin^2 \theta_k + F_l \cos^2 \theta_k + \\ (\dfrac{1}{4}(1-F_l)\sin^2 \theta_k  -F_l\cos^2) \cos 2\theta_l  + S_3 \sin^2 \theta_k \sin^2 \theta_l cos2\phi + \\S_4 \sin2 \theta_k \sin \theta_l \cos\phi + S_5 \sin 2 \theta_k \sin \theta_l \cos \phi +  \\ (S_{6s} \sin^2 \theta_k + S_{6c} \cos^2 \theta_k) \cos \theta_l +  S_7 \sin  2\theta_k \sin \theta_l \sin \phi + \\ S_8 \sin 2 \theta_k \sin 2 \theta_l \sin \phi + S_9 \sin^2 \theta_k  \sin^2 \theta_l \sin 2 \phi) + \\
  304. \textcolor{red}{\dfrac{2}{3} F_s \sin^2 \theta_l + \frac{4}{3} A_s \sin^2 \theta_l \cos \theta_k + I_4 \sin \theta_k \sin 2 \theta_l \cos \phi} \\ \textcolor{red}{+ I_5 \sin \theta_k \sin \theta_l \cos \phi + I_7 \sin \theta_k \sin \theta_l + \sin \phi + I_8 \sin \theta_k \sin 2\theta_l \sin\phi})
  305. \end{multline}
  306.  
  307. \small In this form we ensure normalization.
  308. \end{frame}
  309.  
  310. \begin{frame}\frametitle{How does the dilution work? 1/2}
  311. {~}
  312. \begin{small}
  313. \begin{equation}
  314. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{8}{25}S_x\textcolor{red}{(1-F_s)},
  315. \end{equation}
  316. for $x=3,4,8,9$, and:
  317.  
  318. %%
  319. \begin{equation}
  320. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) f_{S_x}= \frac{2}{5}S_x\textcolor{red}{(1-F_s)},
  321. \end{equation}
  322. for $x=5,6,7$.\\
  323. Not much harm and easy to control.
  324. \end{small}
  325.  
  326. \end{frame}
  327.  
  328. \begin{frame}\frametitle{How does the dilution work? 2/2}
  329. {~}
  330. \begin{small}
  331. Unfortunately $F_l$ and $F_s$ will mix with each other:
  332. \begin{multline}
  333. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_k= \\ \frac{2}{15} (6 + 3 F_l (F_s-1) - F_s)=M_{F_l}
  334. \end{multline}
  335. \begin{multline}
  336. \int_{-1}^{1}d\cos \theta_l \int_{-1}^{1}d\cos \theta_k \int_{-\pi}^{\pi}d\phi PDF_{full}(\cos \theta_k ,\cos \theta_l, \phi) \sin^2 \theta_l= \\ \frac{1}{5} (3 + F_l + F_s - F_l F_s)=M_{F_s}
  337. \end{multline}
  338.  
  339. They can solve this system:
  340.  
  341. $ \begin{cases} F_s = \frac{15}{4} (M_{F_l} + 2 M_{F_s}) \\ F_l = \frac{(15 M_{F_l} + 10 M_{F_s}-18)}{(15 M_{F_l} + 30 M_{F_s}-34)}  \end{cases}$
  342.  
  343.  
  344. \end{small}
  345.  
  346. \end{frame}
  347.  
  348.  
  349. \begin{frame}\frametitle{S-wave moments}
  350. {~}
  351. We can even measure directly the S-wave:
  352. \begin{equation}
  353.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi }sin^2 \theta_l cos \theta_k = \dfrac{32 I_{1b} }{45}
  354. \end{equation}
  355. \begin{equation}
  356.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } sin \theta_k sin 2 \theta_l cos \phi = \dfrac{16 I_4 }{45}
  357. \end{equation}
  358. \begin{equation}
  359.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi } sin\theta_k sin\theta_l cos\phi = \dfrac{4 I_5 }{9}
  360. \end{equation}
  361. \begin{equation}
  362.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin  \theta_k sin 2 \theta_l sin \phi = \dfrac{4 I_7 }{9}
  363. \end{equation}
  364. \begin{equation}
  365.  \dfrac{d^3\Gamma}{\Gamma dcos\theta_k dcos\theta_l d\phi} sin  \theta_k sin2 \theta_l sin \phi = \dfrac{16 I_8 }{45}
  366. \end{equation}
  367.  
  368.  
  369.  
  370. \end{frame}
  371.  
  372.  
  373. \begin{frame}\frametitle{Conclusions}
  374. {~}
  375. \begin{itemize}
  376. \item Method of moments very suitable for $\PBzero \to \PKstar \mu \mu$.
  377. \item The method converge fast and works for the "simple case", i.e. signal only.
  378. \item Method very insensitive to S-wave component, thanks to orthogonality. 
  379. \item Complementary one can measure in-depended S-wave component.
  380. \item No problem with boundary problems.
  381.  
  382. \end{itemize}
  383. What comes in the next talks(stay tuned):
  384. \begin{itemize}
  385. \item Sensitivity will be given tmr.
  386. \item This method reduces the error on unfolding.
  387. \item Systematics easy accessible.
  388. \end{itemize}
  389.  
  390.  
  391. \end{frame}
  392. %%%%%%%%%%%%%%%%%%%%%%%%%55
  393. \begin{frame}\frametitle{~}
  394. {~}
  395. \center \Huge BACKUPS
  396.  
  397. \end{frame}
  398.  
  399. \begin{frame}
  400. {~}
  401.  
  402. \begin{columns}
  403. \column{2.5in}
  404. \includegraphics[scale=0.17]{plots/pool_plots/J1c_50.png}\\
  405. \includegraphics[scale=0.17]{plots/pool_plots/J1c_75.png}\\
  406.  
  407.  
  408. \column{2.5in}
  409. \includegraphics[scale=0.17]{plots/pool_plots/J1c_50E.png}\\
  410. \includegraphics[scale=0.17]{plots/pool_plots/J1c_75E.png}
  411.  
  412.  
  413. \end{columns}
  414. \end{frame}
  415.  
  416. \begin{frame}
  417. {~}
  418.  
  419. \begin{columns}
  420. \column{2.5in}
  421. \includegraphics[scale=0.17]{plots/pool_plots/J1c_125.png}\\
  422. \includegraphics[scale=0.17]{plots/pool_plots/J1c_175.png}\\
  423.  
  424.  
  425. \column{2.5in}
  426. \includegraphics[scale=0.17]{plots/pool_plots/J1c_125E.png}\\
  427. \includegraphics[scale=0.17]{plots/pool_plots/J1c_175E.png}
  428.  
  429.  
  430. \end{columns}
  431. \end{frame}
  432.  
  433. \begin{frame}
  434. {~}
  435.  
  436. \begin{columns}
  437. \column{2.5in}
  438. \includegraphics[scale=0.17]{plots/pool_plots/J1c_225.png}\\
  439. \includegraphics[scale=0.17]{plots/pool_plots/J1c_300.png}\\
  440.  
  441.  
  442. \column{2.5in}
  443. \includegraphics[scale=0.17]{plots/pool_plots/J1c_225E.png}\\
  444. \includegraphics[scale=0.17]{plots/pool_plots/J1c_300E.png}
  445.  
  446.  
  447. \end{columns}
  448. \end{frame}
  449.  
  450.  
  451. \begin{frame}
  452. {~}
  453.  
  454. \begin{columns}
  455. \column{2.5in}
  456. \includegraphics[scale=0.17]{plots/pool_plots/J1c_400.png}\\
  457.  
  458.  
  459.  
  460. \column{2.5in}
  461. \includegraphics[scale=0.17]{plots/pool_plots/J1c_400E.png}\\
  462.  
  463.  
  464.  
  465. \end{columns}
  466. \end{frame}
  467.  
  468. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
  469. \begin{frame}
  470. {~}
  471.  
  472. \begin{columns}
  473. \column{2.5in}
  474. \includegraphics[scale=0.17]{plots/pool_plots/J1s_50.png}\\
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  609. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  750.  
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  753. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  822.  
  823.  
  824. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  894.  
  895. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  964.  
  965. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  1034.  
  1035. %%%%%%%%%%%%%%%%%%%%%%%
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  1059.  
  1060. \column{2.5in}
  1061. \includegraphics[scale=0.2]{plots/J8J5.png}\\
  1062. \includegraphics[scale=0.2]{plots/J8J4.png}\\
  1063.  
  1064. \end{columns}
  1065.  
  1066. \end{frame}
  1067.  
  1068.  
  1069.  
  1070.  
  1071. % ------------------------------------------------------------------------------
  1072. \end{document}