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Patrick@275
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<?xml version="1.0" encoding="utf-8"?>
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Patrick@275
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<publidoc version="1.0">
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Patrick@275
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<topic id="maths" xml:lang="fr">
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Patrick@275
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<head>
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Patrick@275
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<title>Quelques formules mathématiques</title>
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Patrick@281
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<subjectset>
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Patrick@281
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<subject>Mathématiques</subject>
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Patrick@281
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</subjectset>
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Patrick@275
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</head>
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Patrick@275
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Patrick@275
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<section>
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Patrick@275
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<head>
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Patrick@275
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<title>Formule native</title>
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Patrick@275
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</head>
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Patrick@277
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<section>
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Patrick@277
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<head><title>Le dernier théorème de Fermat</title></head>
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Patrick@277
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<p>
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Patrick@277
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Il n'existe pas de nombres entiers non nuls <var>x</var>,
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Patrick@277
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<var>y</var> et <var>z</var> tels que :
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Patrick@280
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<math xml:id="fermat" display="wide">
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Patrick@277
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<var>x</var><sup>n</sup> + <var>y</var><sup>n</sup> = <var>z</var><sup>n</sup>
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Patrick@277
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</math>
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Patrick@277
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dès que <var>n</var> est un entier strictement supérieur à 2.
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Patrick@277
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</p>
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Patrick@277
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</section>
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Patrick@275
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</section>
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Patrick@430
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Patrick@275
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<section>
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Patrick@275
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<head>
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Patrick@308
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<title>Formule en LaTeX</title>
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Patrick@275
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</head>
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Patrick@308
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<section>
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Patrick@308
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<head>
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Patrick@308
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<title>Formule dans le texte</title>
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Patrick@308
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</head>
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Patrick@308
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<p>
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Patrick@308
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On rappelle que la <highlight>moyenne</highlight> de <var>X</var> est
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Patrick@435
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le nombre : <math type="important"><latex>\overline X =
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Patrick@308
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\frac{1}{n}\left( {n_1 x_1 + n_2 x_2 + \dots + n_p x_p } \right)</latex></math>.
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Patrick@308
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</p>
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Patrick@308
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<p>
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Patrick@308
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On appelle <highlight>variance</highlight> de la série
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Patrick@308
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statistique <var>X</var>, le nombre :
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Patrick@308
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<math><latex>V\left( X \right) =
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Patrick@430
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\frac{1}{n}\left( {n_1 \left( {x_1 - \overline X } \right)^2
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Patrick@308
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+ \dots + n_p \left( {x_p - \overline X } \right)^2 }
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Patrick@308
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\right)</latex></math> qu'on réécrit ainsi :
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Patrick@308
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<math><latex>V\left( X \right) =
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Patrick@308
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\frac{1}{n}\sum\limits_{i = 1}^p {n_i \left( {x_i - \overline X }
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Patrick@308
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\right)^2 }</latex></math>.
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Patrick@308
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</p>
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Patrick@308
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<p>
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Patrick@308
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L'<highlight>écart type</highlight> de <var>X</var> est le nombre :
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Patrick@308
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<math><latex>
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Patrick@431
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{\textrm{s}}\left(X\right) = \sqrt{V\left(X\right)}
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Patrick@308
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</latex></math>
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Patrick@308
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</p>
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Patrick@308
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</section>
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Patrick@430
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Patrick@308
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<section>
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Patrick@308
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<head>
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Patrick@308
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<title>Formule mise en évidence</title>
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Patrick@308
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</head>
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Patrick@308
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<p>
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Patrick@308
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Soit la fonction :
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Patrick@308
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<math xml:id="fonction" display="wide">
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Patrick@308
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<latex>f(x) = x^2 + \sqrt[3]{\frac{3x}{2y-3}}</latex>
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Patrick@308
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</math>
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Patrick@308
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</p>
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Patrick@308
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</section>
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Patrick@430
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Patrick@308
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<section>
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Patrick@308
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<head>
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Patrick@308
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<title>Formule encadrée</title>
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Patrick@308
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</head>
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Patrick@308
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<p>
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Patrick@308
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Les transformations de Lorentz :
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Patrick@308
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<math xml:id="lorentz" display="box">
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Patrick@308
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<latex plain="true">
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Patrick@308
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\begin{eqnarray*}
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Patrick@308
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ct' & = & \frac{ct-(v/c)x}{\sqrt{1-v^2/c^2}},\\
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Patrick@308
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x' & = & \frac{x-(v/c)ct}{\sqrt{1-v^2/c^2}},\\
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Patrick@308
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y' & = & y,\\
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Patrick@308
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z' & = & z.
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Patrick@308
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\end{eqnarray*}
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Patrick@308
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</latex>
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Patrick@308
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</math>
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Patrick@308
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</p>
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Patrick@308
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</section>
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Patrick@430
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Patrick@308
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<section>
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Patrick@308
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<head>
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Patrick@308
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<title>Formules numérotées</title>
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Patrick@308
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</head>
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Patrick@308
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<p>
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Patrick@308
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Une expression matricielle :
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Patrick@308
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<math display="numbered">
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Patrick@308
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<latex>
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Patrick@308
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\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] =
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Patrick@308
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\begin{bmatrix} A & B \\ C & D \end{bmatrix}
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Patrick@308
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\times \left[\begin{array}{c} y_1 \\ y_2 \end{array}\right]
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Patrick@308
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</latex>
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Patrick@308
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</math>
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Patrick@308
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</p>
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Patrick@308
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<p>
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Patrick@308
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Une intégrale :
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Patrick@308
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<math display="numbered">
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Patrick@308
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<latex>\int_0^1 \frac{1}{\sqrt{-\ln x}} \; \mathrm dx</latex>
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Patrick@308
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</math>
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Patrick@308
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</p>
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Patrick@308
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</section>
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Patrick@430
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Patrick@308
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<section>
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Patrick@308
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<head>
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Patrick@308
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<title>Formule numérotée et encadrée</title>
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Patrick@308
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</head>
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Patrick@308
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<p>
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Patrick@308
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Les harmoniques sphériques sont des fonctions définies sur la
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Patrick@308
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sphère. Ce sont les fonctions propres du laplacien bidimensionnel à
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Patrick@308
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symétrie sphérique,
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Patrick@308
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<math display="numbered-box">
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Patrick@308
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<latex>
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Patrick@308
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\left({\partial^2\over\partial\theta^2}
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Patrick@308
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+\cot\theta{\partial\over\partial\theta}
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Patrick@308
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+{1\over\sin^2\theta}{\partial^2\over\partial\varphi^2}\right)Y_\ell^m
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Patrick@308
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= -\ell(\ell+1)Y_\ell^m\ .
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Patrick@308
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</latex>
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Patrick@308
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</math>
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Patrick@308
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Les harmoniques sphériques forment un ensemble complet de fonctions des
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Patrick@308
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angles <math><latex>\theta</latex></math> et
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Patrick@308
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<math><latex>\varphi</latex></math> de sorte que toute fonction sur la
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Patrick@308
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sphère peut se décomposer sur une base d'harmoniques sphériques.
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Patrick@308
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</p>
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Patrick@308
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</section>
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iinov@460
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iinov@460
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<section>
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iinov@460
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<head>
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iinov@460
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<title>Formule avec préambule</title>
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iinov@460
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</head>
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iinov@460
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<section xml:lang="en">
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iinov@460
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<p>
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iinov@460
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Let <math><latex>I_+</latex></math> denote the ideal generated by
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iinov@460
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the <math><latex>S_n</latex></math>-invariant homogeneous
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iinov@460
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polynomials of positive degree in <math>
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iinov@460
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<preambule><newcommand name="C">{\mathbb{C}}</newcommand></preambule>
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iinov@460
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<latex>\C[x_1,\dots, x_n,y_1,\dots, y_n]</latex></math> and set
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iinov@460
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</p>
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iinov@460
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<p>
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iinov@460
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<math display="wide">
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iinov@460
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<preambule><newcommand name="C">{\mathbb{C}}</newcommand></preambule>
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iinov@460
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<latex>R_n :=\C[\mathbf{x},\mathbf{y}] / I_+.</latex>
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iinov@460
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</math>
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iinov@460
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</p>
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iinov@460
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<p>
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iinov@460
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It is known for sometime that the bi-graded Frobenius character of
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iinov@460
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<math><latex>R_n</latex></math> is given by the transformation of
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iinov@460
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the elementary symmetric function <math><latex>e_{n}</latex></math>
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iinov@460
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under Bergeron-Garsia's "nabla" operator, <math><latex>\nabla
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iinov@460
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e_n</latex></math>. In other words, the symmetric function
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iinov@460
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<math><latex>\nabla e_n</latex></math> has an underlying
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iinov@460
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<math><latex>S_{n}</latex></math>-representation. Roughly stated,
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iinov@460
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<math><latex>\nabla</latex></math> is a
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iinov@460
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<math>
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iinov@460
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<preambule>
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iinov@460
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<newcommand name="mbb">[1]{\mathbb{#1}}</newcommand></preambule>
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iinov@460
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<latex>\mbb{Q}</latex></math>-linear operator defined on the ring
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iinov@460
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of symmetric functions <math><latex>\Lambda</latex></math> in
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iinov@460
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such a way that the modified Macdonald symmetric functions are
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iinov@460
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the eigenfunctions of <math><latex>\nabla</latex></math> with
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iinov@460
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prescribed eigenvalues.
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iinov@460
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</p>
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iinov@460
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</section>
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iinov@460
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</section>
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Patrick@275
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</section>
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Patrick@275
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</topic>
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Patrick@275
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</publidoc>
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