{"id":7461,"date":"2022-04-14T15:25:00","date_gmt":"2022-04-14T13:25:00","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7461"},"modified":"2023-07-22T10:46:08","modified_gmt":"2023-07-22T08:46:08","slug":"les-carres-de-dirichlet","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/","title":{"rendered":"Les carr\u00e9s de Dirichlet"},"content":{"rendered":"\n<p>Les carr\u00e9s de Dirichlet constituent une famille de carr\u00e9s math\u00e9matiquement int\u00e9ressants.<\/p>\n\n\n\n<p>Nulle question ici de g\u00e9om\u00e9trie, mais plut\u00f4t d&#8217;arithm\u00e9tique&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Carres_de_Dirichlet_definition\" >Carr\u00e9s de Dirichlet: d\u00e9finition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Un_premier_resultat_sur_les_carres_de_Dirichlet\" >Un premier r\u00e9sultat sur les carr\u00e9s de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Unicite_du_carre_de_Dirichlet\" >Unicit\u00e9 du carr\u00e9 de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Une_methode_pour_completer_un_carre_de_Dirichlet\" >Une m\u00e9thode pour compl\u00e9ter un carr\u00e9 de Dirichlet<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_initiale_pour_la_resolution_des_carres_de_Dirichlet\" >\u00c9tape initiale pour la r\u00e9solution des carr\u00e9s de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_1\" >\u00c9tape 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_2\" >\u00c9tape 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Implementation_en_Python\" >Impl\u00e9mentation en Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Pourquoi_cette_methode_fonctionne\" >Pourquoi cette m\u00e9thode fonctionne ?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Epilogue\" >\u00c9pilogue<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Carres_de_Dirichlet_definition\"><\/span>Carr\u00e9s de Dirichlet: d\u00e9finition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>En une phrase, on pourrait d\u00e9finir un carr\u00e9 de Dirichlet comme une grille (carr\u00e9e) de nombres o\u00f9 chacun des nombres est la moyenne des nombres se trouvant au-dessus, en dessous, \u00e0 droite et \u00e0 gauche. Par exemple:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"120\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/07\/carre-dirichlet.png\" alt=\"\" class=\"wp-image-8634\"\/><\/figure>\n<\/div>\n\n\n<p>Le nombre &#8220;4&#8221; est la moyenne arithm\u00e9tique des quatre nombres 7, 3, 0 et 2:$$3 = \\frac{7+3+0+2}{4}.$$<\/p>\n\n\n\n<p>Bien entendu, vous vous en doutez, il y aura un l\u00e9ger soucis aux bords de la grille; c&#8217;est la raison pour laquelle figurent des nombres \u00e0 l&#8217;ext\u00e9rieur du carr\u00e9.<\/p>\n\n\n\n<p>Voici un exemple complet:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"187\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-8.png\" alt=\"carr\u00e9 Dirichlet\" class=\"wp-image-7465\"\/><figcaption class=\"wp-element-caption\">Exemple de carr\u00e9 de Dirichlet<\/figcaption><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{article}\n\\usepackage{array,cellspace}\n\\setlength{\\cellspacetoplimit}{4pt}\n\\setlength{\\cellspacebottomlimit}{4pt}\n\n\\begin{document}\n\n\\begin{tabular}{c|Sc|Sc|Sc|c}\n\\multicolumn{1}{c}{} &amp; \\multicolumn{1}{c}{7} &amp; \\multicolumn{1}{c}{9} &amp; \\multicolumn{1}{c}{9} &amp;  \\\\\\cline{2-4}\n0 &amp; 5 &amp; 7 &amp; 8 &amp; 9 \\\\\\cline{2-4}\n9&amp; 6 &amp; 6 &amp; 7 &amp; 8 \\\\\\cline{2-4}\n0 &amp; 4 &amp; 4 &amp; 6 &amp; 4 \\\\\\cline{2-4}\n\\multicolumn{1}{c}{} &amp; \\multicolumn{1}{c}{6} &amp; \\multicolumn{1}{c}{0} &amp; \\multicolumn{1}{c}{9} &amp; \\\\\n\\end{tabular}\n\n\\end{document}\n<\/pre>\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_premier_resultat_sur_les_carres_de_Dirichlet\"><\/span>Un premier r\u00e9sultat sur les carr\u00e9s de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>Tous les nombres \u00e0 l&#8217;int\u00e9rieur du carr\u00e9 de Dirichlet sont inf\u00e9rieurs ou \u00e9gaux au maximum de ceux qui se trouvent \u00e0 l&#8217;ext\u00e9rieur.<\/p><\/blockquote><\/figure>\n\n\n\n<p>Si l&#8217;on regarde l&#8217;exemple pr\u00e9c\u00e9dent, le maximum des nombres se trouvant \u00e0 l&#8217;ext\u00e9rieur est \u00e9gal \u00e0 9. On constate bien que tous les nombre \u00e0 l&#8217;int\u00e9rieur du carr\u00e9 sont inf\u00e9rieurs ou \u00e9gaux \u00e0 9.<\/p>\n\n\n\n<p>Pourquoi est-ce vrai ? On peut raisonner par l&#8217;absurde. Supposons que le maximum soit \u00e0 l&#8217;int\u00e9rieur, et notons-le &#8220;M&#8221;. Pour plus de facilit\u00e9 dans les explications, je vais d\u00e9signer par &#8220;les nombres autour&#8221; ceux qui se trouvent au-dessus, en dessous, \u00e0 droite et \u00e0 gauche.<\/p>\n\n\n\n<p>Alors, comme moyenne de quatre nombres, M est inf\u00e9rieur ou \u00e9gal aux nombres qui l&#8217;entourent. Mais comme il est le maximum de la grille, il est n\u00e9cessairement \u00e9gal \u00e0 tous les nombres qui l&#8217;entourent. On se retrouve donc localement avec un sch\u00e9ma comme celui-ci:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"172\" height=\"121\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-9.png\" alt=\"\" class=\"wp-image-7467\"\/><\/figure>\n<\/div>\n\n\n<p>Par un raisonnement analogue sur les autres cases (qui contiennent M), on arrive \u00e0 voir que n\u00e9cessairement, le carr\u00e9 de Dirichlet ne comporterait que des M.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Unicite_du_carre_de_Dirichlet\"><\/span>Unicit\u00e9 du carr\u00e9 de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>\u00c9tant donn\u00e9s des nombres \u00e0 l&#8217;ext\u00e9rieur du carr\u00e9 de Dirichlet, il n&#8217;existe qu&#8217;une seule fa\u00e7on de le compl\u00e9ter.<\/p><\/blockquote><\/figure>\n\n\n\n<p>Pour s&#8217;en convaincre, on va consid\u00e9rer deux carr\u00e9s de Dirichlet d&#8217;ordre 2 ayant les m\u00eames nombres ext\u00e9rieurs, mais pas n\u00e9cessairement les m\u00eames nombres int\u00e9rieurs:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"502\" height=\"148\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10.png\" alt=\"carr\u00e9s de Dirichlet\" class=\"wp-image-7468\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10.png 502w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10-300x88.png 300w\" sizes=\"auto, (max-width: 502px) 100vw, 502px\" \/><\/figure>\n<\/div>\n\n\n<p>Consid\u00e9rons maintenant un autre carr\u00e9 de m\u00eames dimensions o\u00f9 tous les nombres sont les diff\u00e9rences de ceux du deuxi\u00e8me et ceux du premier :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"275\" height=\"148\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-11.png\" alt=\"\" class=\"wp-image-7469\"\/><\/figure>\n<\/div>\n\n\n<p>Souvenons-nous maintenant que nous avions les \u00e9galit\u00e9s suivantes:$$\\begin{cases}x = \\frac{A+B+y+z}{4}\\\\x&#8217;=\\frac{A+H+y&#8217;+z&#8217;}{4}\\end{cases}$$Donc:$$x-x&#8217;=\\frac{0 + 0 + (y-y&#8217;) + (z-z&#8217;)}{4}.$$<\/p>\n\n\n\n<p>Donc \\(x-x&#8217;\\) est la moyenne des quatre nombres qui l&#8217;entourent dans le dernier carr\u00e9.<\/p>\n\n\n\n<p>Il en est de m\u00eame pour les trois autres nombres int\u00e9rieurs de ce dernier carr\u00e9. Donc c&#8217;est un carr\u00e9 de Dirichlet.<\/p>\n\n\n\n<p>D&#8217;apr\u00e8s le &#8220;principe du maximum&#8221; vu dans la section pr\u00e9c\u00e9dente, cela signifie que &#8220;0&#8221; est le maximum de tous les nombres de ce carr\u00e9, et donc que \\(x=x&#8217;\\), \\(y=y&#8217;\\), \\(z=z&#8217;\\) et \\(t=t&#8217;\\).<\/p>\n\n\n\n<p>Cela montre donc que la solution d&#8217;un carr\u00e9 de Dirichlet est unique.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_methode_pour_completer_un_carre_de_Dirichlet\"><\/span>Une m\u00e9thode pour compl\u00e9ter un carr\u00e9 de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_initiale_pour_la_resolution_des_carres_de_Dirichlet\"><\/span>\u00c9tape initiale pour la r\u00e9solution des carr\u00e9s de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Nous allons chercher \u00e0 compl\u00e9ter le carr\u00e9 de Dirichlet suivant :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"208\" height=\"187\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-12.png\" alt=\"\" class=\"wp-image-7471\"\/><\/figure>\n<\/div>\n\n\n<p>J&#8217;ai ins\u00e9r\u00e9 des &#8220;0&#8221; \u00e0 l&#8217;int\u00e9rieur par d\u00e9faut, mais ils vont tr\u00e8s vite dispara\u00eetre.<\/p>\n\n\n\n<p>L&#8217;id\u00e9e est de se dire que ceci n&#8217;est que l&#8217;\u00e9tape initiale d&#8217;un (long) processus. Appelons-la l&#8217;\u00e9tape 0.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_1\"><\/span>\u00c9tape 1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>L&#8217;\u00e9tape suivante consiste \u00e0 transformer le premier &#8220;0&#8221; (en haut \u00e0 gauche) en la moyenne des nombres qui l&#8217;entourent.<\/p>\n\n\n\n<p>Il devient alors (6 + 5 + 0 + 0)\/4 = 2,75.<\/p>\n\n\n\n<p>On passe alors au &#8220;0&#8221; qui se trouve \u00e0 sa droite, qui se transforme en (0 + 2,75 + 0 + 0)\/4 = 0,5375.<\/p>\n\n\n\n<p>Ensuite, on passe au &#8220;0&#8221; \u00e0 sa droite, qui devient : (4 + 0,5375 + 0 + 5)\/4.<\/p>\n\n\n\n<p>On parcourt ainsi toute la grille de haut en bas, de gauche \u00e0 droite.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_2\"><\/span>\u00c9tape 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Si les nombres int\u00e9rieurs obtenus \u00e0 l&#8217;\u00e9tape pr\u00e9c\u00e9dente ne sont pas tous \u00e9gaux \u00e0 la moyenne des nombres qui les entourent, on recommence&#8230; jusqu&#8217;\u00e0 obtenir ce que l&#8217;on veut.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation_en_Python\"><\/span>Impl\u00e9mentation en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">square = [[0,  6,  0,  4, 0],\n     [5, 0, 0, 0, 5],\n     [5, 0, 0, 0, 5],\n     [3, 0, 0, 0, 4],\n     [0, 2, 2, 3, 0]]\n\ndef dirichlet_square_solver():\n    end = True\n    for line in range(1,len(square)-1):\n        for col in range(1,len(square)-1):\n            temp = square[line][col]\n            square[line][col] = ( square[line-1][col] + square[line][col-1] + square[line][col+1] + square[line+1][col] ) \/ 4\n            if square[line][col] != temp:\n                end = False\n    \n    if not end:\n        dirichlet_square_solver()\n    else:\n        c = ''\n        for line in range(len(square)):\n            for col in range(len(square)):\n                c += '{:^5}'.format(square[line][col])\n            c += '\\n'\n            \n        print(c)\n            \ndirichlet_square_solver()\n<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>  0    6    0    4    0  \n  5   4.5  3.0  4.0   5  \n  5   4.0  3.5  4.0   5  \n  3   3.0  3.0  3.5   4  \n  0    2    2    3    0  <\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Pourquoi_cette_methode_fonctionne\"><\/span>Pourquoi cette m\u00e9thode fonctionne ?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Notons \\(x^{(k)}_n\\) les suites de nombres d\u00e9finies ainsi:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"296\" height=\"228\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-13.png\" alt=\"\" class=\"wp-image-7473\"\/><\/figure>\n<\/div>\n\n\n<p>\\(x^{(k)}_1\\) est la moyenne des nombres qui l&#8217;entourent, donc sa valeur sera n\u00e9cessairement plus grande que 0. Donc \u00e0 la fin de l&#8217;\u00e9tape 1, toutes les valeurs int\u00e9rieures seront plus grandes que les valeurs initiales.<\/p>\n\n\n\n<p>On comprend alors qu&#8217;\u00e0 la fin de l&#8217;\u00e9tape <em>n<\/em>, toutes les valeurs de \\(x^{(k)}_n\\) seront sup\u00e9rieures \u00e0 celles de l&#8217;\u00e9tape pr\u00e9c\u00e9dente et ainsi de suite (un raisonnement par r\u00e9currence peut nous en convaincre). <\/p>\n\n\n\n<p>Les suites \\( (x^{(k)}_n) \\) sont donc croissantes. De plus, elles sont toutes major\u00e9es (par le maximum, qui est \u00e0 l&#8217;ext\u00e9rieur du carr\u00e9). Donc, elles convergent vers des limites finies, n\u00e9cessairement la moyenne des nombres qui les entourent. Ainsi, le carr\u00e9 des valeurs limites est bien un carr\u00e9 de Dirichlet.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Epilogue\"><\/span>\u00c9pilogue<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Je me suis fortement inspir\u00e9 de la <a href=\"https:\/\/video.math.cnrs.fr\/carres-magiques-de-dirichlet\/\" target=\"_blank\" rel=\"noreferrer noopener\">vid\u00e9o<\/a> d&#8217;Olivier Druet, directeur de recherches au CNRS (Institut Camille Jordan, Universit\u00e9 Lyon 1).<\/p>\n\n\n\n<p>Vous comprendrez, \u00e0 travers l&#8217;introduction cette vid\u00e9o, l&#8217;histoire de ce probl\u00e8me, qui n&#8217;est autre qu&#8217;une histoire de temp\u00e9ratures.<\/p>\n\n\n\n<p>En effet, supposons que la grille initiale (o\u00f9 l&#8217;on ne conna\u00eet que les nombres ext\u00e9rieurs) repr\u00e9sente un groupe de pi\u00e8ces et que les nombres ext\u00e9rieurs repr\u00e9sentent la temp\u00e9rature dans chacune des pi\u00e8ces &#8220;ext\u00e9rieures&#8221;. Alors, le carr\u00e9 de Dirichlet repr\u00e9sente les temp\u00e9ratures de chacune des pi\u00e8ces une fois que toutes les pi\u00e8ces aient une temp\u00e9rature constante, moyenne des temp\u00e9ratures des pi\u00e8ces qui l&#8217;entourent avec lesquelles elle a un mur en commun&#8230; Ouais, je sais, je n&#8217;explique pas super bien&#8230; les transferts thermiques ne sont clairement pas ma sp\u00e9cialit\u00e9 !&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les carr\u00e9s de Dirichlet constituent une famille de carr\u00e9s math\u00e9matiquement int\u00e9ressants. Nulle question ici de g\u00e9om\u00e9trie, mais plut\u00f4t d&#8217;arithm\u00e9tique&#8230;<\/p>\n","protected":false},"author":1,"featured_media":7479,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,6,5],"tags":[341,190,101],"class_list":["post-7461","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-mathematiques","category-python","tag-dirichlet","tag-moyenne","tag-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les carr\u00e9s de Dirichlet - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Que sont les carr\u00e9s de Dirichlet ? Et comment les compl\u00e9ter quand on connait les nombres ext\u00e9rieurs ? 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