From 700f6d235b32c1b81106c130e2b7f719fe082777 Mon Sep 17 00:00:00 2001 From: oskar Date: Sat, 2 May 2026 21:34:24 +0200 Subject: [PATCH] MacBookPro.lan 2026-5-2:21:34:24 --- désintégration audioactive.md | 157 +++++++++++++++++++++++++++++++++- 1 file changed, 155 insertions(+), 2 deletions(-) diff --git a/désintégration audioactive.md b/désintégration audioactive.md index b71ea19f..07af82bb 100644 --- a/désintégration audioactive.md +++ b/désintégration audioactive.md @@ -534,9 +534,162 @@ Conway leur donne des noms d'éléments (de l'hydrogène à l'uranium, ce qui fa > [!proposition]+ Calcul de la valeur de $\lambda$ -> L +> L'annexe 1 fournit le code permettant (entre autres) une approximation de $\lambda$. +> Il est évident que $\lambda$ est un nombre algébrique de degré $92$. + +> [!proposition]+ Théorème cosmologique +> Toute chaine (autre que $[\;]$ et $[22]$) finit, après assez de dérivations, par # Annexes -## Code source utilisé pour le calcul +## Annexe 1 - Code source utilisé pour le calcul +```python +import numpy as np + + +# Liste de dictionnaires représentant les éléments +ELEMENTS = [{"num": 1, "name": "H", "deriv": ["H"]}, #{{{ + {"num": 2, "name": "He", "deriv": ["Hf","Pa","H","Ca","Li"]}, + {"num": 3, "name": "Li", "deriv": ["He"]}, + {"num": 4, "name": "Be", "deriv": ["Ge","Ca","Li"]}, + {"num": 5, "name": "B", "deriv": ["Be"]}, + {"num": 6, "name": "C", "deriv": ["B"]}, + {"num": 7, "name": "N", "deriv": ["C"]}, + {"num": 8, "name": "O", "deriv": ["N"]}, + {"num": 9, "name": "F", "deriv": ["O"]}, + {"num": 10, "name": "Ne", "deriv": ["F"]}, + {"num": 11, "name": "Na", "deriv": ["Ne"]}, + {"num": 12, "name": "Mg", "deriv": ["Pm", "Na"]}, + {"num": 13, "name": "Al", "deriv": ["Mg"]}, + {"num": 14, "name": "Si", "deriv": ["Al"]}, + {"num": 15, "name": "P", "deriv": ["Ho", "Si"]}, + {"num": 16, "name": "S", "deriv": ["P"]}, + {"num": 17, "name": "Cl", "deriv": ["S"]}, + {"num": 18, "name": "Ar", "deriv": ["Cl"]}, + {"num": 19, "name": "K", "deriv": ["Ar"]}, + {"num": 20, "name": "Ca", "deriv": ["K"]}, + {"num": 21, "name": "Sc", "deriv": ["Ho", "Pa", "H", "Ca", "Co"]}, + {"num": 22, "name": "Ti", "deriv": ["Sc"]}, + {"num": 23, "name": "V", "deriv": ["Ti"]}, + {"num": 24, "name": "Cr", "deriv": ["V"]}, + {"num": 25, "name": "Mn", "deriv": ["Cr", "Si"]}, + {"num": 26, "name": "Fe", "deriv": ["Mn"]}, + {"num": 27, "name": "Co", "deriv": ["Fe"]}, + {"num": 28, "name": "Ni", "deriv": ["Zn", "Co"]}, + {"num": 29, "name": "Cu", "deriv": ["Ni"]}, + {"num": 30, "name": "Zn", "deriv": ["Cu"]}, + {"num": 31, "name": "Ga", "deriv": ["Eu", "Ca","Ac","H","Ca","Zn"]}, + {"num": 32, "name": "Ge", "deriv": ["Ho", "Ga"]}, + {"num": 33, "name": "As", "deriv": ["Ge", "Na"]}, + {"num": 34, "name": "Se", "deriv": ["As"]}, + {"num": 35, "name": "Br", "deriv": ["Se"]}, + {"num": 36, "name": "Kr", "deriv": ["Br"]}, + {"num": 37, "name": "Rb", "deriv": ["Kr"]}, + {"num": 38, "name": "Sr", "deriv": ["Rb"]}, + {"num": 39, "name": "Y", "deriv": ["Sr", "U"]}, + {"num": 40, "name": "Zr", "deriv": ["Y","H","Ca","Tc"]}, + {"num": 41, "name": "Nb", "deriv": ["Er", "Zr"]}, + {"num": 42, "name": "Mo", "deriv": ["Nb"]}, + {"num": 43, "name": "Tc", "deriv": ["Mo"]}, + {"num": 44, "name": "Ru", "deriv": ["Eu", "Ca","Tc"]}, + {"num": 45, "name": "Rh", "deriv": ["Ho", "Ru"]}, + {"num": 46, "name": "Pd", "deriv": ["Rh"]}, + {"num": 47, "name": "Ag", "deriv": ["Pd"]}, + {"num": 48, "name": "Cd", "deriv": ["Ag"]}, + {"num": 49, "name": "In", "deriv": ["Cd"]}, + {"num": 50, "name": "Sn", "deriv": ["In"]}, + {"num": 51, "name": "Sb", "deriv": ["Pm", "Sn"]}, + {"num": 52, "name": "Te", "deriv": ["Eu", "Ca","Sb"]}, + {"num": 53, "name": "I", "deriv": ["Ho", "Te"]}, + {"num": 54, "name": "Xe", "deriv": ["I"]}, + {"num": 55, "name": "Cs", "deriv": ["Xe"]}, + {"num": 56, "name": "Ba", "deriv": ["Cs"]}, + {"num": 57, "name": "La", "deriv": ["Ba"]}, + {"num": 58, "name": "Ce", "deriv": ["La", "H","Ca","Co"]}, + {"num": 59, "name": "Pr", "deriv": ["Ce"]}, + {"num": 60, "name": "Nd", "deriv": ["Pr"]}, + {"num": 61, "name": "Pm", "deriv": ["Nd"]}, + {"num": 62, "name": "Sm", "deriv": ["Pm", "Ca","Zn"]}, + {"num": 63, "name": "Eu", "deriv": ["Sm"]}, + {"num": 64, "name": "Gd", "deriv": ["Eu", "Ca","Co"]}, + {"num": 65, "name": "Tb", "deriv": ["Ho", "Gd"]}, + {"num": 66, "name": "Dy", "deriv": ["Tb"]}, + {"num": 67, "name": "Ho", "deriv": ["Dy"]}, + {"num": 68, "name": "Er", "deriv": ["Ho", "Pm"]}, + {"num": 69, "name": "Tm", "deriv": ["Er", "Ca","Co"]}, + {"num": 70, "name": "Yb", "deriv": ["Tm"]}, + {"num": 71, "name": "Lu", "deriv": ["Yb"]}, + {"num": 72, "name": "Hf", "deriv": ["Lu"]}, + {"num": 73, "name": "Ta", "deriv": ["Hf", "Pa","H","Ca","W"]}, + {"num": 74, "name": "W", "deriv": ["Ta"]}, + {"num": 75, "name": "Re", "deriv": ["Ge", "Ca","W"]}, + {"num": 76, "name": "Os", "deriv": ["Re"]}, + {"num": 77, "name": "Ir", "deriv": ["Os"]}, + {"num": 78, "name": "Pt", "deriv": ["Ir"]}, + {"num": 79, "name": "Au", "deriv": ["Pt"]}, + {"num": 80, "name": "Hg", "deriv": ["Au"]}, + {"num": 81, "name": "Tl", "deriv": ["Hg"]}, + {"num": 82, "name": "Pb", "deriv": ["Tl"]}, + {"num": 83, "name": "Bi", "deriv": ["Pm", "Pb"]}, + {"num": 84, "name": "Po", "deriv": ["Bi"]}, + {"num": 85, "name": "At", "deriv": ["Po"]}, + {"num": 86, "name": "Rn", "deriv": ["Ho", "At"]}, + {"num": 87, "name": "Fr", "deriv": ["Rn"]}, + {"num": 88, "name": "Ra", "deriv": ["Fr"]}, + {"num": 89, "name": "Ac", "deriv": ["Ra"]}, + {"num": 90, "name": "Th", "deriv": ["Ac"]}, + {"num": 91, "name": "Pa", "deriv": ["Th"]}, + {"num": 92, "name": "U", "deriv": ["Pa"]}] +#}}} + +# dictionnaire numéro --> élément +# permet de retrouver un élément par son numéro atomique +NUM_OF = {elt["name"]: elt["num"] for elt in ELEMENTS} + +##################################################### +# CRÉATION DE LA MATRICE DES DÉRIVATIONS D'ÉLÉMENTS # +##################################################### +# initialisation +matrix = [[0 for _ in range(len(ELEMENTS))] for _ in range(len(ELEMENTS))] +# remplissage +for elt in ELEMENTS: + num = elt["num"] + for dv_elt in elt["deriv"]: + matrix[num-1][NUM_OF[dv_elt]-1] += 1 +# conversion en tableau numpy +matrix = np.array(matrix) + +# AFFICHER LA MATRICE +print("Matrice 𝑀 :") +print("┏", "━"*92, "┓\n┃", + "┃\n┃".join([''.join([" 12│12─12┼12"[val + 3*(9==col%10) + 6*(9==ln%10)] for col, val in enumerate(line)]) for ln, line in enumerate(matrix)]), + "┃\n┗", "━"*92, "┛", + sep="") + +# CALCUL DE λ +eigvals = np.linalg.eigvals(matrix) +λ = eigvals[np.argmax(np.abs(eigvals))] +print(f"λ = {np.real(λ)} + {np.imag(λ)}𝑖") + +############################################################ +# CALCUL DE LA PLUS PETITE BORNE POUR LE THÉORÈME CHIMIQUE # +############################################################ +def nb_derivations_engendre_tous_les_elements(elt_num: int): + """compte le nombre de dérivations nécessaires pour que l'élément numéro elt_num engendre tous + les autres""" + counts = np.zeros((92,)) + counts[elt_num-1] = 1 + derivations_count = 0 + while np.min(counts) == 0: + counts = counts @ matrix + derivations_count += 1 + return derivations_count + + +min_steps_required = 0 +for elt_num in range(2, 93): + min_steps_required = max(min_steps_required, + nb_derivations_engendre_tous_les_elements(elt_num)) +print("Borne pour le théorème chimique :", min_steps_required) +```