For a radix of 10, I will show that there is no number with a multiplicative persistence ≥ 12 . Résumé: Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.
Les systèmes non linéaires sont plus difficiles à étudier que les systèmes linéaires. Néanmoins, en linéarisant (cas de systèmes linéarisable) un SNL, autour d'un point A de considération finie (situation ou état du système), on obtient un système linéaire qui correspond à une approximation grossière du système non linéaire d’origine.
Cette approche a atteint sa maturité dans le livre de H.W.Bode (1905-1982) à la fin de la IIème guerre mondiale. Les travaux de R.E.Bellman (1920-1984), L.S.Pontryagin et al (1908-1988) surtout de R.Kalman (1930) ont conduit nombre d'automaticiens à privilégier la représentation d‘espace d’état à partir des années 1960.
Un système est non linéaire s’il se comporte non linéairement par rapport à ses composantes intrinsèques.
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Classification. Classification is an important data mining technique based on machine learning which is used to
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advantages and disadvantages
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hardware and software and other resources & applications that can manage themself with high level of human
guidance. Virtualization or Autonomic computing is not a new to the world, but it developed rapidly with Cloud
computing. In this paper there give an overview of various types of computing. There will be discussion on
Cluster, Grid computing, Utility & Cloud Computing. Analysis architecture, differences between them,
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Alternative - Complément au Tramway et 3ème lien de la ville de Québec Daniel Bedard
An update of this presentation has been done with Slide 16 that has been updated and 17 has been added, only.
Cette présentation a été ajournée avec la diapo 16 qui a été modifié et la 17 qui a été ajouté.
Voir ici
https://www.slideshare.net/slideshow/alternative-au-tramway-de-la-ville-de-quebec-rev1-sum-pdf/269691794
CDPQ Infra dévoile un plan de mobilité de 15 G$ sur 15 ans pour la région de Québec. Une alternative plus économique et rapide, ne serait-elle pas posssible?
- Valoriser les infrastructures ferroviaires du CN, en créant un Réseau Express Métropolitain (REM) plutôt qu'un nouveau tramway ou une combinaison des 2.
- Optimiser l'utilisation des rails pour un transport combiné des marchandises et des personnes, en accordant une priorité aux déplacements des personnes aux heures de pointes.
- Intégrer un téléphérique transrives comme 3ème lien urbain dédiés aux piétons et cyclistes avec correspondance avec le REM.
- Le 3 ème lien routier est repensé en intégrant un tunnel routier qui se prolonge avec le nouveau pont de l'Île d'Orléans et quelques réaménagemet de ses chausées.
https://www.linkedin.com/in/bedarddaniel/
English:
CDPQ Infra unveils a $15 billion, 15-year mobility plan for the Quebec region. Wouldn't a more economical and faster alternative be possible?
Leverage CN's railway infrastructure by creating a Metropolitan Express Network (REM) instead of a new tramway or a combination of both.
Optimize the use of rails for combined freight and passenger transport, giving priority to passenger travel during peak hours.
Integrate a cross-river cable car as a third urban link dedicated to pedestrians and cyclists, with connections to the REM.
Rethink the third road link by integrating a road tunnel that extends with the new Île d'Orléans bridge and some reconfiguration of its lanes.
https://www.linkedin.com/in/bedarddaniel/
1. IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 19, Issue 1, Ver. I (Jan.-Feb. 2017), PP 91-94
www.iosrjournals.org
DOI: 10.9790/0661-1901019194 www.iosrjournals.org 91 | Page
Brisure De Symétrie Et Nombres De Lychrel
Latifa Ait abida, Aymane Sghiar, Iliès sghiar, M. sghiar
9 Allée capitaine J. B. Bossu, 21240, Talant, France
Abstract: We show that the existence of the numbers of lychrel is due to a symmetry breaking in the space of
numbers.
Résumé: Nous montrons que l’existence des nombres de lychrel est due à une brisure de symétrie dans l’espace
des nombres.
Keywords: Nombre de lychrel, Théorie des nombres, conjecture de Syracuse.
I. Introduction
Un nombre de Lychrel est un nombre naturel qui ne peut pas former de nombre palindrome lorsqu'il
est soumis au processus itératif qui consiste à l'additionner au nombre formé de l'inversion de ses chiffres en
base 10. Le nom «Lychrel» a été inventé par Wade VanLandingham : il s'agit d'une quasi-anagramme du nom de
sa fiancée, Cheryl. Les nombres de Lychrel étaient considérés comme des nombres théoriques. On n'en
connaît aucun, bien que de nombreux nombres soient suspectés. Le plus petit nombre suspecté d'être de Lychrel
est 196. C'est donc lui qui a attiré le plus d'attention. John Walker a commencé la recherche le 12 août 1987
avec un programme écrit en C qui effectuait les itérations et qui vérifiait si le nombre était un palindrome. Cela
dura presque 3 ans, et finit (comme cela lui était demandé) le 24 mai 1990 avec le message suivant :
Point d'arrêt atteint après 2 415 836 itérations.
Le nombre possède 1 000 000 chiffres. Ainsi 196 a donc atteint un nombre à un million de chiffres
après 2 415 836 itérations sans parvenir à un palindrome. Walker publia ses recherches et en 1995, Tim Irvin
utilisa un superordinateur et atteint un nombre de 2 millions de chiffres en seulement 3 mois sans trouver de
palindrome. Jason Doucette poursuivit la lancée et atteint 12,5 millions de chiffres en mai 2000. Wade
VanLandingham utilisa le programme de Jason Doucette pour atteindre 13 millions de chiffres, un record publié
dans Yes Mag, magazine des sciences pour enfant au Canada. Le 1er mai 2006, VanLandingham atteignit la
barre des 300 millions de chiffres (avec une moyenne d'un million de chiffres tous les 5 ou 7 jours). Malgré cela
aucun palindrome n'a été trouvé.Voir: Jean Paul Delahaye, http://www.lifl.fr/~jdelahay/IAGL/Conjectures.pdf et
http://www.p196.org/ . D'autres candidats comme 879, 1997 et 7059 ont aussi été longuement testés par la
même méthode de brute force en atteignant aussi plusieurs millions d'itérations sans trouver le moindre
palindrome.
Ce problème évoque la fameuse conjecture de Syracuse, nommée aussi problème de Collatz démontré dans
[M. Sghiar] : Dans cette conjecture, N est divisé par 2 s’il est pair, et remplacé par 3N+ 1 s’il est impair. Ce
procédé finit par donner 1.
Par des techniques quasi similaires à la preuve de la conjecture de Syracuse [M. Sghiar], le But de cette
article est de montrer que les nombres de lychrel existent bel et bien, et que leur existence est due à une brisure
de symétrie.
II. Idée De La Preuve
L’idée de la preuve est inspirée de la brisure de symétrie qui joue un rôle important en physique.
on va assimiler les nombres entiers à des particules d’un ensemble ℕ que l’on complète par un ensemble
^ℕ où toute particule a une particule symétrique. Dans
^ℕ il va régner un équilibre stable due à la symétrie de
^ℕ . Et par suite ℕ sera vue comme l’ensemble des particules après la brisure de symétrie.
Cette brisure de symétrie va rendre l’ensemble des particules instables et causera une sorte d’ explosion qui va
éjecter tous les éléments symétriques des particules de
^ℕ∖ ℕ qu’on notera
(^ℕ ∖ ℕ )s
.
On remarque que le pourcentage des éléments de
(^ℕ ∖ ℕ )s
dans l’ensemble ℕ est de 10/100 , ce qui
correspondra à la moyenne des éléments des nombres de lychrel.
III. Définitions
Posons xs
le le symétrique du nombre x : Exemple : 123
s
= 321 , 1000s
= 0001 .
2. Brisure De Symétrie Et Nombres De Lychrel
DOI: 10.9790/0661-1901019194 www.iosrjournals.org 92 | Page
Un nombre x est dit Palindrome si xs
=x .
Remarque : 001∉ℕ , d’où la définition :
Posons
^ℕ= ℕ∪ℕs
où
ℕs
= {xs
,x∈ℕ} .
Soit ϕ la fonction de
^ℕ sur ℕ par : ϕ (x)=x+xs
.
Un nombre entier x est dit un nombre de Lychrel si
∀ i∈ℕ ϕ i
(x) n’est pas un nombre palindrome.
Soit ψ la fonction définie de ℕ sur ℤ par : ψ(x)=x− xs
.
Il est claire qu’un nombre x est Palindrome si et seulement si ψ(x)= 0 .
IV. Enoncée des conjectures
Conjecture du palindrome inévitable :
Tout nombre entier soumis de manière répétée à l’opération renversement-addition finit par produire un
palindrome.
Conjecture du devenir infini de 196 :
Le nombre 196 soumis à l’opération renversement-addition ne produit jamais un palindrome.
V. Existence des nombres de lychrels
Théorème 1:
∀ x∈ℕ, ∃i∈ℕ/(ψ− I )(ϕ i
(x))=− ϕ i
(x)
Preuve :
En assimilant tout nombre x de
^ℕ à une particule, la fonction ϕ induit la force ϕ − I (où I est l’identité) :
(ϕ − I )(x)=ϕ (x)− x=xs
.
De même la fonction ψ induit la force ψ− I (où I est l’identité) :
(ψ− I )(x)=ψ(x)− x=− xs
.
Donc les forces ϕ − I et ψ− I exercent des forces opposées sur chaque particule x, il est donc intéressant
d’étudier l’action de la fonction ψ sur
^ℕ
.
Remarquons d’abord que contrairement à la fonction ϕ qui a tendance à pousser toute particule vers la droite,
l’action de ψ est chaotique : certaines particules sont poussées vers la droite alors que d’autres sont tirées vers
la gauche, d’où l’intérêt de voir si le mouvement agité des particules va se stabiliser.
Effectivement, comme ψ(x)=x− xs
, alors
ψ(xs
)=− ψ (x) , de sorte que la résultante des forces induites
par ψ sur chaque particule est nulle. Il s’ensuit que la résultante des forces induites par ϕ sur chaque
particule est deux fois l’identité. Il en résulte que dans l’orbite de x :
O(x)= {ϕ i
(x), i∈ℕ},
∀ x∈ℕ, ∃i∈ℕ/(ϕ− I )(ϕ i
(x))=ϕ i
(x)puisque ∀ x∈ℕ, ∃i∈ℕ/(ψ− I )(ϕ i
(x))=− ϕ i
(x) ,
soit ψ(ϕ i
(x))= 0
Corollaire 1:
Pour tout élément x dans
^ℕ
, ∃i∈ℕ/ψ (ϕi
(x))= 0
Corollaire 2:
Dans ℕ , il existe des nombres x tels que : ψ(ϕ i
(x))≠ 0 ∀ i∈ℕ .
3. Brisure De Symétrie Et Nombres De Lychrel
DOI: 10.9790/0661-1901019194 www.iosrjournals.org 93 | Page
Preuve :
En effet, en enlevant les particules symétriques des particules de la forme k 10l
on brise la symétrie (ou
l’équilibre), donc les particules se trouvent libre et se déplacent vers la droite (sous l’action de ϕ )sans
s’arrêter, d’où le résultat.
En posant
^ℤ = {± x ,x∈ ^ℕ} , on peut prolonger ψ à
^ℤ par :
ψ(− x)=− ψ(x)
Corollaire 3:
Pour tout élément x dans
^ℤ ,
∃i∈ℕ/ψi
(x)= 0
Preuve :
Analogue à celle du corollaire 1 ci-dessus en considérant ψ au lieu de ϕ .
Corollaire 4:
Dans ℕ , il existe des nombres x tels que
ψi
(x)≠ 0 ∀ i∈ℕ . (Exemple x=69325 ).
Preuve :
Analogue à celle du corollaire 2 ci-dessus.
Pour x=69325 , on trouve : ψ7
(69325)= 2178 et ψ9
(69325)= -2178 , on en déduit que :
∀ i∈ℕ/ψi
(69325)≠ 0 .
Remarques:
i- Les particules symétriques des particules de
^ℕ∖ ℕ représentent un pourcentage moyen de 10/100 dans
ℕ , et cela correspond exactement au pourcentage moyen des nombres de Lychrel suspects. Ce qui explique
que les nombres de lychrels peuvent être assimilées à des particules libres à cause de la brisure de symétrie.
ii- Les nombres de lychrel peuvent s’écrire sous la forme: ∑ ϵi,j ϕ i
(x)
où
ϵi,j∈{− 1,1}
et x un élément
de
(^ℕ ∖ ℕ )s
, autrement dit, comme en mécanique quantique, tout nombre de lychrel est un niveau d’énergie
ou représente un état qui est une combinaison des états ϕ i
(x) .
Exemple:
Voici les premiers nombres de lychrels suspects :
196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1 495, 1 497, 1 585, 1 587, 1 675, 1 677, 1 765,
1 767, 1 855, 1 857, 1 945, 1 947, 1 997…
On a:
196= ϕ5
(10)+ϕ0
(20)= ϕ4
(20)+ϕ0
(20)
493= ϕ4
(30)+ϕ0
(160)
592= ϕ4
(50)+ϕ0
(300)
689= ϕ4
(90)+ϕ0
(400)
887= ϕ4
(80)+ϕ0
(40)
978= ϕ4
(100)+ϕ0
(170)
295= ϕ4
(80)− ϕ4
(50)− ϕ0
(260)
394= ϕ4
(80)− ϕ4
(30)− ϕ0
(120)
4. Brisure De Symétrie Et Nombres De Lychrel
DOI: 10.9790/0661-1901019194 www.iosrjournals.org 94 | Page
691= ϕ4
(80)− ϕ4
(20)+ϕ0
(20)
986= ϕ1
(50)+ϕ0
(1220)− ϕ4
(90)
1495= ϕ4
(70)+ϕ0
(890)
1497= ϕ4
(80)+ϕ0
(650)
1585= ϕ4
(100)+ϕ1
(70)+ϕ0
(700)
1587= ϕ4
(80)+ϕ0
(740)
1675= ϕ1
(50)+ϕ0
(1620)
1677= ϕ1
(70)+ϕ0
(1600)
De même pour les autres nombres suspects (jusqu’au 1997).
Références
[1]. Jean-Paul Delahaye, http://www.lifl.fr/~jdelahay/IAGL/Conjectures.pdf
[2]. http://www.p196.org/
[3]. M. Sghiar, La relativité et la théorie des nombres, prépublication, https://hal.archives-ouvertes.fr/hal-01174146v4