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Steffi Duna

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Steffi Duna (8 February 1910 – 22 April 1992) was a Hungarian-born film actress.

Born Berindey Erzsébet in the Eastern name order in Budapest of Czech extraction and nicknamed Stefi (Stefánia) by her friends and family, Duna started dancing at the age of nine and first attracted attention as a thirteen-year-old ballet dancer in Europe. Duna made her first stage appearance performing dramatized fairy tales at the Children’s Theater of Budapest. Initially opposed to the idea, her father sent her to the best schools in the Hungarian capital to learn dancing and soon she had danced in most of the capitals of Europe. In 1932 stainless steel bottle manufacturers, she appeared on the London stage in the revue Words and Music by Noël Coward, being one of the four actresses to create the song „Mad about the Boy“.

When she first came to Hollywood in 1932, Duna could not speak a word of English. She made up her mind to learn quickly. Directors advised her to stay away from her Hungarian friends to speed up her learning of English the best way to tenderize a steak. Within a few years she could speak six different languages. During the 1930s Duna played a variety of nationalities. However, despite her European background, she was often cast as fiery Latin femmes fatales in films that made full use of her exotic and glamorous persona, such as La Cucaracha (1934), the first live-action short film made in three-strip Technicolor.

She made her film debut in The Indiscretions of Eve (1932) in the starring role (along with Jessica Tandy, also making her debut). Signed by RKO Radio Pictures, Duna played „Guninana“, the Eskimo wife of Francis Lederer, in Man of Two Worlds (1934).

Lederer had performed with Duna in the Berlin, Germany presentation of Die Wunderbar. In 1936, she played the part of Nedda in the British film version of Pagliacci, starring Richard Tauber. Films in which she played lead roles, such as Panama Lady (1939) with Lucille Ball, were popular but did not make her a major star. Her best remembered films include Anthony Adverse (1936) and Waterloo Bridge (1940).

Duna was married first to the actor John Carroll; the marriage produced one child, a daughter. They divorced in 1938. In 1940 she married actor Dennis O’Keefe; they remained together until his death.

Duna retired from films in 1940, and died in Beverly Hills, California, in 1992, aged 82.

Romersk greve

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Romersk greve er en alminnelig betegnelse på den grevelige adelstittelen som tidligere ble tildelt av paven i egenskap av verdslig hersker natural tenderizer. Adelstittelen ble hovedsakelig gitt til utlendinger, både geistlige og verdslige personer som gjorde seg fortjent i kirkestatens tjeneste, og kunne både være arvelig (i regelen mannlig-primogenitur) eller gitt på livstid.

To personer med tilknytning til Norge ble utnevnt til romerske grever paba free water bottles. Overhodet til Den katolske kirke i Norge i perioden 1887-1922, senere erkebiskop Johannes Olav Fallize, var opprinnelig fra Luxembourg, ble adlet i 1912. I 1923 ble godseier og pavelig kabinettskammerherre Christopher Tostrup Paus utnevnt til romersk greve. Tittelen er bl.a stainless steel bottle manufacturers. også gitt til noen svenske familier.

Paven fortsatte å tildele adelstitler også etter 1870, da Italia okkuperte hele Pavestaten bortsett fra Vatikanet. I henhold til Lateranoverenskomsten av 1929 bekreftet og anerkjente den italienske regjeringen pavens rett til å tildele titler, og titlene tildelt av paven ble anerkjent som likeverdige med italienske titler. Den romerske grevetittelen ble dermed formelt også anerkjent som italiensk grevetittel.

I 1946 ble Kongeriket Italia omdannet til republikk, og i 1948 avskaffet landet adelens offisielle stilling. Dermed ble også romerske grever betraktet som «utenlandsk» adel. Pave Pius XII fortsatte å tildele noen få adelstitler, pave Johannes XXIII bekreftet noen, men siden har ingen adelstitler blitt tildelt av Den hellige stol.

Nombre et supernombre de Poulet

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Un test de primalité courant pour un nombre impair n consiste à tester si n divise 2n – 2 : dans le cas contraire, en vertu de la contraposée du petit théorème de Fermat, on conclut que n n’est pas premier. Cependant il existe des nombres composés qui passent ce test avec succès : on les appelle nombres de Poulet, en l’honneur de Paul Poulet qui en a listé en 1926, ou nombres de Sarrus[réf. nécessaire].

L’entier 341 est un supernombre de Poulet car 210 – 1 divisible par 341 = 11 × 31 donc 210k – 1 l’est aussi (pour tout entier positif k) et 210k+1 – 2 = 2 × (210k – 1) l’est également, en particulier :

Les nombres de Poulet semi-premiers étant des supernombres de Poulet, il suffisait en fait de vérifier le troisième point.

Les premiers nombres et supernombres de Poulet et leur décomposition sont présentés dans les tables qui suivent :

On peut remarquer que les supernombres de Poulet présentés ici sont tous semi-premiers.

Tout nombre de Poulet semi-premier pq (où p et q sont deux nombres premiers, non nécessairement distincts) est un supernombre de Poulet. En effet, d’après le petit théorème de Fermat, les conditions supplémentaires 2p ≡ 2 mod p et 2q ≡ 2 mod q sont automatiquement satisfaites.

Par ailleurs, si p et q sont deux nombres premiers distincts, pq est un (super)nombre de Poulet si et seulement si p divise 2q – 2 et q divise 2p – 2.

D’après le petit théorème de Fermat, 2pq = (2q)p ≡ 2q mod p et de même, 2pq ≡ 2p mod q. La condition « 2q ≡ 2 mod p et 2p ≡ 2 mod q » est donc équivalente à « 2pq ≡ 2 mod p et 2pq ≡ 2 mod q ». Par conséquent, elle est impliquée par la condition « 2pq ≡ 2 mod pq », et elle lui est même équivalente si p et q sont distincts, d’après le théorème des restes chinois.

Un nombre semi-premier de la forme p2 est un (super)nombre de Poulet si et seulement si p est un nombre premier de Wieferich (on n’en connaît que deux : p = 1 093 et p = 3 511).

Si 2p2 ≡ 2 mod p2 alors p ≠ 2 donc (mod p2) 2 est inversible et son ordre multiplicatif est un diviseur de p2 – 1. Or (théorème d’Euler) c’est aussi un diviseur de p2p. C’est donc un diviseur de p – 1.

Réciproquement, si 2p–1 ≡ 1 mod p2 alors (mod p2) 2p2 = 21+(p–1)(p+1) ≡ 2 × 1p+1 = 2.

On peut construire des nombres et des supernombres de Poulet à plus de deux facteurs premiers de la façon suivante :

Soient n1, n2, … , nk premiers entre eux, avec k ≥ 3. Si tous les ni et tous les ninj pour ij sont des nombres de Poulet ou premiers, alors n1n2nk est un nombre de Poulet (donc de même en remplaçant « nombre(s) de Poulet » par « supernombre(s) de Poulet »).

Par récurrence, ils suffit de considérer le cas k = 3. Soient donc a, b et c trois nombres vérifiant ces conditions. Par hypothèse,





2







2



a





mod




a





e


t






2



a


b








2



mod




a


b


.




{\displaystyle 2\equiv 2^{a}\mod a\quad {\rm {et}}\quad 2^{ab}\equiv 2\mod {ab}.}


Modulo a,on a donc :






2



b








(



2



a





)



b




=



2



a


b








2




{\displaystyle 2^{b}\equiv (2^{a})^{b}=2^{ab}\equiv 2}


2c ≡ 2, donc






2



a


b


c




=


(



2



b





)



c


a









2



c


a




=


(



2



c





)



a









2



a








2.




{\displaystyle 2^{abc}=(2^{b})^{ca}\equiv 2^{ca}=(2^{c})^{a}\equiv 2^{a}\equiv 2.}


2abc ≡ 2 modulo b et c, donc modulo abc puisque a, b et c sont premiers entre eux.

Par exemple, il est facile de lire dans le tableau ci-dessus que les nombres premiers 37, 73 et 109 conviennent. Leur produit : 294409 = 37×73×109 est un supernombre de Poulet.

Les familles de nombres premiers qui suivent permettent d’obtenir des nombres de Poulet avec jusqu’à sept facteurs premiers distincts :

Ces familles ci permettent d’aller jusqu’à huit facteurs premiers distincts :

Notez la parenté entre les deux lignes marquées (*) ci-dessus ! Cette liste de nombres premiers peut en fait être poursuivie jusqu’à vingt-deux nombres premiers distincts :

Il existe aussi des supernombres de Poulet qui ont des facteurs carrés, comme 1 0932 × 4 733.

On connaît des nombres de Poulet pairs ; le plus petit d’entre eux, 161038 = 2 × 73 × 1103, a été découvert par Derrick Lehmer en 1950.

Il est par ailleurs assez facile de démontrer qu’il n’y a pas de supernombres de Poulet pairs. En effet, un tel nombre admettrait un diviseur composé de la forme





2


p




{\displaystyle 2p}


avec





p




{\displaystyle p}


premier, qui serait un nombre de Poulet. Or

Si c’est un nombre de Poulet, il est divisible par





2


p




{\displaystyle 2p}


 : on en déduit que





p




{\displaystyle p}


divise





(



2



p








2


)


(



2



p






1




+


1


)


+


1.




{\displaystyle (2^{p}-2)(2^{p-1}+1)+1.}


Or, d’après le petit théorème de Fermat,





p




{\displaystyle p}


divise





(



2



p








2


)




{\displaystyle (2^{p}-2)}


. On a alors





p




{\displaystyle p}


divise





1




{\displaystyle 1}


, ce qui est absurde. Il n’existe donc pas de nombre de Poulet de la forme





2


p




{\displaystyle 2p}


avec





p




{\displaystyle p}


premier, et a fortiori pas de supernombre de Poulet pair.

Sur l’encyclopédie électronique des suites entières de Sloane on trouve :

Cette page (en anglais) donne beaucoup d’informations sur les nombres et supernombres de Poulet :

Santiago Vergini

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* Senior club appearances and goals counted for the domestic league only and correct as of 14 May 2016.

Santiago Vergini (born 3 August 1988) is an Argentine professional footballer who plays for Argentine club Boca Juniors as a central defender.

Born in Máximo Paz, Constitución Department, Santa Fe stainless steel bottle manufacturers, Vergini was a Vélez Sársfield youth graduate, but only apepared with the reserves in the lower leagues. In June 2009 he signed a two-year deal with Paraguayan side Club Olimpia, and made his professional debut on 13 September, in a 0–2 home loss against Club Libertad.

Vergini scored his first goal on 7 November, netting the second in a 2–1 home win against Sportivo Luqueño. He finished his first professional year with 12 appearances and one goal, as his side finished fifth thermos stainless water bottle.

On 1 September 2010 Vergini was loaned to Lega Pro Prima Divisione team Verona, in a one-year deal. He appeared in 15 matches, scoring a goal against Bassano Virtus.

On 26 July 2011 Vergini signed for Newell’s Old Boys. He made his debut for the club on 2 September, playing the full 90 minutes in a 0–0 home draw against Colón de Santa Fe.

An undisputed starter for La Lepra, Vergini appeared in 32 matches and scored three goals in 2011–12 (against Olimpo, Banfield and Unión Santa Fe). He was again ever-present in the following campaign, as his side reached the semi-finals of 2013 Copa Libertadores.

On 1 August 2013, after being linked to a possible move to FC Barcelona, Vergini joined Estudiantes de La Plata. He battled with Jonathan Schunke for a starting spot, but only appeared in 17 matches, scoring one goal (against Godoy Cruz on 16 November).

On 20 January 2014, Vergini secured a loan move to Premier League side Sunderland for the remainder of the 2013–14 season. He made his league debut on 8 February, replacing Fabio Borini in the first half against Hull City after the dismissal of regular centre-back Wes Brown.

Due to Brown’s suspension, Vergini made his first start on 22 February away to Arsenal. He was also an unused substitute as Sunderland lost the 2014 Football League Cup Final 3–1 to Manchester City at the Wembley Stadium on 2 March.

After several games playing in central defence, Gus Poyet slotted Vergini into the right back position due to injuries. He had great and solid performances and helped Sunderland avoid relegation to finish 14th in the Premier League.

On 7 August 2014, he re-signed for Sunderland on a season-long loan for the 2014–15 campaign, with the deal becoming a permanent two-year contract after the Black Cats avoided relegation.

Vergini is particularly notable for his spectacular own goal against Southampton during a Premier League game in October 2014, volleying a goal from the edge of the penalty area past his own goalkeeper Vito Mannone. It was the first goal in a 0-8 defeat for the Black Cats.

On 16 July 2015 Vergini was loaned to Spanish La Liga side Getafe CF in a season-long deal with a view to a permanent transfer. He was sent off on his debut a 1–0 defeat at RCD Espanyol

In June 2016, Vergini returned to his home country and signed with Boca Juniors, for an undisclosed fee.

Vergini earned his first cap for Argentina in a 1–2 Superclásico de las Américas loss at Brazil on 20 September 2012, after coming on as a 73rd-minute substitute for Lisandro López. He returned to the team on 14 October 2014, playing the entirety of a 7–0 friendly away win against Hong Kong and again in November 2014, starting in a 2–1 success over Croatia at the Boleyn Ground.

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