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SUMMARY / RELATED TOPICS

Well-order

In mathematics, a well-order on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder and wellordering or well order, well ordered, well ordering; every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor, namely the least element of the subset of all elements greater than s. There may be elements besides the least element. In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the least element of the subset of all upper bounds of T in S. If ≤ is a non-strict well ordering < is a strict well ordering. A relation is a strict well ordering if and only; the distinction between strict and non-strict well orders is ignored since they are interconvertible.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set; the observation that the natural numbers are well ordered by the usual less-than relation is called the well-ordering principle. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set; the position of each element within the ordered set is given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects; the size of a finite set is equal to the order type.

Counting in the everyday sense starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects, thus for finite n, the expression "n-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can be an infinite ordinal, it will count from zero. For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types. For a countably infinite set, the set of possible order types is uncountable; the standard ordering ≤ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor. Another well ordering of the natural numbers is given by defining that all numbers are less than all odd numbers, the usual ordering applies within the evens and the odds: 0 2 4 6 8... 1 3 5 7 9...

This is a well-ordered set of order type ω + ω. Every element has a successor. Two elements lack a predecessor: 0 and 1. Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, for example, the set of negative integers does not contain a least element; the following relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds: x = 0 x is positive, y is negative x and y are both positive, x ≤ y x and y are both negative, |x| ≤ |y|This relation R can be visualized as follows: 0 1 2 3 4... −1 −2 −3... R is isomorphic to the ordinal number ω + ω. Another relation for well ordering the integers is the following definition: x ≤z y iff; this well order can be visualized as follows: 0 −1 1 −2 2 −3 3 −4 4... This has the order type ω; the standard ordering ≤ of any real interval is not a well ordering, for example, the open interval ⊆ does not contain a least element. From the ZFC axioms of set theory one can show.

Wacław Sierpiński proved that ZF + GCH imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable well order of the reals; however it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that V=L, it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set. An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose X is a subset of R well ordered by ≤. For each x in X, let s be the successor of x in ≤ ordering on X. Let A = whose elements are nonempty and disjoint intervals; each such interval contains at least one rational number, so there is an injective function from A to Q. There is an injection from X to A, and it is well known that there is an injection from Q to the natural numbers

Frances, Countess of PĂ©rigord

Frances de Châtillon was Countess of Périgord, Viscountess of Limoges, Dame of Avesnes and Châlus. She was the eldest daughter of Viscount of Limoges and Isabelle de La Tour d'Auvergne. In 1470, she married Alain the Great, Count of Graves and Viscount of Tartas, son of Jean I of Albret and Catherine of Rohan. Through her father, Frances had a claim on the throne of Brittany through the Penthièvre line. Frances had seven surviving children with Alain, including: John, king iure uxoris of Navarre until 1516, who married in 1484 Catherine, Queen of Navarre. Gabriel, lord of Avesnes-sur-Helpe Charlotte of Albret, Dame of Châlus, who married in 1500 Cesare Borgia Amanieu d'Albret, bishop of Pamiers and Lescar, a cardinal Pierre, Count of Périgord Louise, Viscountess of Limoges, who married in 1495 Charles I de Croÿ Isabelle, who married Gaston II, Captal de Buch De Guibours, Anselme. Histoire généalogique et chronologique de la maison royale de France, des pairs, grands officiers de la Couronne, de la Maison du Roy et des anciens barons du royaume.

6. Paris. P. 105. Retrieved 23 November 2018. CS1 maint: date format Duchesne, André. Histoire de la Maison de Chastillon sur Marne avec les genealogies et armes des illustres Familles de France & des Pays-bas, lesquelles y ont esté alliées. Paris: Sabastien Cramoisy. Pp. 148–158. Retrieved 23 November 2018

Dominic Fursey Bodkin

Dominic Fursey Bodkin was a Christian Brother and a Catholic educator in Australia. Bodkin was born near Tuam, County Galway, Ireland in 1843, he attended the Christian Brothers school in Tuam. Aged 16 years, Bodkin entered the novitiate of the Christian Brothers in Dublin, he taught in a number of schools during his training, but in Waterford. Archbishop of Melbourne James Alipius Goold believed there was a need for religious teachers for schools in his diocese. In 1867 Goold visited Europe and with the assistance of the Cardinal Prefect of Propaganda he obtained the assistance of a group of Irish Christian Brothers. Bodkin, John Barnabas Lynch, Patrick Joseph Nolan and Patrick Ambrose Treacy left Ireland for Melbourne to establish the Christian Brothers in Australia, arriving in Port Philip Bay on the Donald Mckay on 18 November 1868. In 1869 they established their first school at the rear of St Francis Church in Elizabeth Street, Melbourne, their monastery and second school, Parade College, was established in Victoria Parade, East Melbourne.

On 24 April 1876, Bodkin opened a school in Rattray Street, South Island, New Zealand, becoming its first principal. In 1989 this school merged with other Catholic schools in Dunedin to become Kavanagh College. Bodkin was the first principal of St Joseph's College, Nudgee in Brisbane, which opened in 1891, he returned to Dunedin and about 1904 moved to Western Australia where he worked with orphans on farm schools. Aged 85 years, Bodkin died on 20 February 1929 at Western Australia. Media related to Dominic Fursey Bodkin at Wikimedia Commons

Al-Dumayr offensive (April 2016)

The al-Dumayr offensive was a military offensive launched in April 2016 by the Islamic State of Iraq and the Levant near the town of al-Dumayr, east of Damascus, Syria. The attack is notable for the abduction of hundreds of cement plant workers by ISIL. On 4 April, ISIL attacked areas around the city of al-Dumayr, northeast of Damascus, resulting in 250–300 cement plant workers being abducted from a factory by ISIL. ISIL massacred 175 of them afterwards, while 75 escaped. Druze employees were murdered. On 6 April, ISIL launched an attack on the Dumayr Airbase, outside the town, sending five car bombs and killing 12 Syrian soldiers; the attack was repelled by the National Defence Forces. Due to a partial ceasefire, the Syrian Armed Forces allowed some Jaysh al-Islam militants from Ghouta to cross into al-Dumayr in order to fight ISIL; the Syrian Air Force hit ISIL targets in the front with the rebels. On 9 April, a new ISIL attack on the airport and power plant was repelled, after which the airport was declared secured.

On 11 April, a Syrian Air Force plane was shot down by ISIL near the airbase. Two days yet another attack on the airbase was repelled, although Army checkpoints outside the base had sustained heavy damage. On 14 April, the Army launched a counter-attack recapturing several hilltops, by the following day, they had retaken control of the Khan Abu Shamat base and the Badia cement plant. On 16 April, the military continued to advance and recaptured the Battalion 559 base, Al-Sini Factory, Al-Safa Station and the triangle Baghdad-Palmyra-Jordan checkpoint, ending the ISIL offensive

Defensores de Belgrano de Villa Ramallo

Club Atlético y Social Defensores de Belgrano known as Defensores de Belgrano de Villa Ramallo, is an Argentine sports club located in the Ramallo Partido of Buenos Aires Province. The football team plays in the Torneo Argentino A, the regionalised third division of Argentine football league system. Other sports practised at the institution are basketball, field hockey, handball, roller skating, swimming and volleyball. Básquet:una categoría sub 13 increíble con jugadores que hicieron historia para el club. DT:Martín Blanco. Jugadores: Torriani Martín, Leguizamo Martín, filanti Aseff Joaquín, Passciulo Vito, Tomatis Facundo, Gonzales Francisco, Rossi Francisco, Sangasis Rueda, Juan Ignacio, Costoya Mauricio, Sica Amadeo, Sica Santino, Martínez Máximo. Que supo llegar a la final del final four de la copa de Oro de la ABSN 2019 perdiendo la final con Regatas de San nicolas por un parcial de 63-39 y así ganar el segundo puesto en la liga. También lograron clasificar por primera vez en la historia del club a nivel inferiores a un zonal de clubes de la zona norte de BS.

AS de la categoría. En ese mismo plantel hubo ocho preseleccionados para la selección de San Nicolás y fueron: Torriani Martín, Rossi Francisco, Leguizamo Martín, Tomatis Facundo, Gonzales Francisco, Filanti aseff Joaquín, Passciulo Vito, Sica Santino, por la cual quedaron seleccionados solo dos jugadores para jugar el zonal de selecciones de zona norte de BS. AS en la cual quedaron segundos en el triangular ganándole a la selección de pergamino 70-63 y perdiendo con la selección de Zárate-campana por 57-58 que clasifico al provincial de selecciones. Official site

Chris Dixon

Chris Dixon is an American internet entrepreneur and investor. He is a general partner at the venture capital firm Andreessen Horowitz, worked at eBay, he is the co-founder and former CEO of Hunch, a website. Dixon earned a BA and an MA from Columbia University, majoring in philosophy, has an MBA from Harvard Business School, his early college days were at Wesleyan University. In the late 1990s, Dixon spent three years as a software programmer at Arbitrade, a hedge fund focused on high-frequency trading, he joined the venture capital firm Bessemer Venture Partners. In 2005, Dixon co-founded SiteAdvisor, a web-security startup, bought by security company McAfee in 2006. In 2009, he founded Hunch with Caterina Fake and Tom Pinckney, acquired by eBay in 2011, he co-founded Founder Collective, a seed-stage venture capital fund and became an investor in BuzzFeed, Betaworks, Venmo and Hotel Tonight. Dixon is a general partner at a venture capital firm in Menlo Park, California. Since joining the firm in January 2013, Dixon has led a variety of investments for the firm including FiftyThree Soylent, Nootrobox and he sits on the boards of drone startup Airware, 3D printing startup Shapeways, digital Bitcoin wallet Coinbase.

Dixon led the firm's investment and sits on the board of Oculus VR, acquired by Facebook in March 2014. In 2010, Bloomberg L. P. named Dixon the top angel investor in the technology industry. Dixon won the 2012 Crunchie "Angel of the Year" award. Official website