In mathematics, permutation is the act of arranging the members of a set into a sequence or order, or, if the set is ordered, rearranging its elements—a process called permuting. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set, namely:, and; these are all the possible orderings of this three-element set. Anagrams of words whose letters are different are permutations: the letters are ordered in the original word, the anagram is a reordering of the letters; the study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are studied in every branch of mathematics, in many other fields of science. In computer science, they are used for analyzing sorting algorithms; the number of permutations of n distinct objects is n factorial written as n!, which means the product of all positive integers less than or equal to n.

In algebra, in group theory, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs once as an image value; this is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f. For example, the permutation mentioned above is described by the function α defined as: α = 3, α = 1, α = 2; the collection of such permutations form a group called the symmetric group of S. The key to this group's structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols; the set used is S = N n =, but there is no restriction on the set. In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set; when k is equal to the size of the set, these are the permutations of the set.

Al-Khalil, an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels; the rule to determine the number of permutations of n objects was known in Indian culture around 1150. The Lilavati by the Indian mathematician Bhaskara II contains a passage that translates to: The product of multiplication of the arithmetical series beginning and increasing by unity and continued to the number of places, will be the variations of number with specific figures. Fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells: "first, two must be admitted to be varied in two ways" which he illustrates by showing 1 2 and 2 1, he explains that with three bells there are "three times two figures to be produced out of three" which again is illustrated. His explanation involves "cast away 3, 1.2 will remain.

He moves on to four bells and repeats the casting away argument showing that there will be four different sets of three. This is a recursive process, he continues with five bells using the "casting away" method and tabulates the resulting 120 combinations. At this point he gives up and remarks: Now the nature of these methods is such, that the changes on one number comprehends the changes on all lesser numbers... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body. A first case in which unrelated mathematical questions were studied with the help of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that properties of the permutations of the roots of an equation are related to the possibilities to solve it; this line of work resulted, through the work of Évariste Galois, in Galois theory, which gives a complete description of what is possible and impossible with respect to solving polynomial equations by radicals.

In modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. In mathematics texts it is customary to denote permutations using lowercase Greek letters. Either α and β, or σ, τ and π are used. Permutations can be defined as bijections from a set S onto itself. All permutations of a set with n elements form a symmetric group, denoted S n, where the group operation is function composition

List of Spanish football transfers summer 2017

This is a list of Spanish football transfers for the summer sale in 2017–18 season. Only moves from La Liga and Segunda División are listed; the summer transfer window will begin on 1 July 2017, although a few transfers took place prior to that date. The window will close at midnight on 1 September 2017. Players without a club can join one during or in between transfer windows. Clubs below La Liga level can sign players on loan at any time. If needed, clubs can sign a goalkeeper on an emergency loan. Manager: Luis Zubeldía Manager: José Ángel Ziganda Manager: Diego Simeone Manager: Ernesto Valverde Manager: Juan Carlos Unzué Manager: Pepe Mel Manager: José Luis Mendilibar Manager: Quique Sánchez Flores Manager: José Bordalás Manager: Pablo Machín Manager: Manolo Márquez Manager: Asier Garitano Manager: Juan Muñiz Manager: Míchel Manager: Quique Setién Manager: Zinedine Zidane Manager: Eusebio Sacristán Manager: Eduardo Berizzo Manager: Marcelino Manager: Fran Escribá Manager: José Manuel Aira Manager: Julio Velázquez Manager: Luis Miguel Ramis Manager: Gerard López Manager: Álvaro Cervera Manager: Luis Carrión Manager: Rubén de la Barrera Manager: Lluís Carreras Manager: José Luis Oltra Manager: Rubi Manager: Curro Torres Manager: Francisco Manager: Jagoba Arrasate Manager: Diego Martínez Manager: Juan Antonio Anquela Manager: Míchel Manager: Aritz López Garai Manager: Luis Tevenet Manager: Paco Herrera Manager: José Luis Martí Manager: Luis César Sampedro Manager: Natxo González

Raphaelle Peale

Raphaelle Peale is considered the first professional American painter of still-life. Peale was born in Annapolis, the fifth child, though eldest surviving, of the painter Charles Willson Peale and his first wife Rachel Brewer, he grew up in Philadelphia, spent his life there in a home at the corner of 3rd and Lombard. Like his siblings, Raphaelle was trained by his father as an artist. Early in his career, the pair collaborated on portraits. On some commissions, Raphaelle painted miniatures while his brother, painted full-size portraits. In 1793, he made a trip to South America in order to collect specimens for the Peale Museum founded by his father, he exhibited five portraits and eight other paintings still lifes, at the Columbianum, Philadelphia in 1794. His first professional exhibition was in 1795 at the age of 21. In 1797, with his brother Rembrandt, he traveled to Charleston, South Carolina, where they attempted to establish another museum; the plan fell through and Raphaelle returned to painting miniatures.

He married Martha McGlathery at the age of twenty, with her had eight children. For about two years beginning in 1803, Peale toured Virginia with the "physiognotrace", a profile making machine, with which he was successful. By 1806 he had begun to suffer the symptoms of arsenic and mercury poisoning brought on by his work as a taxidermist in his father's museum. In August 1809 he was hospitalized with delirium, for the rest of his life he suffered debilitating attacks yearly—which his father ascribed to "gout of the stomach" caused by consumption of pickles and excessive drinking. From 1810, Peale concentrated on still-life painting exclusively, becoming America's first professional still-life painter, he exhibited at the Pennsylvania Academy of the Fine Arts and elsewhere from 1814–18. By 1813, he was unable to walk without crutches. After the downturn in his health, in an era when most artists considered still life a subject worthy only of amateurs, he devoted himself exclusively to still life painting.

It is. His work was on frequent exhibit at the Pennsylvania Academy of the Fine Arts between 1814 and 1818. After indulging in a night of heavy drinking, his health destroyed, he died on March 4, 1825, at age 51 at his home in Philadelphia. Alfred Frankenstein has called Raphaelle Peale "the first distinguished still-life specialist to emerge in this country, he is one of the four major still-life painters of the nineteenth century in the United States." His style may have been influenced by Spanish still life paintings he saw on his trip to Mexico and by the two works by Juan Sanchez Cotan, exhibited at the Pennsylvania Academy in 1816. Most of Peale's paintings are small in scale, depict a few objects—usually foodstuffs—arranged on a tabletop before a darkened background. A notable exception is Venus Rising from the Sea -- A Deception, it was his nephew George Escol Sellers's opinion that Raphaelle Peale was the most talented of Charles Wilson Peale's artist children and that "it was the Revolution that made him the wreck he was".

Blackberries, c. 1813 Melons and Morning Glories, 1813 A Dessert, 1814 Still Life with Orange and Book, 1815 Fruit and Pretzel, unknown Bowl of Peaches, 1816 Still Life with Fruit and Wine, 1821 Still Life with Peaches, 1822 Lemons and Sugar, unknown Frankenstein, The Reality of Appearance, Greenwich: New York Graphic Society, 1970. ISBN 0- 8212-0357-6 Lauren Lessing and Mary Schafer, “Unveiling Raphaelle Peale’s Venus Rising from the Sea – A Deception,” Winterthur Portfolio 43, 229–59. Http:// 32 pages Phoebe Lloyd, "Philadelphia Story", Art in America, 154–171, 195–200. Margaret C. Conrads, ed; the Collections of The Nelson-Atkins Museum of Art: American Paintings to 1945, vol. 1: Margaret C. Conrads, ed; the Collections of The Nelson-Atkins Museum of Art: American Paintings to 1945, vol. 2: Raphaelle Peale at "Raphaelle Peale Artwork Images, Reviews", World Wide Arts Resources, 2007, webpage: WWAR-RPeale.

"The Metropolitan Museum of Art – Works of Art: American Paintings", Metropolitan Museum of Art, 2007, webpage: MMA-RPeale. Birmingham Museum of Art, A Portrait of Margaret George McGlathery, 1817 Raphaelle Peale from the Smithsonian American Art Museum