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Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra and functional programming; the term was introduced by Benjamin Peirce in the context of elements of algebras that remain invariant when raised to a positive integer power, means " the same power", from idem + potence. An element x of a magma is said to be idempotent if: x • x = x. If all elements are idempotent with respect to • • is called idempotent; the formula ∀x, x • x = x is called the idempotency law for •. The natural number 1 is an idempotent element with respect to multiplication, so is 0, but no other natural number is. For the latter reason, multiplication of natural numbers is not an idempotent operation. More formally, in the monoid, idempotent elements are just 0 and 1. In a magma, an identity element e or an absorbing element a, if it exists, is idempotent.

Indeed, e • e = e and a • a = a. In a group, the identity element e is the only idempotent element. Indeed, if x is an element of G such that x • x = x x • x = x • e and x = e by multiplying on the left by the inverse element of x. Taking the intersection x∩y of two sets x and y is an idempotent operation, since x∩x always equals x; this means that the idempotency law ∀ x ∩ x = x is true. Taking the union of two sets is an idempotent operation. Formally, in the monoids and of the power set of the set E with the set union ∪ and set intersection ∩ all elements are idempotent. In the monoids and of the Boolean domain with the logical disjunction ∨ and the logical conjunction ∧ all elements are idempotent. In a Boolean ring, multiplication is idempotent. In a Tropical semiring, addition is idempotent. In the monoid of the functions from a set E to itself with the function composition ∘, idempotent elements are the functions f: E → E such that f ∘ f = f, in other words such that for all x in E, f = f.

For example: Taking the absolute value abs of an integer number x is an idempotent function for the following reason: abs = abs is true for each integer number x. This means that abs ∘ abs = abs holds, that is, abs is an idempotent element in the set of all functions with respect to function composition. Therefore, abs satisfies the above definition of an idempotent function. Other examples include: the identity function is idempotent. If the set E has n elements, we can partition it into k chosen fixed points and n − k non-fixed points under f, kn−k is the number of different idempotent functions. Hence, taking into account all possible partitions, ∑ k = 0 n k n − k is the total number of possible idempotent functions on the set; the integer sequence of the number of idempotent functions as given by the sum above for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, … starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, …. Neither the property of being idempotent nor that of being not is preserved under function composition.

As an example for the former, f = x mod 3 and g = max are both idempotent, but f ∘ g is not, although g ∘ f happens to be. As an example for the latter, the negation function ¬ on the Boolean domain is not idempotent, but ¬ ∘ ¬ is. Unary negation − of real numbers is not idempotent, but − ∘ − is. In computer science, the term idempotence may have a different meaning depending on the context in which it is applied: in imperative programming, a subroutine with side effects is idempotent if the system state remains the same after one or several calls, in other words if the function from the system state space to itself associated to the subroutine is idempotent in the mathematical sense given in the definition; this is a useful property in many situations, as it means that an operation can be repeated or retried as as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was performed or not. A function looking up a customer's name and address in a database is idempotent, since this will not cause the database to change.

Changing a customer's address to XYZ is idempotent, because the final address will be the same no matter how many times XYZ is submitted. However, placing an order for a cart for the customer is typica

Suzanne Alaywan is a poet and painter. She has admitted to Joan Miró as being inspiration to her work, in addition to Japanese art, she writes predominantly in Arabic. Born in Beirut to a Lebanese father and Iraqi mother. Due to the Lebanese civil war, Alaywan spent her adolescent years between Andalus and Cairo, she attended the American School of Paris, graduating in 1992. In 1997 she graduated from the faculty of Media in the American University of Cairo. Since 1994, Alaywan has published nine collections of poetry. Café Bird Angels’ Hideout Unique Temporary Sun No Way Out Provisional Sun A Presence Called Love Blind Lantern Envisioning the Scene Junk Words; this has been translated as "A Clutter of Words". All Roads Lead to Salah Salem; the Gazelle ` s Throw Her works have been included in other collected volumes. A poem titled "About Fear" appeared in Issue 14 of A Gathering of the Tribes, a publication focused on art and culture from a diverse perspective. Three poems titled "Montmartre", "Draft of a City", "Degree Zero of the Desert" appeared in the winter 2013 issue of ArteEast Shadehat called City of Translation.

Her work appeared in Diván de poetisas árabes contemporáneas, a spanish language book which highlights the importance of poetic history beginning in the mid-20th century. The volume includes work by ten women poets including Alaywan. A poem titled " A House made of Sugar" appeared in Lisan, a German-language publication focused on Arabic poetry and fiction in translation. Three poems from her collection "A Clutter of Words" were published in Transference Vol. 5, an academic journal that focuses on poetic translations. Official site in Arabic Official Blog

Pinion Pines is a census-designated place located in Mohave County, United States. The population was 186 at the 2010 census. Pinion Pines is located at 35°08′48″N 113°54′21″W. According to the United States Geological Survey, the CDP has a total area of 1.5 square miles, all of it land. As of the 2010 census, there were 186 people living in the CDP: 99 female. 35 were 19 years old or younger, 12 were ages 20–34, 33 were between the ages of 35 and 49, 57 were between 50 and 64, the remaining 49 were aged 65 and above. The median age was 54.7 years. The racial makeup of the CDP was 95.2% White, 0.5% Asian, 3.8 Other, 0.5% two or more races. 5.4 % of the population were Latino of any race. There were 80 households in the CDP, 58 family households and 22 non-family households, with an average household size of 2.33. Of the family households, 48 were married couples living together, while there were 4 single fathers and 6 single mothers; the CDP contained 102 housing units, of which 80 were occupied and 22 were vacant.

As of July 2016, the average home value in Pinion Pines was \$338,487. The average household income was \$69,544, with a per capita income of \$30,311

Mega Brands America, Inc. known as Rose Art Industries, LLC, is an American arts and stationery company based in Irvine, California. It sells products under the brands RoseArt, The Board Dudes, The Write Dudes, USA Gold, Moon Products, MEGA Puzzles and Fuzzy Poster, it is owned by parent company MEGA Brands Inc. a Montreal-based company. It has additional operations in Washington. 2013 marked the 90th anniversary of RoseArt. The company was established when Isidor Rosen founded the Rosebud Art Company in New York City in 1923. During the 1970s, the company renamed itself RoseArt. RoseArt has been a subsidiary of Mega Brands since 2005. On January 24, 2010, Mega Bloks Inc. agreed to acquire The Board Dudes, Inc. a held company based in Corona, California. The Board Dudes designs and distributes a range of products for the school and office supply markets. Primary product categories include dry erase boards, cork boards and locker products, writing instruments, novelty items and storage products sold under the Board Dudes, Locker Dudes, Write Dudes, related brands.

These goods are sold in mass and specialty channels throughout the U. S. and Canada. The Board Dudes founders and principals, Ben Hoch and Michael Cerillo, both maintain senior managerial roles within the arts and stationery division of MEGA Brands. In September 2006, the former owners of Rose Art Industries filed suit against Mega Brands, alleging insider trading by company executives in 2005 and 2006. MEGA Brands counter sued and in November 2009 the parties settled; the Rosens agreed to forgo claims of an additional \$54.8 million. Since the beginning, RoseArt has sold many different art supplies. Over the years, RoseArt supplies have come in many different colors. Below is a list of art supplies. Name – Issued – Retired – Notes Apricot – 1949 Black – 1949 – Blue – 1949 Blue Berry – 1992 Blue Green – 1949 Blue Violet – 1949 Brown – 1949 Green – 1949 Green Blue – 1949 Green Yellow – 1949 Grey – 1949 Orange – 1949 Orange Red – 1949 Orange Yellow – 1949 Peach – 1949 Pink – 1949 Purple – 1949 Red – 1949 Red Orange – 1949 Red Violet – 1949 Rose – 1949 Sky Blue – 1958 Teal – 1958 Turquoise – 1958 Violet – 1949 Violet Blue – 1949 White – 1949 Yellow – 1949 Yellow Green – 1949 Yellow Orange – 1949 Name – Issued – Retired – Notes Apricot – 1946 Black – 1923 – Blue – 1923 Blue Green – 1934 Blue Violet – 1934 Brown – 1923 Cerulean Dandelion Forest Green Gold Golden Rod Green – 1923 Green Blue – 1934 Green Spring Green Yellow – 1934 Grey – 1923 Indigo - 2010 Lavender Lemon Yellow Mahogany Maize Melon Navy Blue Orange – 1923 Orange Red – 1934 Orange Yellow – 1934 Orchid Peach – 1946 Pink – 1923 Purple – 1923 Red – 1923 Red Orange – 1934 Red Violet – 1934 Rose – 1934 Rust Red Salmon Scarlet - 2010 Sweet Bitter Teal Blue Thistle Turquoise Blue Turtle Green Violet – 1923 Violet Blue – 1934 Violet Red – 1934 White – 1923 Yellow – 1923 Yellow Green – 1934 Yellow Orange – 1934 Crayola MEGA Brands home page

Thomas Jordan was an English poet and actor, born in London or Eynsham in Oxfordshire about 1612 or 1614. Jordan was a boy actor in the King's Revels Company, which played at the Salisbury Court and Fortune theatres, continued with the company as an adult, he is known to have performed the part of Lepida, the mother of Messalina, in Thomas Rawlins's Messalina some time between 1634 and 1636. In 1637, Jordan published his earliest known work, Poeticall Varieties, or Variety of Fancies, which shows his theatrical background, it was dedicated to John Ford of a cousin of John Ford the dramatist. His connection with the King's Revels Company ceased in 1636, his activities in the late 1630s are not known. Lynn Hulse suggests as "an attractive possibility" that he may have been attached to the Werburgh Street Theatre in Dublin. Details that support an Irish connection include a commendatory verse signed "T. I." in one of the plays of James Shirley, the Werburgh Street house dramatist, the dedication of Jordan's miscellany Sacred Poems to James Ussher, archbishop of Armagh and primate of all Ireland.

By 1641 he was acting with the King's Company at their playhouse in Clerkenwell. His second comedy, "Youths Figaries", was written for the troupe that year and was "publikely Acted 19. Days together, with extraordinary Applause". In the period of the official closing of the theatres during the Commonwealth, 1642–60, Jordan was involved in some of the clandestine theatrical activities at the Red Bull Theatre. In a raid on the playhouse in September 1655, several actors were arrested, including one Thomas Jay, alias Thomas Jordan. Jordan also supported himself and his family for some time by writing dedications, commendatory verses, panegyrics. According to Thomas Seccombe's Dictionary of National Biography article, these were remarkable for their brazen plagiarisms: "His plan seems to have been to print a book with the dedication in blank, to fill in the name afterwards by means of a small press worked by himself". For example, Wit in a Wildernesse was dedicated to at least five different individuals.

Jordan reissued both his own and other people's already-published works with new title-pages. Jordan was a prolific writer of verse, collections of poetry and sundries. Being a fervent supporter of the Royalist cause, he produced a stream of both prose and poetry in support of the cause, both before and after the Restoration in 1660; the miscellany A Royal Arbor of Loyal Poesie, written in 1663, was reissued five times. When the theatres were reopened at the Restoration in 1660, Jordan again focused his energies on the drama, writing two plays and a masque as well as numerous drolls, afterpieces and epilogues, doing some acting, he played the part of Captain Penniless in his own play Money is an Ass. Jordan started writing civic verse in the late 1650s, including an eclogue in four parts for the Lord Mayor, Sir Thomas Allen, a jig, "The Cheaters Cheated", for the sheriffs of London, he wrote speeches and songs for at least five of the great livery company feasts given in honour of General Monck in the spring of 1660.

In 1671 he was chosen to be poet of the corporation of London. The chief duties of the city laureates were to invent pageants for the successive Lord Mayor's Shows, to compose a yearly panegyric upon the Lord Mayor elect. Jordan conducted the annual civic ceremonies successfully for fourteen years, adapting each to the changing political situation, his civic entertainments were according to S. J. Owen characterised by "moderation and peace, coupled with Protestantism, the promotion of trade, a inclusive vision in which the city and its government have an important role in the nation", he was succeeded by Matthew Taubman in 1685, assumed to be the year of Jordan's death. Several of his contemporaries wrote scornfully of Jordan; the biographer William Winstanley ranks him with John Tatham as "indulging his Muse more to vulgar fancies than the high-flying wits of those times" and Samuel Wesley in his Maggots calls Jordan's muse the inspirer of dullness. Readers have been more appreciative. William Hazlitt credited Jordan with a greater share of poetical merit than most of his profession.

Jordan's carpe diem-themed poem "Coronemus nos Rosis antequam marcescant", alternatively "Let us drink and be merry", was anthologised in the early 20th century, the text of the poem is available on several internet sites. Works by Thomas Jordan at LibriVox

Andrea Pisani was a Venetian noble who served as Captain General of the Sea during the Seventh Ottoman–Venetian War. Andrea Pisani was born in Venice to a noble family belonging to the Venetian patriciate, he was the son of Paolina Contarini. During his youth, he was banished from Venice on the orders of the Council of Ten, for having perpetrated indecent acts against the Sisters of Sant Catherine the Virgin Martyr in Brescia. In order to redeem himself, he enlisted as a volunteer in the Imperial army operating in Hungary during the Siege of Buda. In the next year, he returned to Venice, enlisting in the Venetian navy under the captain Pietro Zaguri. In 1693, he was appointed paymaster under the Doge and Captain General of the Sea Francesco Morosini. In 1695, he took part in the naval Battle of the Oinousses Islands, under the command of Captain General Giovanni Antonio Zeno against the Ottoman fleet, in the next year he fought in the Battle of Andros under Captain General Alessandro Molin. On his return to Venice he was elected to the Venetian Senate.

As senator, he occupied several different magistratures, until returning to military duties in 1715, with the outbreak of the Seventh Ottoman–Venetian War, when he was appointed Provveditore Generale delle Isole, Captain General of the Sea. In 1716, he participated in the successful defence of Corfu, recaptured Butrint and Santa Maura. In 1717, he distinguished himself in the battle off Passavas, along with his brother Carlo: aboard a small felucca, Pisani passed through the enemy ships in full action, reorganized his own ships of the line, encouraged his troops. Once back in Santa Maura, he busied himself with its re-fortification, alongside Count Johann Matthias von der Schulenburg, he recovered Preveza and Vonitsa, for which he was awarded by the Senate the Knighthood of the Order of the Golden Stola. In 1718, he was besieging Dulcigno, he returned with the fleet to Corfu. On 21 September 1718 he was killed in an explosion caused when a thunderbolt struck a gunpowder magazine, his body was transported to Venice.

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