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Complement (set theory)

In set theory, the complement of a set A refers to elements not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A; the relative complement of A with respect to a set B termed the difference of sets A and B, written B \ A, is the set of elements in B but not in A. If A is a set the absolute complement of A is the set of elements not in A, within a larger set, implicitly defined. In other words, let U be a set that contains all the elements under study. Formally: A c =; the absolute complement of A is denoted by A c. Other notations include A ¯, A ′, ∁ U A, ∁ A. Assume that the universe is the set of integers. If A is the set of odd numbers the complement of A is the set of numbers. If B is the set of multiples of 3 the complement of B is the set of numbers congruent to 1 or 2 modulo 3. Assume that the universe is the standard 52-card deck. If the set A is the suit of spades the complement of A is the union of the suits of clubs and hearts.

If the set B is the union of the suits of clubs and diamonds the complement of B is the union of the suits of hearts and spades. Let A and B be two sets in a universe U; the following identities capture important properties of absolute complements: De Morgan's laws: c = A c ∩ B c. C = A c ∪ B c. Complement laws: A ∪ A c = U. A ∩ A c = ∅. ∅ c = U. U c = ∅. If A ⊆ B B c ⊆ A c.. Involution or double complement law: c = A. Relationships between relative and absolute complements: A ∖ B = A ∩ B c. C = A c ∪ B. Relationship with set difference: A c ∖ B c = B ∖ A; the first two complement laws above show that if A is a non-empty, proper subset of U is a partition of U. If A and B are sets the relative complement of A in B termed the set difference of B and A, is the set of elements in B but not in A; the relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard. It is sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements b − a, where b is taken from B and a from A.

Formally: B ∖ A =. ∖ =. ∖ =. If R is the set of real numbers and Q is the set of rational numbers R ∖ Q is the set of irrational numbers. Let A, B, C be three sets; the following identities capture notable properties of relative complements: C ∖ = ∪

County of Toulouse

The County of Toulouse was a territory in southern France consisting of the city of Toulouse and its environs, ruled by the Count of Toulouse from the late 9th century until the late 13th century. The territory is the center of a region known as Occitania. Under the Carolingians and dukes were appointed by the royal court; this office became hereditary. The counts of Toulouse ruled the city of Toulouse and its surrounding county from the late 9th century until 1271. At times, the counts of Toulouse or family members were counts of Quercy, Albi, Nîmes and marquesses of Gothia; the Roman province of Gallia Narbonensis fell to the Visigothic Kingdom in the 5th century. Septimania, the Visigothic province corresponding to the county of Toulouse, fell to the Emirate of Córdoba in the 750s before it was conquered into the Kingdom of the Franks by Pippin the Short in 759 following the Siege of Narbonne. Septimania within the Frankish realm would become known as Gothia or Marca Gothica by the end of the 9th century.

Pippin was followed by his sons Charlemagne and Carloman. Because of this event, Hunald II, son of the late Duke Waifer, raised an insurrection against Frankish power in Aquitaine. Charlemagne soon defeated him. In 771, Carloman died and Charlemagne was left as the only ruler of the Frankish realm. In 778, Charlemagne led his army into Spain against the Arabs. On his way back the famous event of Roncesvalles occurred: Charlemagne's rear-guard was attacked in the pass of the same name by some Basque warriors; this led him to realize that Frankish power in Gascony and Aquitaine was still feeble, that the local populations were not loyal to the Franks. That same year, he reorganized the administration of the region: direct Frankish administration was imposed, Frankish counts were established in key cities, such as Toulouse. In 781, Charlemagne set up the Kingdom of Aquitaine, comprising the whole of Aquitaine plus the Mediterranean coast from Narbonne to Nîmes, gave the crown of Aquitaine to his three-year-old son Louis.

Other such sub-kingdoms were created inside the wider Carolingian empire in places such as Bavaria or Lombardy. They were meant to ensure the loyalty of local populations in territories freshly conquered and with strong local idiosyncrasies. Crowns were given to the sons of Charlemagne. General supervision of this Basque frontier seems to have been placed in the hands of Chorson, count or duke of Toulouse; these politics displeased the Basques, in 787 or 789 Chorson was captured by Odalric "the Basque" son of Duke Lupus, who forced Chorson into an agreement which Charlemagne considered so shameful that he replaced him by the Count William in 790. Toulouse was a major Carolingian military stronghold close to Muslim Spain. Military campaigns against the Muslims were launched from Toulouse every year during Charlemagne's reign. Barcelona was conquered in 801, as well as a large part of Catalonia. Together with the northern areas of Aragon and Navarre along the Pyrenees, the region became the southern march of the Frankish empire.

In 814, Charlemagne died, his only surviving son was Louis, king of Aquitaine, who became Emperor Louis the Pious. The Kingdom of Aquitaine was transmitted to the second son of Louis the Pious. Gothia was detached from the Kingdom of Aquitaine and administered directly by the emperor, thus recreating the limits of the former duchy of Aquitaine. Problems soon arose. Louis the Pious had three sons, in 817 he arranged an early allocation of the shares in the future inheritance of the empire: Pippin was confirmed king in Aquitaine, Louis the German was made king in Bavaria, while the eldest son Lothair was made co-emperor with future authority over his brothers. In 823, Charles the Bald was born from the second wife of Louis the Pious. Soon enough, she wished to place her son in the line of succession. Louis the Pious was rather weak, a fight started between the three sons on one side, their father and his new wife on the other side, which would lead to the total collapse of the Frankish empire. Louis the Pious was toppled from power re-installed toppled re-installed again.

In 838 Pippin I of Aquitaine died, Louis the Pious and his wife managed to install Charles the Bald as the new king of Aquitaine. At the Assembly of Worms in 839 the empire was re-divided: Charles the Bald was given the western part of the empire, Lothar the central and eastern part, while Louis the German was keeping only Bavaria. Pippin II of Aquitaine, the son of Pippin I, was not going to accept such a decision, he was hailed king by the Aquitanians, he resisted his grandfather. Louis the German in Bavaria opposed his father's decision. Louis the Pious died in 840. Lothar, the eldest son, claimed a general war broke out. First allied with his nephew Pippin II, Louis the German soon allied with his half-brother Charles the Bald and they jointly defeated Lothar. In August 843, they signed the most important treaty in European history, the Treaty of Verdun; the empire was divided in three: Charles the Bald was given the western part, Francia Occidentalis, Louis the German was given the eastern part, Francia Orientalis, while Lothar was given the central part, soon to be conquered and divided by his two brothers.

Following the Treaty of Verdun, Charles the Bald moved so

James Robson (poet and songwriter)

James Robson was a Northumbrian landowner, songwriter, "political criminal" and one time Jacobite rebel. James Robson was born in Northumberland in, or near, Thropton, a small village near to Rothbury and was described as a “freeholder”, in the village, he became a Jacobite, siding with James Francis Edward Stuart and took part in the First Jacobite Rising. During this uprising, General Carpenter, after marching his men and horses into Scotland, returned to Newcastle tired and weary, but was ordered to meet the Jacobite "rebels" at Lancaster, he met the rebels at Preston, where the rebels, after defending the place for some time, surrendered to the King's troops. Among the rebels was Mr James Robson of Throston, imprisoned in Preston Jail, he was a poet, but whether this was by natural bent, or because of his incarceration, is not clear. It appears that he may have been a musician, as many of the historical documents describe him as "a leader of a band in the Pretender’s army", although these does not make clear whether this means a "group of musicians" or just a "group of rebels", John Bell in his Rhymes of Northern Bards, states that "James Robson ….. was at that time a musician in the rebel army".

According to "Archaeologia Aeliana" he married Mary, who died in 1723. According to “Archaeologia Aeliana” James Robson had been freed. In the Rothbury Parish Records, in the section devoted to Papists and Dissenters, the baptismal records appertaining to Robson, regular, ceased in 1714, but in 1723, in the burial records, is a “Mary, wife of James Robson, Thropton”, on 6 May 1757 is the burial of “James Robson of Thropton” These include:- Satyr Upon Women – described by Bell as “This song is imperfectly compiled from part of a Satire upon Women wrote in Preston prison in 1715, by Mr James Robson, a freeholder in Thropton, near Rothbury, Northumberland, at that time a musician in the rebel army”; the two descriptions of this song in The Local Historian's Table Book and An historical view of the County by Eneas Mackenzie are identical, read “Mr James Robson, stone-mason of Thropton, was leader of the band in the Pretender’s army in 1715. He wrote a satire on women, several other poetical pieces, while confined prisoner at Preston in Lancashire.

It is said that he sang the satire at an iron-barred window, looking into a garden, where a lady and her maid were walking. When the song was finished, the former observed, "That young man seems severe upon our sex, but he is singing more from oppression than pleasure. “A Song – composed by Mr James Wilson, of Cawsey Park, on Mr Coughron and family leaving Hebron Hill”. This Mr Coughron was the brother of the celebrated mathematician George Coughron who died of smallpox in Newcastle 7 January 1774 aged 21 Hobby Elliott – This song is said to have been written by Mr James Robson, Stone Mason, at Thropton, near Rothbury, leader of the band in the Pretender's army, in 1715. Geordie dialect words John Bell Rhymes of Northern Bards History and gazetteer of the Counties of Durham and Northumberland ……by Wm Parson and Wm White published 1828 page 484 Bell’s Rhymes of Northern Bards 1812 Archaeologia Aelians – volume XVI of 1844 – page 110 The Local Historian's Table Book of remarkable occurrences, historical facts, traditions and descriptive ballads &c. &c. connected with the Counties of Newcastle-upon-Tyne and Durham by Moses Aaron Richardson volume 1 published 1841 – Page 350 The Local Historian's Table Book of remarkable occurrences, historical facts, traditions and descriptive ballads &c. &c. connected with the Counties of Newcastle-upon-Tyne and Durham by Moses Aaron Richardson volume 2 published 1841 – page 225 An historical and descriptive view of the County of Northumberland and of those parts of the County of Durham situated Nborth of the River Eneas Mackenzie published in 1825 – see page 69 Allan’s Illustrated Edition of Tyneside songs and readings

Necropolis of Carenque

The Necropolis of Carenque is an archaeological site consisting of three well-preserved late Neolithic collective sepulchres or tombs, dating back to about 3000 BCE, which are dug into smooth limestone outcrops. It is situated in the municipality of Amadora in the Lisbon District of Portugal; the site was discovered and excavated in 1932 and was classified as a National Monument in 1936. The short time between discovery in 1932 and this classification owed much to the important role played at the time by the archaeologist responsible for the caves' discovery, Manuel Heleno, acting-Director of what is now the National Archaeology Museum of Portugal. Tombs such as these are designated as'artificial caves' or hypogea because they were excavated in the rock, they are part of a Mediterranean funeral tradition, practised in the area of what is now known as Portugal. Other examples include the Artificial caves of Casal do Pardo and the Artificial caves of Alapraia, there are many examples of megalithic dolmen, where burials were made inside structures made of large stones.

The first depositions of corpses in the Necropolis of Carenque go back to the end of the Neolithic period, when there was an agro-pastoral economy in the area, as opposed to hunting and gathering. There is evidence for its use in the Chalcolithic during the Copper Age until around 2000 BCE. Dug into Albian-Cenomanian limestone, the three caves each have an access corridor that communicates with the funeral chamber through a small portal; the chamber has a hole, or skylight, in its roof and this and the corridor were closed with limestone slabs. In his field notebooks Heleno refers to a small fourth cave, almost destroyed. Little is known of the dead buried there, such as, it appears the first corpses may have been placed against the walls, in a fetal position, surrounded by votive offerings. The use of the Necropolis in the Copper Age disturbed the original arrangement of the bones and artifacts, making it difficult to reconstitute the arrangements inside the tomb. Items found during the excavation included numerous human bones as well as ceramics, including those from the beaker culture, lithic and metallic materials, such as arrowheads and daggers.

Votive offerings identified included limestone idols, such as smooth or decorated cylinders, as well as adzes, bone pins and miscellaneous beads. Most of the artifacts and human remains are on deposit at the National Archaeology Museum in Lisbon and have been extensively studied; the necropolis is one of the best-preserved of its kind. The similarity between the design of the three caves indicates that the builders were following a clear plan. Cave 1, to the east, has a circular chamber with a diameter of about 4.5 metres. The skylight was 1.30 metres high. The narrow entrance corridor was around 5 metres long and was originally covered with limestone slabs; the entrance to this cave faces northeast, while the entrances to the other two face southeast. Cave 2 has a circular chamber with a diameter of 4.5 metres and a corridor of 5.5 metres. The chamber is about 1.75 metres high and the skylight has a diameter of 1.60 metres. Cave 3, the smallest, has a chamber with a diameter of just under 4 metres.

The skylight appears to be larger than those of the other caves but this is because it has eroded over the years. Media related to Necrópole de Carenque at Wikimedia Commons

Chile Con Soul

Chile Con Soul is the ninth album by The Jazz Crusaders recorded in 1965 and released on the Pacific Jazz label. AllMusic rated the album with 4 stars. "Agua Dulce" - 5:25 "Soul Bourgeoisie" - 7:44 "Ontem a Note" - 4:19 "Tough Talk" - 2:38 "Tacos" - 4:08 "Latin Bit" - 4:08 "The Breeze and I" - 5:08 "Dulzura" - 4:37 Wayne Henderson - trombone Wilton Felder - tenor saxophone Hubert Laws - flute Joe Sample - piano Clare Fischer - organ Al McKibbon - bass Stix Hooper - drums Carlos Vidal - congas Hungaria Garcia - timbales, cowbells

Adela Investment Company

The Adela Investment Company was a private investment corporation created by multinational companies to promote economic and social development in Latin America and the Caribbean. Adela operated from 1965 to 1980 and was dissolved in 1994. During that period, Adela financed over a thousand private ventures throughout the region, its model for development was replicated in Asia. Its investment policies were adapted by hundreds of investment funds operating throughout the developing world. In 1962, President John F. Kennedy proposed the Alliance for Progress between the United States and the developing countries of Latin America and the Caribbean; the idea was to encourage Latin America to undertake social reforms. In exchange, the United States and the international aid agencies would provide the funding to support the reforms. A Republican Senator from New York, Jacob Javits, proposed that the world's business community join together to form a new corporation that would underwrite private sector investments throughout Latin America.

He wanted the private sector to complement governmental programs under the Alliance for Progress. He presented his ideas to a NATO parliamentary meeting in Paris in November 1963, his proposal was endorsed by the delegates, who passed a resolution recommending that the Secretary Generals of the Organization of Economic Cooperation and Development, the Organization of American States and the Inter-American Development Bank give their support. A task force was created and funded by the Ford Foundation, led by George S. Moore, President of the First National City Bank, Hermann Abs of Germany's Deutsche Bank and Gianni Agnelli of Fiat, among others; the group proposed to create ` Adela' for short. They were joined by Senator Javits in inviting private corporations from Europe, North America and Japan to capitalize this new venture. Adela was incorporated in Luxembourg on September 24, 1964, it hired a Swiss engineer, Ernst Keller, as its first Managing Director and opened its operations center in Lima, Peru in early 1965.

Adela's founders set out five objectives: To make a significant contribution to the economic development of Latin America. To encourage local and foreign investment throughout the region. To earn a reasonable return on its investments. To encourage Latin American countries to maintain a favorable climate for foreign investment. To work with its shareholders. A total of 242 private banks and companies invested in Adela. No one shareholder was allowed to own more than 1% of the company so that no single investor could exert excessive influence over its operations; as a bloc, Europe provided the largest number of shareholders, although the United States was the most important single country. Shareholders came from Japan and six Latin American nations; the combined net worth of Adela's shareholders was the highest of any private corporation ever. Adela's founders included many of the best known company presidents and chairmen, including men such as Marcus Wallenburg Sr. Chairman of Stockholms Enskilda Bank, Mogens Pagh, Chairman of Denmark's East Asiatic Company, Friederick Philips, Chairman of Gloeilampen Fabrieken of Eindhoven of the Netherlands, W. Earle McLaughlin, Chairman of the Royal Bank of Canada.

Henry Ford II of the Ford Motor Company, William Blackie of Caterpillar, Sam Carpenter III of du Pont, Max Eisenring, Chairman of Swiss Reinsurance, Thomas J. Watson Jr. of International Business Machines, Yoshinzane Miki of The Fuji Bank, Oskar Nathan of Dresdner Bank and Samuel Schweiser of Swiss Bank Corporation. Adela's Managing Director, Ernst Keller, was advised on business issues by Peter Drucker of the United States, on political matters by Alejandro Orfila of Argentina, a two time Secretary General of the Organization of American States, on shareholder relations with Marcus Wallenburg Sr. of Sweden. Adela was designed to make minority investments in private corporations in the developing countries of Latin America; the company sought to divest its holdings once they became profitable and would reinvest the proceeds in a new business. The company made medium term loans for specific projects, but its primary function was to provide risk capital to Latin American businesses. On occasion, Adela acted as an entrepreneur by taking the lead in creating a new business.

Adela worked with the Inter-American Development Bank, the International Finance Corporation and the United States Agency for International Development. The IDB and the IFC both made long term loans to Adela, while USAID provided long-term funding to financial institutions created by Adela with local Latin American investors. Adela funded over 1,000 private companies for a total amount of $2.5 billion. The size of these investments varied from small printing companies to large steel mills and paper, hotels and chemicals, it helped start local development finance companies, which themselves financed more businesses. At its peak, assets reached half a billion dollars; the Adela model was replicated in Asia based in Tokyo, in Africa based in Geneva and in agriculture based in Miami. All were funded investments in private enterprise. Adela earned a profit in its first year of operations and continued posting strong earnings for its first decade. However, a growing number of its investments were failing, which forced the company to cease writing new business in 1980.

Control of the company was taken over by a creditors committee, m