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Abstract algebra

In algebra, a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, fields, vector spaces and algebras; the term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject. For example, the structure of groups is a single object in universal algebra, called variety of groups; as in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many – most – of these problems were in some way related to the theory of algebraic equations. Major themes include: Solving of systems of linear equations, which led to linear algebra Attempts to find formulas for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry Arithmetical investigations of quadratic and higher degree forms and diophantine equations, that directly produced the notions of a ring and ideal.

Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and proceed to establish their properties. This creates a false impression that in algebra axioms had come first and served as a motivation and as a basis of further study; the true order of historical development was exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theory. There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, geometry.

Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work. In 1882, considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element, sufficient in his context.

Permutations were studied by Joseph-Louis Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Lagrange's goal was to understand why equations of third and fourth degree admit formulas for solutions, he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the abstract view of the roots, i.e. as symbols and not as numbers. However, he did not consider composition of permutations. Serendipitously, the first edition of Edward Waring's Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the fundamental theorem of symmetric polynomials, specially considered the relation between the roots of a quartic equation and its resolvent cubic. Mémoire sur la résolution des équations of Alexandre Vandermonde developed the theory of symmetric functions from a different angle, but like Lagrange, with the goal of understanding solvability of algebraic equations.

Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde. Cauchy states quite that Vandermonde had priority over Lagrange for this remarkable idea, which led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups, like his predecessors in the context of solving algebraic equations, his goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four. En route to this goal he introduced the notion of the order of an element of a group, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as if G is a subgroup of S5 whose order is divisible by 5 G contains an element of order 5. However, he got by without formalizing the concept of a group, or of a permutation group; the next step was taken by Évariste Galois in 1832, although his work remained

List of birds of Corsica

This list of birds of Corsica includes the 323 bird species that have been recorded on the island. Of these 102 breed regularly. Corsica is a French island in the Mediterranean Sea located west of the Italian Peninsula, southeast of the French mainland, north of the Italian island of Sardinia. Mountains make up two-thirds of the island; the island measures 183 km in length and 83 km east to west. The status of each species is based on the annotated list by Jean-Claude Thibault and Gilles Bonaccorsi published in 1999. There are 69 species in the "accidental visitor" category for those species that have been recorded on less than ten occasions. Divers are a group of aquatic birds found in many parts of Northern Europe, they are the size of a large duck or small goose, which they somewhat resemble in shape when swimming, but to which they are unrelated. In particular, divers' legs are set far back which assists swimming underwater but makes walking on land difficult. There are five species. Black-throated diver - Gavia arctica - Winter visitor Red-throated diver - Gavia stellata - Accidental visitor Grebes are small to medium-large freshwater diving birds.

They are excellent swimmers and divers. However, they have their feet placed far back on the body. There are 23 species worldwide. Black-necked grebe - Podiceps nigricollis - Passage migrant, winter visitor and occasional breeder Great crested grebe - Podiceps cristatus - Resident breeder, passage migrant and winter visitor Horned grebe - Podiceps auritus - Accidental visitor Little grebe - Tachybaptus ruficollis - Resident breeder and winter visitor Red-necked grebe - Podiceps grisegena - Passage migrant and winter visitor The procellariids are the main group of medium-sized "true petrels", characterised by united nostrils with medium septum and a long outer functional primary. There are two species. Scopoli's shearwater - Calonectis diomedea - Breeding visitor and passage migrant Yelkouan shearwater - Puffinus yelkouan - Winter visitor, passage migrant and occasional breeder The family Hydrobatidae is the northern storm-petrels, small pelagic petrels with a fluttering flight which follow ships.

Storm petrel - Hydrobates pelagicus - Breeding visitor The sulids comprise the gannets and boobies. Both groups are medium to large coastal seabirds. There are ten species worldwide only one of, seen off the coast of Corsica. Gannet - Morus bassanus - Winter visitor and passage migrant Phalacrocoracidae is a family of medium-to-large fish-eating seabirds that includes cormorants and shags. Plumage colouration varies, with the majority having dark plumage. There are 41 species worldwide: two species have been recorded in Corsica. Common shag - Phalacrocorax aristotelis - Resident breeder Cormorant - Phalacrocorax carbo - Winter visitor and passage migrant The Pelecanidae are a family of large water birds, they have a large throat pouch used for catching prey. There are eight species worldwide. Great white pelican - Pelecanus onocrotalus - Accidental visitor The family Ardeidae contains bitterns and egrets. Herons and egrets are medium to large wading birds with long legs. Bitterns tend to be more wary.

Members of Ardeidae fly with their necks retracted, unlike other long-necked birds such as storks and spoonbills. There are 72 species worldwide. Bittern - Botaurus stellata - Passage migrant and winter visitor Cattle egret - Bubulcus ibis - Winter visitor, passage migrant and occasional breeder Great white egret - Ardea alba - Winter visitor Grey heron - Ardea cinerea - Winter visitor Little bittern - Ixobrychus minutus - Breeding visitor and passage migrant Little egret - Egretta garzetta - Passage migrant, winter visitor and occasional breeder Night heron - Nycticorax nycticorax - Passage migrant Purple heron - Ardea purpurea - Breeding visitor Squacco heron - Ardeola ralloides - Passage migrant Western reef heron - Egretta gularis - Accidental visitor Threskiornithidae is a family of large terrestrial and wading birds which includes the ibises and spoonbills, they have broad wings with 11 primary and about 20 secondary feathers. They are strong fliers and, despite their size and weight capable soarers.

There are 36 species worldwide, two occur in Corsica. Common spoonbill - Platalea leucorodia - Passage migrant Glossy ibis - Plegadis falcinellus - Passage migrant Storks are large, long-legged, long-necked, wading birds with long, stout bills. Storks are mute, their nests may be reused for many years. Many species are migratory. There are 19 species. Black stork - Ciconia nigra - Passage migrant White stork - Ciconia ciconia - Passage migrant and winter visitor Flamingos are gregarious wading birds 3 to 5 feet high, found in both

1962 in the Vietnam War

The Viet Cong insurgency expanded in South Vietnam in 1962. U. S. military personnel flew combat missions and accompanied South Vietnamese soldiers in ground operations to find and defeat the insurgents. Secrecy was the official U. S. policy concerning the extent of U. S. military involvement in South Vietnam. The U. S.'s commanding general of MACV, Paul D. Harkins, projected optimism that progress was being made in the war, but that optimism was refuted by the concerns expressed by a large number of more junior officers and civilians. Several prominent magazines and politicians in the U. S. questioned the military strategy the U. S. was pursuing in support of the South Vietnamese government of President Ngô Đình Diệm. Diệm created the Strategic Hamlet Program as his top priority to defeat the Viet Cong; the program intended to cluster South Vietnam's rural dwellers into defended villages where they would be provided with government social services. North Vietnam increased its support to the Viet Cong, infiltrating men and supplies into South Vietnam via the Ho Chi Minh trail.

North Vietnam proposed negotiations to neutralize South Vietnam as had been done in neighboring Laos and Cambodia, but the failure of the Laotian neutrality agreement doomed that initiative. US analyses and statements about progress and problems with the war conflicted or contradicted each other, reflected in this article. 3 JanuaryThe first U. S. military transport aircraft arrived in South Vietnam. The aircraft would be used to transport South Vietnamese soldiers. 4 JanuaryDeputy Secretary of Defense Roswell Gilpatric recommended to General Lyman Lemnitzer, the Chairman of the Joint Chiefs of Staff, that for military operations involving Americans in South Vietnam The Pentagon develop a "suitable cover story, or stories, a public explanation, a statement of no comment...for approval of the Secretary of Defense." 10 JanuaryThe first Operation Ranch Hand mission began. Defoliants were sprayed from U. S. aircraft along several miles of Highway 15 leading from the port of Vũng Tàu to the airforce base at Biên Hòa near Saigon.

Although the United States wished to keep the use of defoliants secret, the South Vietnamese government announced publicly that defoliants supplied by the U. S. were being used to kill vegetation near highway routes. 12 JanuaryOperation Chopper was the first combat operation for American soldiers in Vietnam. U. S. pilots transported about 1,000 South Vietnamese soldiers by helicopter to land and attack Viet Cong guerrillas about 10 miles west of Saigon. The operation was deemed a success. Chopper heralded a new era of air mobility for the U. S. Army, growing as a concept since the Army formed twelve helicopter battalions in 1952 during the Korean War. U. S. President John F. Kennedy said only that the U. S. was helping the South Vietnamese army with "training and transportation." He declined to offer more details about Operation Chopper to avoid "assisting the enemy." 15 JanuarySecretary of Defense Robert McNamara met with his top military advisers. CINCPAC intelligence told him that the Viet Cong now numbered 20,000 to 25,000 and were increasing by 1,000 per month after casualties.

South Vietnam's armed forces had suffered more than 1,000 casualties in the previous month, most by the paramilitary Self Defense Corps. McNamara ordered sending 40,000 M-1 carbines to Vietnam to arm the Self Defense Corps and the Civil Guard, although those two organizations were the sources of many of the Viet Cong's captured weapons. McNamara pressed for a "hold" operation in a single South Vietnamese province. Clear and hold envisioned the ARVN securing the province followed by civic and political action to exclude the Viet Cong permanently. Military Assistance Advisory Group chief General Lionel C. McGarr proposed instead using 2 ARVN divisions in a conventional military sweep focused on killing Viet Cong but without the followup to hold the area. 20 JanuaryAdmiral Harry D. Felt, CINCPAC commander, authorized American advisers to accompany South Vietnamese military forces on combat operations. 2 FebruaryRoger Hilsman, a U. S. State Department official with World War II experience in guerrilla war, submitted a paper entitled "A Strategic Concept for South Vietnam" to President Kennedy and General Taylor.

Drawing on British adviser Robert Grainger Ker Thompson's plan for strategic hamlets, Hilsman said "the struggle for South Vietnam...is a battle for control of the villages." He stated that "the problem presented by the Viet Cong is a political and not a military problem, that to be effective counterinsurgency "must provide the people and the villages with protection and physical security." Hilsman's solution to this problem was similar to that of Thompson's. Hilsman advocated that the South Vietnamese army adopt tactics of mobility and small unit operations. Conventional warfare such as the use of artillery or aerial bombardment to soften up the enemy will "only give advance warning of an operation, permit the Viet Cong to escape, result in the death of uncommitted or wavering civilians whose support is essential for the Viet Cong's ultimate defeat." 3 FebruaryPresident Diệm created by presidential decree the strategic hamlet program. President Diệm's brother, Ngô Đình Nhu, headed the program.

The plans called for rural people to provide manpower and labor to build and defend the strategic hamlets. It was an ambitious program which projected that 7,000 strategic hamlets would be built by the end of 1962 and 12,000 by the end of 1963, thus consolidating nearly all the rural population of South Vietnam. 8 FebruaryThe Military Assistance Command, Vietnam was created to support and assist South Vietnam in defeating the Viet Cong insurgency. MAAG continued to exist, but on