Co

Co, symbol for the element cobalt.

Organization owned by and operated for the benefit of those using its services. Cooperatives have been successful in such fields as the processing and marketing of farm products and the purchasing of other kinds of equipment and raw materials, and in the wholesaling, retailing, electric power, credit and banking, and housing industries. The modern consumer cooperative traces its roots to Britain's Rochdale Society of Equitable Pioneers (1844); the movement spread quickly in northern Europe. In the U.S., agricultural marketing cooperatives developed in rural areas in the 19th century; other contemporary examples include consumer and housing cooperatives. Seealso credit union.

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International organization founded in 1961 to stimulate economic progress and world trade. Based in Paris, the OECD serves as a consultative assembly and a clearinghouse for economic data, and it also coordinates economic aid to developing countries. Its members include Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Luxembourg, Mexico, The Netherlands, New Zealand, Norway, Poland, Portugal, South Korea, Spain, Sweden, Switzerland, Turkey, the U.K., and the U.S.

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Organization of Englishmen and Virginia colonists, established in 1748 to promote trade with American Indians and secure British control of the Ohio River valley for settlement. Activity in the area claimed by France led to the last French and Indian War (1754). A separate organization, the Ohio Co. of Associates (1786), founded Marietta, Ohio, the first permanent settlement north of the Ohio River.

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(1783–1821) British-Canadian fur-trading company. Its operations were centred around the Lake Superior region and the valleys of the Red, Assiniboine, and Saskatchewan rivers. It later spread north and west to the Arctic and Pacific oceans. When its competitor, the Hudson's Bay Co., established a colony on the Red River (1811–12), North West workers destroyed the colony in the Seven Oaks Massacre. Hudson's Bay workers retaliated by destroying the North West post at Fort Gibraltar. The British government pressured the two companies to merge in 1821 as the Hudson's Bay Co.

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Corporation prominent in Canadian economic and political history. It was incorporated in England (May 2, 1670) to seek the Northwest Passage to the Pacific, to occupy lands adjacent to Hudson Bay, and to carry on commerce. The lands granted to the company, known as Rupert's Land, extended from Labrador west to the Rocky Mountains and from the headwaters of the Red River on the southern Canadian border north to Chesterfield Inlet on Hudson Bay. The company first engaged in the fur trade and established trading posts around Hudson Bay. By 1783 competitors had formed the North West Co., and armed clashes continued until the two companies merged in 1821. The company was given exclusive fur-trade rights until 1858, when the monopoly was not renewed and independent companies entered the fur trade. In 1870 the company sold its territories to the government in exchange for £300,000 and mineral rights to lands around the posts and a fertile portion of western Canada. It remained a large fur-collecting and marketing agency until 1991, with extensive real-estate interests and many department stores.

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International organization founded in 1961 to stimulate economic progress and world trade. Based in Paris, the OECD serves as a consultative assembly and a clearinghouse for economic data, and it also coordinates economic aid to developing countries. Its members include Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Luxembourg, Mexico, The Netherlands, New Zealand, Norway, Poland, Portugal, South Korea, Spain, Sweden, Switzerland, Turkey, the U.K., and the U.S.

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Trading company founded by Jean-Baptiste Colbert in 1664, and its successors, established to oversee French commerce with India, East Africa, and other territories of the Indian Ocean and the East Indies. In constant competition with the already-established Dutch East India Co., it mounted expensive expeditions that were often harassed by the Dutch. It also suffered in the French economic crash of 1720, and by 1740 the value of its trade with India was half that of the English East India Co. Its monopoly over French trade with India was ended in 1769, and it languished until its disappearance in the French Revolution.

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Trading company founded by the Dutch in 1602 to protect their trade in the Indian Ocean and to assist in their war of independence from Spain. The Dutch government granted it a trade monopoly in the waters between the Cape of Good Hope and the Straits of Magellan. Under the administration of forceful governors-general, it was able to defeat the British fleet and largely displace the Portuguese in the East Indies. It prospered through most of the 17th century but then began to decline as a trading and sea power; it was dissolved in 1799. Seealso East India Co., French East India Co.

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or English East India Co.

English chartered company formed for trade with East and Southeast Asia and India, incorporated in 1600. It began as a monopolistic trading body, establishing early trading stations at Surat, Madras (now Chennai), Bombay (Mumbai), and Calcutta (Kolkata). Trade in spices was its original focus; this broadened to include cotton, silk, and other goods. In 1708 it merged with a rival and was renamed the United Co. of Merchants of England Trading to the East Indies. Becoming involved in politics, it acted as the chief agent of British imperialism in India in the 18th–19th century, exercising substantial power over much of the subcontinent. The company's activities in China in the 19th century served as a catalyst for the expansion of British influence there; its financing of the tea trade with illegal opium exports led to the first Opium War (1839–42). From the late 18th century it gradually lost both commercial and political control; its autonomy diminished after two acts of Parliament (1773, 1774) established a regulatory board responsible to Parliament, though the act gave the company supreme authority in its domains. It ceased to exist as a legal entity in 1873. See also Dutch East India Co., French East India Co.

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Political party prominent in western Canada in the 1930s and '40s. It was founded in Calgary, Alta., in 1932 by a federation of farm, labour, and socialist parties to transform the capitalist system into a “cooperative commonwealth” by democratic means. It called for the socialization of banks and public ownership of transportation, communication, and natural resources. It won the general election in Saskatchewan in 1944 and took over the provincial government. It won further Saskatchewan elections but declined elsewhere. In 1961 it merged with the New Democratic Party.

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Metallic chemical element, one of the transition elements, chemical symbol Co, atomic number 27. Widely dispersed in small amounts in many minerals and ores, this magnetic, silvery white metal with a faint bluish tinge is used mostly for special alloys (e.g., alnico, tool steel) with exacting applications. At valence 2 or 3 it forms numerous coordination complexes. One is vitamin B₁₂ (cyanocobalamin; see vitamin B complex). Cobalt and its compounds are used in electroplating and colouring ceramics and glass and as lamp filaments, catalysts, a trace element in fertilizers, and paint and varnish driers. The pigment cobalt blue has a variable composition, roughly that of cobalt oxide plus alumina. A radioactive isotope of cobalt emits penetrating gamma rays that are used in radiation therapy.

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Corporation created by the Canadian government in 1918 to operate a number of nationalized railroads (including the old Grand Trunk lines, Intercolonial Railway, National Transcontinental Railway, and Canadian Northern Railway) as one of Canada's two transcontinental railroad systems. Its passenger services were taken over by VIA Rail Canada in 1978, and the company was privatized in 1995. The Canadian National Railway stretches across Canada from Nova Scotia to Vancouver. It bought the Illinois Central Corp. in 1999, thus acquiring a railroad network that links Canada to the Gulf of Mexico.

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Enterprise formed by John Jacob Astor in 1808 that dominated the U.S. fur trade early in the 19th century. The company, considered the first U.S. business monopoly, absorbed or drove out rivals throughout the central and western U.S. Exploration by its trappers and traders helped open the frontier to settlement. By 1834, when Astor sold his company, it had become the largest commercial organization in the U.S.

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In computational complexity theory, co-NP is a complexity class. A problem $mathcal\{X\}$ is a member of co-NP if and only if its complement $overline\{mathcal\{X\}\}$ is in complexity class NP. In simple terms, co-NP is the class of problems for which efficiently verifiable proofs of no instances, sometimes called counterexamples, exist.

An example of an NP-complete problem is the subset sum problem: given a finite set of integers is there a non-empty subset which sums to zero? The complementary problem is in co-NP and asks: "given a finite set of integers, does every non-empty subset have a nonzero sum?" To give a proof of a "no" instance one must specify a non-empty subset which does sum to zero. This proof is then easy to verify.

P, the class of polynomial time solvable problems, is a subset of both NP and co-NP. P is thought to be a strict subset in both cases (and demonstrably cannot be strict in one case but not the other). NP and co-NP are also thought to be unequal. If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP.

This can be shown as follows. Assume that there is an NP-complete problem that is in co-NP. Since all problems in NP can be reduced to this problem it follows that for all problems in NP we can construct a non-deterministic Turing machine that decides the complement of the problem in polynomial time, i.e., NP is a subset of co-NP. From this it follows that the set of complements of the problems in NP is a subset of the set of complements of the problems in co-NP, i.e., co-NP is a subset of NP. Since we already knew that NP is a subset of co-NP it follows that they are the same. The proof for the fact that no co-NP-complete problem can be in NP is symmetrical.

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete (since otherwise NP = co-NP).

An example of a problem which is known to be in NP and in co-NP is integer factorization: given positive integers m and n determine if m has a factor less than n and greater than one. Membership in NP is clear; if m does have such a factor then the factor itself is a certificate. Membership in co-NP is more subtle; one must list the prime factors of m and provide a primality certificate for each one.

Integer factorization is often confused with the closely related primality problem. Both primality testing and factorization have long been known to be NP and co-NP problems. The AKS primality test, published in 2002, proves that primality testing also lies in P, while factorization may or may not have a polynomial-time algorithm.

References

External links

• Complexity Zoo: coNP