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In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic structures such as rings; the term identity element is shortened to identity, when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Let be a set S equipped with a binary operation ∗. An element e of S is called a left identity if e ∗ a = a for all a in S, a right identity if a ∗ e = a for all a in S. If e is both a left identity and a right identity it is called a two-sided identity, or an identity. An identity with respect to addition is called an additive identity and an identity with respect to multiplication is called a multiplicative identity; these need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. The distinction is used most for sets that support both binary operations, such as rings, integral domains, fields.

The multiplicative identity is called unity in the latter context. This should not be confused with a unit in ring theory, any element having a multiplicative inverse. By its own definition, unity itself is a unit; as the last example shows, it is possible for to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities, but if there is both a right identity and a left identity they must be equal, resulting in a single two-sided identity. To see this, note that if l is a left identity and r is a right identity l = l ∗ r = r. In particular, there can never be more than one two-sided identity: if there were two, say e and f e ∗ f would have to be equal to both e and f, it is quite possible for to have no identity element, such as the case of integers under the multiplication operation. Another common example is the cross product of vectors, where the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied.

That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of group without identity element involves the additive semigroup of positive natural numbers. Absorbing element Additive inverse Generalized inverse Identity Inverse element Monoid Pseudo-ring Quasigroup Unital Beauregard, Raymond A.. A First Course In Linear Algebra: with Optional Introduction to Groups and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X Fraleigh, John B. A First Course In Abstract Algebra, Reading: Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N. Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 McCoy, Neal H. Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225 M. Kilp, U. Knauer, A. V. Mikhalev, Monoids and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 14–15

Henderson Haverfield Carson was a U. S. Representative from Ohio. Born on a farm near Cadiz, Carson attended the public and high schools. Cleveland Law School and Baldwin-Wallace College at Berea, Ohio, LL. B. 1919. He became affiliated with the legal department of the Pennsylvania Railroad Co. in 1915. Enlisted in the Field Artillery in 1918, he was transferred to Base Hospital, One Hundred and Nineteenth Unit, Camp Zachary Taylor and served there until honorably discharged in 1919 as a corporal. He was admitted to the bar in 1919 and commenced practice in Canton, Ohio, in 1922, he served as member of the faculty of McKinley Law School 1926-1942, where he received his J. D. degree. Carson was elected as a Republican to the Seventy-eighth Congress, he was an unsuccessful candidate for reelection in 1944 to the Seventy-ninth Congress. Carson was elected to the Eightieth Congress, he was an unsuccessful candidate for reelection in 1948 to the Eighty-first Congress. He resumed the practice of law in Canton and Washington, D.

C.. Resided in Canton, where he died October 5, 1971, he was interred in West Lawn Cemetery. United States Congress. "Henderson H. Carson". Biographical Directory of the United States Congress; this article incorporates public domain material from the Biographical Directory of the United States Congress website http://bioguide.congress.gov

USS Ranger was a sloop-of-war in the Continental Navy in active service in 1777–1780, the first to bear her name. Built in Portsmouth, New Hampshire, she is famed for the one-ship guerilla campaign waged by her captain, Captain John Paul Jones, against the British during the American Revolution. In six months spent in British waters she captured five prizes, staged a single failed attack on the English mainland at Whitehaven, sent the Royal Navy seeking to run her down in the Irish Sea. Jones was detached in Brest, France to take charge of Bonhomme Richard, turning over command of Ranger to his first officer, Lieutenant Thomas Simpson. Under Simpson Ranger went on to capture twenty-four more prizes abroad the Atlantic and along the U. S. coast during 1778 and 1779. Sent to the South in late 1779 to aid the U. S. garrison at Charleston, South Carolina, during the British siege, she continued her predatory ways until forced to take station on the Cooper River, was captured on May 11, 1780 with the fall of the city.

She was brought into the Royal Navy as HMS Halifax. Decommissioned in 1781 in Portsmouth, she was sold that year as a merchant ship. Ranger was launched 10 May 1777 by James Hackett, master shipbuilder, at the shipyard of John Langdon on what is now called Badger's Island in Kittery, Maine. After fitting out, she sailed for France on 1 November 1777, carrying dispatches telling of General Burgoyne's surrender to the commissioners in Paris. On the voyage over, two British prizes were captured. Ranger arrived at Nantes, France, 2 December, where Jones sold the prizes and delivered the news of the victory at Saratoga to Benjamin Franklin. On 14 February 1778, Ranger received a nine-gun salute to the new American flag, the "Stars and Stripes" from the ship of the line Robuste, under Lamotte-Picquet, at Quiberon Bay; this was the first salute from a warship and, the second to an American fighting vessel by a foreign power. Ranger sailed from Brest 10 April 1778, for the Irish Sea and four days captured a prize between the Scilly Isles and Cape Clear.

On 17 April, she sent her back to France. Captain Jones led a raid on the British port of Whitehaven, 23 April, spiking the guns of the fortress, but failing to burn the ships in the harbor. Sailing across the bay to St. Mary's Isle, the American captain planned to seize the Earl of Selkirk and hold him as a hostage to obtain better treatment for American prisoners of war. However, since the Earl was absent, the plan failed. Several Royal Navy vessels were searching for Ranger, Captain Jones sailed across the North Channel to Carrickfergus, Ireland, to induce HMS Drake of 14 guns, to come out and fight. Drake came out against the wind and tide, after an hour's battle, the battered Drake struck her colors, with three Americans and five British killed in the combat. Having made temporary repairs, with a prize crew on Drake, Ranger continued around the west coast of Ireland, capturing a stores ship, arrived at Brest with her prizes on 8 May. Captain Jones was detached to command Bonhomme Richard, leaving Lieutenant Simpson, his first officer, in command.

Ranger departed Brest 21 August, reaching Portsmouth, New Hampshire on 15 October, in company with Providence and Boston, plus three prizes taken in the Atlantic. The sloop departed Portsmouth on 24 February 1779 joining with the Continental Navy ships Queen of France and Warren in preying on British shipping in the North Atlantic. Seven prizes were captured early in April, brought safely into port for sale. On 18 June, Ranger was underway again with Providence and Queen of France, capturing two Jamaicamen in July and nine more vessels off the Grand Banks of Newfoundland. Of the 11 prizes, three were recaptured, but the remaining eight, with their cargoes, were worth over a million dollars when sold in Boston. Underway on 23 November, Ranger was ordered to Commodore Whipple's squadron, arriving at Charleston on 23 December, to support the garrison there under siege by the British. On 24 January 1780, Ranger and Providence, in a short cruise down the coast, captured three transports, loaded with supplies, near Tybee, Georgia.

The British assault force was discovered in the area. Ranger and Providence sailed back to Charleston with the news. Shortly afterwards the British commenced the final push. Although the channel and harbor configuration made naval operations and support difficult, Ranger took a station in the Cooper River, was captured when Charleston fell on 11 May 1780. Ranger was commissioned under the name HMS Halifax, she was decommissioned in Portsmouth, England, in 1781 sold as a merchant vessel for about 3 percent of her original cost. Ranger's specifications were: Begun:January 11, 1777 Launched:May 10, 1777 into the Piscataqua River Location:Rising Castle, now Badger's Island, Maine Departed:Nov 1, 1777 Builder:John Langdon Designer:James Hackett Yard Boss:Tobias Lear IV Officers: John Paul Jones, Captain Thomas Simpson, Portsmouth, 1st Lt. Elijah Hall, Portsmouth, 2nd Lt. Samuel Wallingford, Lt of Marines Dr Ezrah Green, Surgeon Mr Joseph Frazer, Sr Officer of Marines Capt Matthew Parke Crew:145 men including nearly half from Piscataqua area Cost:\$65,000 Continental dollars Rating:Sloop of war Rigging:Square rigged on all three masts with royals, a full set of studding sails Arms:18 nine-pounder guns Painting:Topside black with broad yellow stripe and masthead Dimensions:(Recorded by Royal Navy after capture

The Waddensea of Hamburg between Elbe and Weser is a German Biosphere reserve. It was added in 1992 by the UNESCO in their "man and biosphere"; this site is a part of the Wadden Sea on the North Sea coast, about 40 km north of the city of Bremerhaven in the Land of Hamburg. Situated close to the mouth of the Elbe River, it represents an estuary system, the habitat for the seal Phoca vitulina and a large diversity of birds and fish; the nutrient-rich waters of the Elbe support a high biomass production and are important for fish spawning. The site includes sand and mudflats with channels and saltmarshes; the site has been designated as a National Park, Ramsar site and EU Special Protection Area for wild birds. 38 people are living in the biosphere reserve. The main human activities at the site are recreation and tourism and some agricultural practices such as livestock grazing; the adjacent waters north to the biosphere reserve are intensively used by ships, so that potential oil spills are a major threat to the site.

Pollution stemming from the Elbe River impacts the ecosystem. An information centre on the Neuwerk Island provides an exhibition and educational materials for tourists; the major ecosystem type is temperate coastal/marine zone. The major habitats & land cover types seagrass; this article incorporates text from a free content work. License statement: UNESCO - MAB Biosphere Reserves Directory, UNESCO, UNESCO. To learn how to add open license text to Wikipedia articles, please see this how-to page. For information on reusing text from Wikipedia, please see the terms of use

Saint Peter Basseterre is one of 14 administrative parishes that make up Saint Kitts and Nevis. It is located on the main island of Saint Kitts and the parish capital is Monkey Hill; the parish's 8 square miles consists of forest-covered rugged hills and mountains in the North west and interior, climaxing at the near 3,000 ft Olivee's Mountain. Another notable feature is Monkey Hill, it lies just South-East of Olivee's Mountain. The ridge of hills in the North continues Eastward to the Canada hills before breaking to the Conaree Hills. South of this Northern ridge lies the flat and fertile Basseterre Valley, home to the Monkey Hill Village and most of the other settlements of the parish, it is home to many peasant farms and the Robert L. Bradshaw International Airport, which serves the island; the coastline is marked by a 6-mile long continuous strip of sand that stretches from Saint Mary Cayon, into Saint Peter Basseterre and into Saint George Basseterre, the sands lightening in hue as one descends southward.

Notable areas along the stretch are Conaree Beach. The beaches of Saint Peter Basseterre are used by Leatherback turtles to lay their eggs and have large reef formations offshore. Capital – Monkey Hill Other Villages: Bayford's Canada Conaree John England Village La Fontaine Morgan Heights New Road Saint Peter's Ogee's Parry's StapletonThe parish capital is Monkey Hill and is located in the center of the parish. Other villages located in the central area are New Road, St. Peter's, Parry's, Ogee's, Bayford's and John England Village. Conaree is near the seashore; the fertile lands are in the central area and on the hillsides of the South East Range mountains. Saint Peter's parish's economy is industrial; the Canada Estate area is home to the island's quarry site, many subsidiary industries which manufacture blocks and ready-mixed concrete. Many of the island's trucking and heavy-equipment services are headquartered there; the parish is home to the island's waste disposal area, located North of the village of Conaree.

The parish is home to the island's airport, the larger of the two in the federation. Near the village of Conaree lies two of the island's best-kept sporting facilities, used for cricket and football; the Conaree area was once home to many small inns, which hugged Conaree beach, though many have been deserted. It once housed a gospel radio station but it was moved to Nevis in 1990. Agriculture dominates the Basseterre Valley, peasant farmers rear livestock and grow various fruits and vegetables for local consumption on their small holdings. Tourism is not a key industry in the parish, though it is home to the White House Inn and the ruins of the Château de la Montagne at La Fontaine, once the elaborate and lavish home of St. Kitts' famous former French governor, Phillippe de Longvilliers de Poincy; the parish is home to the Robert L. Bradshaw International Airport, which serves the island of Saint Kitts, is credited as being the finest mid-sized airport in the Caribbean. There are no seaports in the parish due to the harsh Atlantic waves along the coast and the many dangerous reef formations.

There is but one community festival: the St. Peter's Fest, celebrated in October. Many persons from outlying villages such as Monkey Hill and Ogee's watches the splendour in this riveting festival; this festival features calypso shows, queen shows, street jam sessions and late night limes

Muruwari is the Australian Aboriginal language of the Muruwari people, an isolate within the Pama–Nyungan family. Poorly attested Barranbinja may have been a dialect. Muruwari means'to fall with a fighting club in one's hand'; the Muruwari language was collated from many tapes of language material recorded by Jimmy Barker of Brewarrina, Emily Horneville and Shillin Jackson of Goodooga, Robin Campbell of Weilmoringle. The Murawari language was first published by R. H. Mathews in the early 1900s and again by Ian Sims, Judy Trefry, Janet Mathews, Lynette F. Oates. "Pitara yaan Muruwariki" Meaning: "Muruwari is good, sweet talk". Murrawarri Republic