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Abelian group

In abstract algebra, an abelian group called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers, they are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces, are developed; the theory of abelian groups is simpler than that of their non-abelian counterparts, finite abelian groups are well understood. On the other hand, the theory of infinite abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b; the symbol • is a general placeholder for a concretely given operation.

To qualify as an abelian group, the set and operation, must satisfy five requirements known as the abelian group axioms: Closure For all a, b in A, the result of the operation a • b is in A. Associativity For all a, b, c in A, the equation • c = a • holds. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element. Commutativity For all a, b in A, a • b = b • a. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group". There are two main notational conventions for abelian groups -- multiplicative; the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and ordered groups, where an operation is written additively when non-abelian.

To verify that a finite group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this table contains the product gi ⋅ gj; the group is only if this table is symmetric about the main diagonal. This is true since if the group is abelian gi ⋅ gj = gj ⋅ gi; this implies that the th entry of the table equals the th entry, thus the table is symmetric about the main diagonal. For the integers and the operation addition "+", the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer n has an additive inverse, −n, the addition operation is commutative since m + n = n + m for any two integers m and n; every cyclic group G is abelian, because if x, y are in G xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ; every ring is an abelian group with respect to its addition operation.

In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, the nonzero real numbers are an abelian group under multiplication; every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups and direct sums of abelian groups are again abelian; the finite simple abelian groups are the cyclic groups of prime order. The concepts of abelian group and Z-module agree. More every Z-module is an abelian group with its operation of addition, every abelian group is a module over the ring of integers Z in a unique way. In general, matrices invertible matrices, do not form an abelian group under multiplication because matrix multiplication is not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of 2×2 rotation matrices. Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.

See Section 6.5 of Cox for more information on the historical background. If n is a natural number and x is an element of an abelian group G written additively nx can be defined as x + x +... + x and x = −. In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. Theorems about abelian groups can be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups, a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group; the former may be written as a direct sum of finitely many groups of the form Z/pkZ for p prime, the latter is a direct sum of finitely many copies of Z. If f, g: G → H are two group homomorphisms between abelian groups their sum f + g, defined by = f + g, is again a homomorphism.

The set Hom of all group homomorphisms from G to H thus turns in

James MacRitchie

James MacRitchie was Municipal Engineer to the Singapore Municipal Commission from 1883 to 1895. Singapore's oldest reservoir Macritchie Reservoir was named after him in 1922. MacRitchie was born in Southampton on the son of the P&O superintendent engineer. Educated in the Dollar Academy the Dollar Institution, Scotland and at the Universities of Glasgow and Edinburgh, MacRitchie worked as a civil engineer on bridges and docks in Scotland, including the Albert Bridge, Glasgow where he gained experience on the use of concrete foundations encased in iron caissons. MacRitchie went to Calcutta in 1867 to work as an assistant engineer on the Calcutta water project the Palta Water Works, built between 1865 and 1868 and is reported to be'the first modern waterworks in Asia'. Since renamed the Indira Gandhi Water Treatment Works, it had a capacity of 6 million gallons per day in 1865 but has since been expanded significantly. MacRitchie married Miss Cameron, daughter of the editor of The Straits Times, in 1883.

Upon completion of the Calcutta water project, MacRitchie spent seven years working for the Japan government as Lighthouse Engineer. According to his obituary, published in Singapore, he was responsible for designing and building lighthouses around the coast of Japan and is said to have been the last foreigner to have held this position. MacRitchie worked as an assistant to Richard Henry Brunton the so-called'father of Japanese lighthouses'. See also'Building Japan 1868-1876' by Richard Henry Brunton. After Japan, he moved to Brazil where he worked on water works, gas works and tramways before arriving in Singapore in 1883. MacRitchie took over as Municipal Engineer from Thomas Cargill in 1883, his report to the Municipal Commission in 1884 demonstrates the wide range of duties he had to undertake in Singapore as well as his diligence. During his career he was responsible for numerous works and immense improvements in the streets, public markets and other infrastructure in Singapore. MacRitiche was the architect for the new cast-iron octagonal-shaped Lau Pa Sat market, relocated to its present position in 1894.

MacRitchie followed the original style of George Drumgoole Coleman's 1838 market. His first major Singapore project was the iron Coleman Street Bridge which replaced a rickety wooden bridge, his design of small iron bridges first used in Singapore was soon being used throughout Malaya. MacRitchie applied himself to the problem of night soil and studied various alternatives for Singapore including a pneumatic system and conversion to'poudrette'; the Municipal Commission sent him on a 3-month fact finding trip to India in 1893 to inspect sanitation, night soil and water supply systems and to bring best practices back to Singapore. The report he prepared had'all the data and, in many cases, the worked out detail for dealing with the sewage and refuse of Singapore' and was described as'a valuable contribution to municipal engineering in the East'. Water supply was a priority for MacRitchie, he replaced water mains and introduced a water filtration plant after which the water supply was said to be second to none in the region.

He drew on best practice from England and experimented with a new'Polarite' filtration material for Singapore. His experience was called upon in Penang where he was asked to advise the Municipal Commission on a new reservoir in 1890 It may have been on one of these visits that he worked on the design of the reservoir of the Penang Botanic Gardens waterfall completed in 1892, he is best remembered for the Thomson Road or the Impounding Reservoir, known today as the MacRitchie Reservoir. Construction proved a challenge and was delayed when the works were flooded four times during the 1891-94 construction period; when completed, the'magnificent waterworks', one and a half miles in length and five and a half miles in circumference, provided the city with 650 million gallons of storage. MacRitchie calculated that this gave Singapore 130 days storage and supply based on daily consumption of 3.2 million gallons of water per day in 1895. MacRitchie noted that the demand for water had more than doubled from 1.4 million gallons per day in 1885 which explains the enormous value and importance of the Impounding Reservoir.

The historic timeline of the MacRitchie Reservoir Park is shown in the footnoteThree Reservoirs in Singapore are named after influential Municipal Engineers: James MacRitchie, Robert Peirce and David J. Murnane. MacRitchie became embroiled in a public debate about the use of electricity for public lighting in Singapore with the Colonial Engineer, Major Henry McCallum, in favour of electric light. MacRitchie held that'the extra light was uncalled for' and the Municipality could not afford the cost of electric light. One commentator at the time was concerned that Singapore might fall behind other'Eastern Towns', including Rangoon, which had introduced electricity. Although MacRitchie was an expert in water works, he had less experience with electricity. Many of the assumptions in his 1892 report on the use of electric light, as an alternative to gas, were subject to extensively-reported criticism by the electrical engineer O. V. Thomas. However, Thomas had an obvious commercial interest as he was a tenderer to supply the power plant and overhead cables.

Another commentator defended MacRitchie as'a competent civil engineer, an excellent judge of costs and estimates and the difficulties of labour in the East' MacRitchie died 26 April 1895, aged 47, at his home'Woodside', Grange Road, Singapore. The esteem in which

Bloody Bones (novel)

Bloody Bones is a horror/mystery novel by American writer Laurell K. Hamilton, the fifth book in the Anita Blake: Vampire Hunter series. Bloody Bones continues the adventures of Anita Blake; this time, Anita travels to Branson, Missouri and is enmeshed in a series of supernatural murders and disappearances that she and her vampire would-be lover, Jean-Claude must resolve. As with its predecessors, Bloody Bones blends elements of supernatural and police procedural fiction. Within the book, "Bloody Bones" is the name of a restaurant, operated by two of the principal characters in the novel and Dorcas Bouvier. Hamilton employed the practice of naming each novel after a fictional location within the story for most of the Anita Blake series. In this case, the restaurant itself is named after a character in the novel and Bloody Bones, making the title somewhat eponymous. Bloody Bones begins on Saint Patrick's Day, shortly after the events of the previous Anita Blake novel, The Lunatic Cafe. Like the previous novels, the novel opens with Anita considering a possible job.

This time, her manager, Bert, is calculating a possible bid for a mass zombie raising in Branson, Missouri. Bert explains that a law firm is soliciting bids to raise an entire graveyard in order to determine who owns a piece of land needed for a resort complex; the graves are unmarked and may contain corpses at least 100 years old, Anita finds out there are some that are much older than that, which will make the raising difficult. In Anita's opinion, she is the only person in the world who might be able to raise that many ancient unmarked graves without a human sacrifice, she agrees to take the job, takes Larry along to boost her powers, as a training experience.. Arriving in Branson, Anita meets Raymond Stirling, the lawyer in charge of the development project and his assistants, Lionel Bayard, Ms Harrison and Beau, learns that Stirling is in a dispute with Magnus and Dorcas Bouvier, two siblings who claim to own the land at issue and refuse to sell. If the corpses on the land confirm that it belongs to the Bouviers, Stirling's project will be unable to continue.

After reviewing the site and making plans to explore the site further that evening, Anita receives a call from Dolph. Dolph asks Anita for advice on a crime scene back in St. Louis and asks her to assist the local police with a nearby crime scene. Anita and Larry drive to the scene and meet Sergeant Freemont, who appears to want to crack the case herself and resents their intrusion. Anita inspects the murder victims—three teen-aged or younger boys cut apart with a blade; each of the boys' faces have been disfigured or removed, Freemont reveals that a teenaged boy and girl were murdered earlier, with similar wounds. Anita warns Freemont that in her opinion, the boys were cut apart by a sword wielded by something as fast and strong as a vampire, with enough mental power to hold two of the boys motionless while killing the third. Larry is shaken by viewing his first murder scene. Anita and Larry go to the Bouviers' restaurant, named "Bloody Bones," to investigate the land dispute and to get dinner.

There, they meet Dorrie, each of whom is part-fey. Magnus is using glamour to host a date night. By touching the restaurant patrons, he makes them irresistibly attractive for one night, in return for drawing some power for himself. After trying unsuccessfully to seduce Anita, Magnus is coy about why the Bouviers refuse to sell their land. Magnus admits to destroying several trees outside the restaurant while in a drunken rage, causing Anita to consider him as a suspect for the recent killings. During dinner, Dolph pages Anita again, asks her to assist on another possible local vampire crime. Anita tells Dolph that Magnus is part-fey and a potential suspect Anita and Larry drive to the home of Mr. and Mrs. Quinlan. There, they meet Sheriff David St. John, his wife Beth, Deputy Zack Coltraine, Mr. and Mrs. Quinlan, their son Jeff. Jeff's older sister, Ellie is lying in dead of a vampire bite. Anita and Larry deduce that her death was voluntary, learn that Ellie's boyfriend Andy disappeared, they guess that Ellie's boyfriend has been raised as a vampire and turned her as well, but Mr. Quinlan refuses to believe them and demands that Anita stake Ellie to prevent her from rising.

Anita asks him to wait twenty-four hours to "cool off" and promises to stake Ellie if her father demands it after that time. After instructing the Quinlans to place the Host at each doorway to prevent any vampires from reentering the home, Anita explains that the vampire that turned Ellie has a resting place nearby, that they may catch it if they attempt a nighttime hunt, she heads out into the woods after it, together with Larry, Sheriff St. John, Deputy Coltraine, two other police officers and Granger. During the hunt, Anita learns that Wallace was a survivor of a vampire attack and shows him her own scars in an effort to put him at ease. During the hunt and the others are ambushed by a pack of vampires. In the fight, Anita kills two vampires, but Granger is bitten, Wallace's arm is broken, Xavier kills Coltrain with a sword. While the hunters regroup, now under vampire control, attempts to shoot Larry, Anita is forced to kill him; the group hears screams from the Quinlan home, St. John and Anita run for the house, leaving Larry and Wallace to bring up the rear.

When Anita gets to the house, Beth St. John is dead and Jeff has been

Lucky Star (2013 film)

Lucky Star is a 2013 Malayalam comedy-drama film written and directed by Deepu Anthikkad in his directorial debut. The film stars Mukesh alongside Rachana Narayanankutty, who debuts in cinema; the film is about. Lucky Star released on 8 March 2013, it was well received in theatres. Jayaram as Ranjith Mukesh as Dr. John Chitillapally Rachana Narayanankutty as Janaki Viroj Dasani as Lucky Pooja Ramachandran as Swapna Mamukoya as Pappan T. G. Ravi as Bhaskaran Ammu Ramachandran as Sumithra Sreekumar Oneindia gave the film 3 stars out of 5 and stated "the first half is funny and entertaining, the second half gets a little serious." While Indiaglitz called it " A light-hearted entertainer that's an easy watch." Times of India gave 3.5 stars out of 5. Sify wrote "this film is not bad and has its moments as well, but it could have been much better.” Rediff gave 2.5 stars out of 5 and wrote "Lucky Star could have been a much better film if a little more thought had been given to how the film should progress in the second half."

The film's soundtrack contains all composed by Ratheesh Vegha. Lyrics by Rafeeq Ahamed and Arjun Vinod Varma. Lucky Star on IMDb

Don Rowe

Donald Howard Rowe was an American player and pitching coach in professional baseball. A left-handed pitcher, Rowe had a 14-year professional career and spent only one partial season in Major League Baseball as a member of the 1963 New York Mets. Rowe was a native of Brawley and attended Long Beach State University, he signed with the Pittsburgh Pirates in 1954, in his tenth pro season, he debuted with the Mets on April 9, 1963. His final appearance was on July 18, 1963. After retiring from playing, Rowe became the pitching coach for the Chicago White Sox in 1988 and the Milwaukee Brewers from 1992 to 1998, worked as a pitching coach in the farm systems of the California Angels, San Francisco Giants, White Sox and Brewers, he coached football and tennis at Golden West College, Huntington Beach, California. Rowe died from Parkinson's disease in Newport Beach, California, at the age of 69. Career statistics and player information from Baseball-Reference Baseball Reference Baseball Gauge Retrosheet Venezuelan Professional Baseball League

John W. Dorsey

John Wesley Dorsey, Jr. served as the acting President of the University of Maryland, College Park from 1974 to 1975 and as the acting chancellor of the university from August 1974 to June 1975. Dorsey was born in Maryland in 1936 and graduated from the University of Maryland with a B. S. in economics in 1958. He continued to attend graduate school at Harvard University receiving an M. A. in 1962 and a Ph. D. in economics in 1964 and receiving a certificate from the London School of Economics. Dorsey started teaching at College Park as an assistant professor in 1963 and became director of the Bureau of Business and Economic Research in 1966. Dorsey served as the vice chancellor for administrative affairs from 1970 to 1977, became the chancellor of the University of Maryland, Baltimore County. From 1986 to 1989, he was Special Assistant to the President of the University. In 1989, he returned to the Department of Economics at the University of Maryland from which he retired as an emeritus professor in 2001.

Dorsey has worked as a SECU Board member since 1975, serving as Chairman for four years and as Vice Chairman for six years. He died of respiratory failure on July 28, 2014. Records of the Office of the President, University of Maryland