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In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. In the nineteenth century number systems called quaternions, coquaternions and octonions became established concepts in mathematical literature, added to the real and complex numbers; the concept of a hypercomplex number covered them all, called for a discipline to explain and classify them. The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, was carried forward by his son Charles Sanders Peirce. Most they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications; the Cayley–Dickson construction used involutions to generate complex numbers and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, the octonions, the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, ℍ.

In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8, it was matrix algebra. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices. Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by matrices, or direct sums of systems of matrices. From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn's thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number; as Hawkins explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory".

In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers, translated in 1989. Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers, including the role of such luminaries as Theodor Molien and Eduard Study. For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra. A definition of a hypercomplex number is given by Kantor & Solodovnikov as an element of a finite-dimensional algebra over the real numbers, unital and distributive. Elements are generated with real number coefficients for a basis. Where possible, it is conventional to choose the basis so that i k 2 ∈. A technical approach to hypercomplex numbers directs attention first to those of dimension two. Theorem: Up to isomorphism, there are three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, the dual numbers. In particular, every 2-dimensional unital algebra over the reals is associative.

Proof: Since the algebra is 2-dimensional, we can pick a basis. Since the algebra is closed under squaring, the non-real basis element u squares to a linear combination of 1 and u: u 2 = a 0 + a 1 u for some real numbers a0 and a1. Using the common method of completing the square by subtracting a1u and adding the quadratic complement a21 / 4 to both sides yields u 2 − a 1 u + a 1 2 4 = a 0 + a 1 2 4, thus 2 = u ~ 2 where u ~ 2 = a 0 + a 1 2 4. The three cases depend on this real value: If 4a0 = −a12, the above formula yields ũ2 = 0. Hence, ũ can directly be identified with the nilpotent element ϵ of the basis of the dual numbers. If 4a0 > −a12, the above formula yields ũ2 > 0. This leads to the split-complex numbers which have normalized basis with j 2 = + 1. To obtain

Beth Sullivan is an American film and television writer and producer, best known as the creator and executive producer of the CBS series Doctor Quinn, Medicine Woman. Sullivan served as the sole executive producer of Doctor Quinn, Medicine Woman. In doing so, she was the first woman to succeed in a singular capacity in the traditionally male arena of one-hour drama "showrunners." The series received numerous Emmy and People’s Choice Award nominations and won several of each, plus a Golden Globe Award. In addition, the show attained widespread community acknowledgment, receiving the Heroes Memorial Foundation of the United States of America Founder’s Award for honorable recognition of Native Americans, the Genesis Award for spotlighting animal issues, the Family Film Award for promoting family values, the Environmental Media Award for raising environmental awareness, as well as a citation from the Library of Congress for the promotion of literacy. Prior to Doctor Quinn, Sullivan created and served as co-executive producer of the Emmy and Golden Globe Award-winning one-hour drama series The Trials of Rosie O'Neill.

Sullivan brought her interest in drama to prime time made-for-television movies, as well. In the 1989-90 season, two of Sullivan’s telefilms, both of which dealt with social issues, aired on NBC and CBS in November sweeps. Most notably, A Cry For Help: The Tracey Thurman Story dramatized the landmark federal lawsuit that resulted in legislation to strengthen police responsibility toward battered wives, it earned the highest movie rating of the season. In addition, it received a special commendation from the Los Angeles County Board of Supervisors for its use as a training instrument for law enforcement officers. Sullivan was honored for her work by the Los Angeles County Domestic Violence Council. After Doctor Quinn, Medicine Woman, Sullivan created and executive produced the one-hour drama series Ponderosa for NBC/PAX from January 2000 until May 2002, when she was catastrophically injured in a car crash, she was in long-term recovery for many years, while caring for her twins and Jack, who were only six at the time of her traumatic injury.

They're now in college, Beth has resumed her writing and producing career. Sullivan is a past member of the Writers Guild of America - West Board of Directors and a former Trustee of the Guild Foundation, she is a graduate of the UCLA School of Theater and Television and a former development and production executive in the television division of 20th Century Fox Studio. Sullivan has taught at the American Film Institute. For her contributions to the television industry, Sullivan received one of its highest honors, the Genii Award from the American Women in Radio and Television—now called Alliance for Women in Media—organization; as well, she has received top honors from the Women’s American Medical Association, the National Organization of the Daughters of the American Revolution, the YWCA USA, Catholics In the Media, Big Brothers Big Sisters of America and an Emmy citation from the Academy of Television Arts & Sciences for her contributions to the medium. Beth Sullivan on IMDb

Lal Bangla are two imperial late-Mughal mausoleums located in Delhi, that are that protected monument under the Archaeological Survey of India. Lal Bangla consists of two tombs made out of yellow sandstone. One of the tombs contains two graves, believed to be that of Lal Kunwar, the mother of Shah Alam II and Begum Jaan, his daughter. Both mausoleums consist of square rooms at diagonals with oblong halls between them; the mausoleum stands on a red sandstone platform with rooms at corners. The dome of the mausoleum has a pinnacle at the top; the mausoleums share architectural similarities with the use of red and yellow sandstone of the Tomb of Safdarjung. The adjoining enclosure has three tombs belong to the family of emperor Akbar II; the buildings are within the premises of the Delhi Golf Club. Located within the complex is the tomb of Syed Abid, built in 1036 AH. Sir Syed Ahmed Khan's seminal work on the monuments of Delhi, Aasar Us Sanaadeed mentions Syed Abid as an associate of Khan Dauran Khan, one of Shahjahan's leading soldiers.

List of Monuments of National Importance in Delhi Tomb of Mariam-uz-Zamani, resting place of Empress Mariam uz-Zamani, consort of Emperor Akbar Tomb of Nur Jahan in Lahore, resting place of Empress Nur Jahan, consort of Emperor Jahangir Roshanara Bagh in Delhi, resting place of Roshanara Begum, second daughter of the Emperor Shah Jahan Media related to Lal Bangla at Wikimedia Commons

Áras an Uachtaráin the Viceregal Lodge, is the official residence and principal workplace of the President of Ireland. It is located off Chesterfield Avenue in the Phoenix Park in Dublin; the building, which has ninety-five rooms, was designed by Nathaniel Clements and completed in 1751. The original house was designed by park ranger and amateur architect, Nathaniel Clements, in the mid-eighteenth century, it was bought by the administration of the British Lord Lieutenant of Ireland to become his summer residence in the 1780s. His official residence was in the Viceregal Apartments in Dublin Castle; the house in the park became the Viceregal Lodge, the "out of season" residence of the Lord Lieutenant, where he lived for most of the year from the 1820s onwards. During the Social Season, he lived in state in Dublin Castle. Phoenix Park once contained three official state residences; the Viceregal Lodge, the Chief Secretary's Lodge and the Under Secretary's Lodge. The Chief Secretary's Lodge, now called Deerfield, is the residence of the United States Ambassador to Ireland.

The Under Secretary's Lodge, now demolished, served for many years as the Apostolic Nunciature. Some historians have claimed that the garden front portico of Áras an Uachtaráin was used as a model by Irish architect James Hoban, who designed the White House in Washington, D. C. However, the porticoes were not part of Hoban's original design and were, in fact, added to the White House at a date by Benjamin Henry Latrobe. In 1882, its grounds were the location of the Phoenix Park Murders; the Chief Secretary for Ireland, Lord Frederick Cavendish, his Undersecretary, Thomas Henry Burke, were stabbed to death with surgical knives while walking back to the residence from Dublin Castle. A small insurgent group called; the Lord Lieutenant, the 5th Earl Spencer, heard the victims' screams from a window in the ground floor drawing room. In 1911, the house underwent a large extension for the visit of Queen Mary. With the creation of the Irish Free State in 1922, the office of Lord Lieutenant was abolished.

The new state planned to place the new representative of the Crown, Governor-General Tim Healy, in a new, smaller residence, but because of death threats from the anti-treaty IRA, he was installed in the Viceregal Lodge temporarily. The building was at the time nicknamed "Uncle Tim's Cabin" after him, in imitation of the famous US novel Uncle Tom's Cabin by the US author Harriet Beecher Stowe, it remained the residence of the Governor-General of the Irish Free State until 1932, when the new Governor-General, Domhnall Ua Buachalla, was installed in a specially hired private mansion in the southside of Dublin. The house was left empty for some years, until the office of President of Ireland was created in 1937. In 1938, the first President, Douglas Hyde lived there temporarily while plans were made to build a new presidential palace on the grounds; the outbreak of World War II saved the building, renamed Áras an Uachtaráin, from demolition, as plans for its demolition and the design of a new residence were put on hold.

By 1945 it had become too identified with the presidency of Ireland to be demolished, though its poor condition meant that extensive demolition and rebuilding of parts of the building were necessary, notably the kitchens, servants' quarters and chapel. Since further restoration work has been carried out from time to time; the first President, Douglas Hyde lived in the residential quarters on the first floor of the main building. Presidents moved to the new residential wing attached to the main house, built on for the visit of King George V in 1911. However, in 1990 Mary Robinson moved back to the older main building, her successor, Mary McAleese lived in the 1911 wing. Though Áras an Uachtaráin is not as palatial as other European royal and presidential palaces, with only a handful of state rooms, it is a comfortable state residence. All Taoisigh as well as government ministers receive their seal of office from the President at Áras an Uachtaráin as do judges, the Attorney General, the Comptroller and Auditor General, senior commissioned officers of the Defence Forces.

It is the venue for the meetings of the Presidential Commission and the Council of State. Áras an Uachtaráin houses the headquarters of the Garda Mounted Unit. The Office of Public Works furnishes the private quarters of Áras an Uachtaráin for the presidential family. Various visiting British monarchs stayed at the Viceregal Lodge while Ireland was part of the United Kingdom, notably Queen Victoria and George V. American presidents hosted there include John F. Kennedy, Richard Nixon, Ronald Reagan, Bill Clinton, Barack Obama. Former Secretary of State Hillary Clinton has visited. Other famous visitors to Áras an Uachtaráin have been Nelson Mandela, Aung Sang Suu Kyi, Princess Grace of Monaco and her husband, Prince Rainier III. On 17 May 2011, Elizabeth II became the first British monarch to visit

Gmina Dzierzkowice is a rural gmina in Kraśnik County, Lublin Voivodeship, in eastern Poland. Its seat is Dzierzkowice, a location, divided into several sołectwos; the offices of the gmina are in fact in Terpentyna, which lies 11 kilometres north-west of Kraśnik and 47 km south-west of the regional capital Lublin. The gmina covers an area of 86.81 square kilometres, as of 2006 its total population is 5,401. Gmina Dzierzkowice contains the villages and settlements of Dębina, Dzierzkowice-Góry, Dzierzkowice-Podwody, Dzierzkowice-Rynek, Dzierzkowice-Wola, Dzierzkowice-Zastawie, Ludmiłówka, Sosnowa Wola, Terpentyna, Wyżnianka, Wyżnianka-Kolonia, Wyżnica and Wyżnica-Kolonia. Gmina Dzierzkowice is bordered by the town of Kraśnik and by the gminas of Annopol, Gościeradów, Józefów nad Wisłą, Kraśnik, Trzydnik Duży and Urzędów. Polish official population figures 2006

The 2009 ICC Champions Trophy was a One Day International cricket tournament held in South Africa between 22 September and 5 October, at Wanderers Stadium and Centurion Park, both in the Gauteng province. It was the sixth ICC Champions Trophy, was known as the ICC Knock-out. Two teams from two groups of four qualified for the semi-finals, the final was staged in Centurion on 5 October. Australia defended the title by beating New Zealand by six wickets in the final; the Champions Trophy was the brainchild of Jagmohan Dalmiya, ICC president in the late 1990s. It had a dual aim of spreading the game to emerging nations and raising money for the ICC in between World Cups, thus enabling it to pump more cash into those fledgling cricket countries; the first tournament, labelled as a mini World Cup, was staged in Dhaka in October 1998 and raised more than £10 million. The second, in Nairobi, was a commercial success. By the time the 2002 event was held – and there was disquiet as it was so close to the World Cup five months – the idea of playing in developing nations had been ditched and as revenue-generation was the main raison d'etre, it needed to be in one of the main countries as this allowed the format to be expanded.

In 2004 the jamboree moved to England and it became clear the format of group games led to too many meaningless games. By the time the 2006 tournament in India came into view, the event was under fire from some quarters, at one time there were hints that India might decline to take part in 2008; the tournament was scheduled to be held in Pakistan between 11 and 28 September 2008 in Lahore and Karachi. The ICC postponed the tournament due to security fears expressed by several participating countries. On 24 July 2008, the International Cricket Council announced that the tournament would take place in Pakistan after all despite players from Australia, South Africa and New Zealand raising concerns over touring the country. On 22 August 2008, South Africa announced that it would not take part in the Champions Trophy due to security concerns. Two days on 24 August 2008, after speculation that the tournament would be held elsewhere, the ICC announced that the tournament would be postponed until October 2009.

At its meeting in February 2009, the ICC board decided to move the tournament out of Pakistan on security concerns. At the time, Sri Lanka was the favoured alternate host. In March 2009, the ICC Chief Executives' Committee recommended to the ICC board that the tournament be held in South Africa as there were concerns that the weather in Sri Lanka during September and October could result in too many games being washed out; the ICC board ratified the recommendation, the event took place in South Africa between 22 September and 5 October 2009. Matches will be played both in the Johannesburg area; the 2009 ICC Champions Trophy was contested by the top eight teams, seeded and divided into two groups. No associates nations participated in this tournament; each team played every other team in its group once. Points were allocated for each match in accordance with the system described below which applied throughout the competition. Following the group stage, the top two teams from each group progressed to the semi-finals, where the winner of Group A played the runner up of Group B and the winner of Group B played the runner up of Group A.

The winners of the semi-finals contested the final. Most runs Most wickets ICC Champions Trophy Official website Tournament schedule at Cricinfo.com