In mathematics, Minkowski's theorem is the statement that every convex set in R n, symmetric with respect to the origin and which has volume greater than 2 n contains a non-zero integer point. The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers, it can be extended from the integers to any lattice L and to any symmetric convex set with volume greater than 2 n d, where d denotes the covolume of the lattice. Suppose that L is a lattice of determinant d in the n-dimensional real vector space ℝn and S is a convex subset of ℝn, symmetric with respect to the origin, meaning that if x is in S −x is in S. Minkowski's theorem states that if the volume of S is greater than 2n d S must contain at least one lattice point other than the origin; the simplest example of a lattice is the integer lattice ℤn of all points with integer coefficients. For n = 2, the theorem claims that a convex figure in the Euclidean plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin.
The area bound is sharp: if S is the interior of the square with vertices S is symmetric and convex, has area 4, but the only lattice point it contains is the origin. This example, showing that the bound of the theorem is sharp, generalizes to hypercubes in every dimension n; the following argument proves Minkowski's theorem for the specific case of L = ℤ2. It can be generalized to arbitrary lattices in arbitrary dimensions. Consider the map f: S → R 2 / 2 L, ↦ Intuitively, this map cuts the plane into 2 by 2 squares stacks the squares on top of each other. F has area less than or equal to 4, because this set lies within a 2 by 2 square. Assume for a contradiction that f could be injective, which means the pieces of S cut out by the squares stack up in a non-overlapping way; because f is locally area-preserving, this non-overlapping property would make it area-preserving for all of S, so the area of f would be the same as that of S, greater than 4. That is not the case, so the assumption must be false: f is not injective, meaning that there exist at least two distinct points p1, p2 in S that are mapped by f to the same point: f = f.
Because of the way f was defined, the only way that f can equal f is for p2 to equal p1 + for some integers i and j, not both zero. That is, the coordinates of the two points differ by two integers. Since S is symmetric about the origin, −p1 is a point in S. Since S is convex, the line segment between −p1 and p2 lies in S, in particular the midpoint of that segment lies in S. In other words, 1 2 = 1 2 = is a point in S, but this point is an integer point, is not the origin since i and j are not both zero. Therefore, S contains a nonzero integer point. An application of this theorem is the result that every class in the ideal class group of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K, called Minkowski's bound: the finiteness of the class number of an algebraic number field follows immediately. Minkowski's theorem is useful to prove Lagrange's four-square theorem, which states that every natural number can be written as the sum of the squares of four natural numbers.
Danzer set Pick's theorem Dirichlet's unit theorem Minkowski's second theorem "Minkowski's theorem". PlanetMath. Stevenhagen, Peter. Number Rings. Malyshev, A. V. "Minkowski theorem", in Hazewinkel, Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Hazewinkel, Michiel, ed. "Geometry of numbers", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Peter Forbes Ricketts, Baron Ricketts, is a retired British senior diplomat and a life peer. He sits as a crossbencher in the House of Lords. Ricketts attended Bishop Vesey’s Grammar School, Sutton Coldfield, Pembroke College, Oxford where he read English Literature, he married Suzanne Horlington. Ricketts replaced Peter Westmacott as HM Ambassador to France effective January 2012, with Kim Darroch taking Ricketts's old role as National Security Adviser. In December 2015 the Foreign and Commonwealth Office announced that he was to retire from the Diplomatic Service in January 2016. Prior to his appointment as National Security Adviser, Ricketts had been the Permanent Secretary in the Foreign and Commonwealth Office. Before he took over that position in July 2006, he served as the Permanent Representative to NATO in Brussels, he was previously the Chairman of the Joint Intelligence Committee, leading him to give evidence to The Iraq Inquiry in November 2009. He began his career in the Foreign and Commonwealth Office in 1974 and served as the Assistant Private Secretary to former Foreign Secretary Geoffrey Howe.
Apart from Brussels, he has been posted to Singapore, Washington D. C. and Paris. Ricketts retired from HM Diplomatic Service in January 2016. In 2016 he took appointments as Strategic Adviser to Lockheed Martin UK and Non Executive Director of Engie, he was appointed CMG in the 1999 Birthday Honours, Knight Commander of the Order of St Michael and St George in 2003, Knight Grand Cross of the Order of St Michael and St George in the 2011 New Year Honours, Knight Grand Cross of the Royal Victorian Order in 2014. He was nominated for a life peerage in the 2016 Prime Minister's Resignation Honours and was created Baron Ricketts, of Shortlands in the County of Kent, on 17 October. Politics of the United Kingdom
Sir Abraham Elton, 2nd Baronet of Bristol and Clevedon Court, was a British merchant and Whig politician, who sat in the House of Commons for Taunton between 1724 and 1727, for Bristol from 1727 until his death in 1742. He served as the High Sheriff of Bristol from 1710 to 1711, was Mayor of Bristol for the year 1719 to 1720. Elton was the eldest son of Abraham Elton, his wife Mary Jefferies, his date of birth is not known, but he was baptised on 30 June 1679. He married Abigail Bayly, the daughter of Zachary Bayly of Charlcot House, near Westbury and Northwood Park, near Glastonbury, Somerset, on 14 May 1702. Elton was a merchant and industrialist, like his father before him, he served as the High Sheriff of Bristol from 1710–11, he invested in slave ships with his brothers and Jacob. He was the Master of the Society of Merchant Venturers in 1719 and Mayor of Bristol from 1719–20, but in 1720, he was made bankrupt during the "South Sea Bubble"; as soon as he completed his term as Mayor, he left Bristol and travelled to France, did not return until his father paid off his debts.
Elton returned to England by 1724, stood in the Taunton by-election of 1724 for the Whigs, as an unexpected late entrant. He was duly elected to serve as a Member of Parliament for Taunton, though one of the other candidates, George Deane, filed a petition against his election; the petition was rejected by a vote of 151 to 104. He only served Taunton until the general election, in 1727, when his father vacated his seat in Bristol. At the resulting election, Elton paid his Tory opponent £1,000 to withdraw from the election, allowing him to be returned unopposed. In Parliament, he became a member of the gaols committee. Upon his father's death on 9 February 1728, Elton became Sir Abraham Elton, 2nd Baronet, inherited Clevedon Court. In February 1730 he spoke against the Royal African Company’s petition to be spared the cost of maintaining their forts, he petitioned government on mercantile issues, amongst them. He was said to have made a "bantering speech" against the proposed Excise Bill of 1733.
Elton topped the poll in a contest at the 1734 British general election. He continued raising petitions on mercantile issues, voted with the Opposition in all recorded divisions, he was returned unopposed at the 1741 British general election. Elton died on 20 October 1742, leaving three daughters; the baronetage passed to his eldest son, who became Sir Abraham Elton, 3rd Baronet but died without issue. The baronetcy passed to his brother Sir Isaac Elton, 4th Baronet. Another of Elton's sons, Jacob was killed in a sea battle. Elton's daughters Mary and Elizabeth married
Bridgeport is an elevated station on the Canada Line of Metro Vancouver's SkyTrain rapid transit system. It is located in Richmond, British Columbia, south of Vancouver; the Canada Line branches outbound at this station, with one branch heading westward to YVR–Airport station at the Vancouver International Airport and the other heading south to Richmond–Brighouse station in the commercial centre of Richmond. Bridgeport station is located near the intersection of River Road and Great Canadian Way—north of Bridgeport Road and in the same general area as the River Rock Casino—and is the northernmost SkyTrain station in Richmond; the Canada Line's Operations and Maintenance Centre is located northeast of the station. There is a large park-and-ride facility adjacent to the station; the City of Richmond anticipates that the area surrounding this station will be redeveloped, proposals include the building of office suites and hotels. The Canada Line splits just southwest of Bridgeport station, with the main line continuing southward through Richmond to its terminus at Richmond–Brighouse station.
A branch line heads westward across Sea Island to YVR–Airport station at the Vancouver International Airport. Passengers traveling between Richmond and Sea Island must transfer at Bridgeport to complete their journey; the station serves as the inbound terminus for express buses from Delta and White Rock. In the past, prior to the opening of the Canada Line, these buses continued to Downtown Vancouver; this station serves as the inbound terminus for some Canada Line trains from Richmond–Brighouse and YVR–Airport during weekdays. Bus bay assignments
Ken Kelley is a former American football linebacker who played two seasons in the United States Football League with the Philadelphia Stars, Chicago Blitz and Birmingham Stallions. He was drafted by the Philadelphia Stars in the 1983 USFL Territorial Draft, he played college football at Penn State University and attended Sterling High School in Somerdale, New Jersey. Raised in Stratford, New Jersey, Kelley played high school football for the Sterling High School Silver Knights, he was a safety and linebacker for the Silver Knights before converting to quarterback his junior year. The team garnered a number one ranking in New Jersey. In his four years playing for the Silver Knights, they won four Colonial Conference titles and reached the first four South Jersey Group 3 championship games, winning three of them, he was inducted into the Camden County Sports Hall of Fame in 2013. Kelley played college football for the Penn State Nittany Lions, he was captain of the 1982 national championship team.
He redshirted his freshman year. Kelley was drafted by the Philadelphia Stars in the 1983 USFL Territorial Draft, he played for the Stars during the 1984 season. Kelley played for the Chicago Blitz in 1984. Kelley played for the Birmingham Stallions in 1985. Just Sports Stats College stats
Raghunath Krishna Phadke was a famous Indian sculptor from the state of Maharashtra, India. He is a winner of 1961 Padma Shri Award. During the first half of the 20th century, the Maharaja of Dhar was a patron of art and would invite several artists to his kingdom. Phadke was one of those. On his request, Phadke started an art studio in a city in Madhya Pradesh, he settled in Dhar in 1933. It is located on the outdoors of Dhar Fort; the museum today has sculptures of many important people from Indian History such as Mahatma Gandhi, Jawaharlal Nehru, Bal Gangadhar Tilak, Ram Mohan Roy, among others. There are bust statues of kings, local chieftains and spiritual leaders; the studio has all busts lined up in a row in an academic style. Phadke's own personal legacy of arts can be seen in busts installed in public places in Dhar and Ujjain; the Government Institute of Fine Arts, Dhar was founded under Phadke's guidance on 24 November 1939. It is affiliated to Sir J J School of Arts and Indira Kala Sangeet Vishwavidyalaya, Rajnadgaon, Chhattisgarh.
Since 2002, the institute is working under the Higher Education of MP Government