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Geometric algebra

The geometric algebra of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which contains both the scalars F and the vector space V. Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form. Clifford's contribution was to define a new product, the geometric product, that united the Grassmann and Hamilton algebras into a single structure. Adding the dual of the Grassmann exterior product allows the use of the Grassmann–Cayley algebra, a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra providing a framework for classical geometries. In practice and several derived operations allow a correspondence of elements and operations of the algebra with geometric interpretations; the scalars and vectors have their usual interpretation, make up distinct subspaces of a GA. Bivectors provide a more natural representation of pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, angular momentum, electromagnetic field and the Poynting vector.

A trivector can represent an oriented volume, so on. An element called a blade may be used to represent a subspace of V and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike vector algebra, a GA accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis, differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has found use as a computational tool in computer graphics and robotics.

The geometric product was first mentioned by Hermann Grassmann, chiefly interested in developing the related exterior algebra. In 1878, William Kingdon Clifford expanded on Grassmann's work to form what are now called Clifford algebras in his honor. For several decades, geometric algebras went somewhat ignored eclipsed by the vector calculus newly developed to describe electromagnetism; the term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics. There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic, "full of geometric significance" and equivalent to the universal Clifford algebra. Given a finite-dimensional quadratic space V over a field F with a symmetric bilinear form g: V × V → F, the geometric algebra for this quadratic space is the Clifford algebra Cl ⁡; as usual in this domain, for the remainder of this article, only the real case, F = R, will be considered. The notation G will be used to denote a geometric algebra for which the bilinear form g has the signature.

The essential product in the algebra is called the geometric product, the product in the contained exterior algebra is called the exterior product. It is standard to denote these by juxtaposition and the symbol ∧; the above definition of the geometric algebra is abstract, so we summarize the properties of the geometric product by the following set of axioms. The geometric product has the following properties, for A, B, C ∈ G: A B ∈ G 1 A = A 1 = A, where 1 is the identity element A = C A =

1988 AFC Asian Cup qualification

Qualification for the Asian Football Confederation's 1988 AFC Asian Cup finals held in Qatar between 2 and 18 December. Saudi Arabia defeated Republic of Korea in the final match in Doha. Qatar qualified automatically as host Saudi Arabia qualified automatically as defending championsThe other 8 qualifying teams were: Bahrain China PR Iran Japan Kuwait South Korea Syria United Arab Emirates * Withdrew RSSSF details "Stiff test for Lions in 1988"The Straits Times, 13 December 1986, Page 38

Jesse Lloyd

Jesse Lloyd was the founder of Lloydtown, Ontario and a leader in the Upper Canada Rebellion of 1837. Born in Springfield Township, Pennsylvania, he was the third son of Quakers William Lloyd and Susannah Heacock; the Lloyds, who were United Empire Loyalists came to Canada at Niagara in 1788 but soon returned to the United States. They immigrated permanently to Upper Canada in 1808. Upon arrival, they crossed the Niagara gorge and migrated north to settle in the 10th concession of King Township. By 1824, Jesse Lloyd had established a sawmill in Tecumseth Township, he was active in King Township where he bought and sold lots, built several mills and, in the process, established the village of Lloydtown, Ontario|. In the days of the Family Compact in Upper Canada, agitation grew from year to year. Meetings for reform were held all over the home district and in some remote parts of the province. Lloyd was a local leader in public affairs in his district. During the Upper Canada Rebellion in 1837, a proclamation was issued and a reward of 500 pounds was put on Lloyd's head.

He was forced to flee the country and he went to Tippecanoe County, where he lived out the last few months until his death in 1838. Lloyd was the only family member to flee with the rest remaining in Canada. Rea, J. Edgar. "Rebellion in Upper Canada, 1837". MHS Transactions. 3. Manitoba Historical Society

Campbell Clark

Dr Archibald Campbell Clark FFPSG was a nineteenth-century Scottish physician who made major advances in mental health care philosophies. He was born at Tarbert, Loch Fyne, the son of Donald Clark, a merchant, his wife Margaret Campbell, his father died when he was young and they moved to Lochgilphead. He was educated there at the Free Church School. From around 1867 he assisted at the local asylum, he worked for some years as a warehouseman in Glasgow studied medicine at Edinburgh University graduating MB ChB in 1878 and gaining his doctorate in 1886. He was assistant medical officer at the Melrose Asylum in the Scottish Borders before joining the Edinburgh Asylum under Dr Thomas Clouston. Around 1890 he became Medical Superintendent of the Glasgow District Asylum at Bothwell. In 1895 he was appointed Chief Medical Superintendent of the newly completed asylum - Hartwood Hospital, serving Lanark. With over 2500 patients it was the largest asylum in Europe. Controversially by today's standards he employed electroconvulsive therapy and was the first person in Scotland to perform a lobotomy in attempts to control behaviour.

He was the first to advocate professional training of all staff, had a strong reputation for improving the actual conditions of the inmates. He was president of the Caledonian Medical Society, he died of influenza on 28 November 1901 at his house in Hartwood Village near the hospital. He was buried in the hospital cemetery in Hartwood. Hartwood closed in 1995 following the rolling out of "care in the community" in the Community Care Act 1990, he is remembered on the Pinel Memorial at the Royal Edinburgh Hospital. He had two sons and one daughter; the Special Training of Asylum Attendants Essays on Hallucinations by Asylum attendants Handbook for Instruction of Asylum Attendants Experimental Dietetics in Lunacy Practice The Sexual and Reproductive Functions and Perverted, in Relation to Insanity Etiology and Treatment of Puerperal Insanity The Future of Asylum Service A Clinical Manual of Mental Diseases The Therapeutic Value of Spleen Removal On Epileptic Speech


Repelinosaurus is an extinct genus of basal kannemeyeriiform dicynodont from the area of Luang Prabang in Laos, Southeast Asia that lived at around the time of the Permian-Triassic boundary and dates to the earliest Early Triassic. Its type and only known species is R. robustus. Repelinosaurus represents the earliest known kannemeyeriiform, it supports the idea of a more rapid radiation of the Triassic kannemeyeriiform dicynodonts during the Early Triassic following the Permian mass extinction; the discovery of an early kannemeyeriiform in an understudied locality like Laos highlights the importance of such places in dicynodont research, focused on important localities such as the Karoo Basin of South Africa. Repelinosaurus was a medium-sized dicynodont known only from skulls missing lower jaws and the rest of the skeleton. However, it resembled other kannemeyeriiform dicynodonts, so was a built, stocky-limbed quadruped with a short tail and a large head with nearly toothless jaws and a tortoise-like beak, sporting a pair of prominent tusks.

The skull of Repelinosaurus is narrow for a dicynodont, the snout in front of the eyes is short—among the shortest of any dicynodont—but proportionately wide, tapering to a flat, squared-off beak. The bony nostrils are large, occupy half of the length of the short snout; the upper surfaces of the snout are rugose on the premaxilla, this textured surface ends abruptly at the contact of the nasal and frontal bones. Like other Kannemeyeriiformes, the nasals sport bony bosses, however uniquely they form as a single central swelling on the snout while most other Kannemeyeriiformes have a distinct pair; this boss stops just above the edge of the nostrils, separated from the frontals by a notch, although the prefrontal bones sport their own smaller, weakly developed boss. The rough texture of the snout implies it was covered like the beak; the maxilla is robust, more so in one specimen than the other, in which the caniniform process housing the tusk is more developed. The caniniform process is vertical, so the prominent tusks point directly downwards.

The palatine bones are wide at the front and form rough, rugose pads on the roof of the mouth likely covered in keratinous horn. The frontals are wide, so the eyes sit down on the side of the head and face outwards; the postorbital bars closing off the back of the orbits are short, so the skull appear narrow from above for a dicynodont. The pineal foramen, bordered by the preparietal in front and the elongated parietal bones behind, is oval and sits flat on the skull, noticeably varies in size between the two known specimens; the two skulls differ in other ways, some of which appear to be related ontogeny, such as the larger skull being more robust, having a more prominent caniniform process and better developed ornamentation with nasal bosses that extend further out to the sides of the snout. These differences appear to be due to changes in ontogeny or sexual dimorphism, as has been observed in some other dicynodonts, but this cannot be confirmed for Repelinosaurus. Both specimens of Repelinosaurus were discovered in the Purple Claystone Formation of the Luang Prabang Basin in northern Laos.

This sedimentary unit consists of purple silty-claystones mixed with layers of conglomerates and sandstone, as well as volcaniclastic sediments. Estimated dates for the age of the formation have ranged from the Late Permian to the Late Triassic or the earliest Jurassic period. More radiometric dating using U—Pb geochronology from detrital zircon has yielded a maximum age for deposition of 251.0 ± 1.4 Ma. However, the mixing and reworking of the sediments implies that the actual depositional age of the formation is younger than this placing it in the Early Triassic; the first dicynodont remains to be discovered in the Purple Claystone Formation was a single, poorly preserved partial skull discussed by French geologist Jean-Baptiste-Henri Counillon in 1896. This skull was described in 1923 by another French geologist, Joseph Répelin, who named it as a new species of Dicynodon, "Dicynodon incisivum"; the incomplete and damaged nature of the skull made identification difficult, it has been variously attributed to Dicynodon and Lystrosaurus due to a supposed resemblance to the latter.

The specimen has since been lost, the poor quality of the remaining illustrations of the skull are unsuitable for supporting the validity of the species, "D. incisivum" has since been considered a nomen dubium. As such, its relationships to other Purple Claystone dicynodonts like Repelinosaurus remain unknown. More dicynodont remains were recovered by a Franco-Laotian expedition between 1993 and 2003 lead by palaeontologist Philippe Taquet. Three skulls in particular were studied and described in 2009 and were assigned to Dicynodon, tentatively as a new species, although this relationship was not tested and remained uncertain. In 2019, the three skulls were more described in full detail and were recognised as representing two distinct new taxa. Two of these skulls, specimens LPB 1993-2 and LPB 1993-9, were assigned to the new genus Repelinosaurus; the third skull was assigned to Counillonia. The specimens were temporarily stored and studied at the Muséum National d'Histoire Naturelle in Paris, is permanently housed at the Savannakhet Dinosaur Museum in Laos.

The larger of the two specimens, LPB 1993-2, was made the holotype of Repelinosaurus. It is a partial skull missing porti

John Harington, 2nd Baron Harington of Exton

John Harington, 2nd Baron Harington of Exton, of Burley-on-the-Hill, Rutland was a young English peer and politician. He was Baron Harington of Exton, he was the surviving son of Sir John Harington and his wife, Anne Keilway, daughter of Robert Keilway, Surveyor of the Court of Wards and Liveries, was born at Combe Abbey, near Coventry, Warwickshire, in April 1592. He was admitted in 1607 to Sidney Sussex College, founded by Frances Sidney, his father's aunt, to which he and his father were benefactors, he was educated with the Prince of Wales and they remained close friends until the prince's death. He succeeded his father as Baron in August 1613, his distant descendants include the actor Kit Harington. Friend and companion of Henry Frederick, Prince of Wales, on 5 January 1604 he was created, along with The Duke of York and others, a Knight of the Bath. In September he went a foreign tour with a master of the free school at Guildford. While abroad he corresponded in French and Latin with Prince Henry.

After seven weeks in the Low Countries, where he visited the universities, courts of three princes, military fortifications, Harington went to Italy in 1608. He wrote from Venice announcing his intention of returning through France to spend the rest of his life with his royal friend. Henry's death grieved him. On his return to Coventry Harington became the Member of Parliament for Coventry for a brief period when the incumbent John Rogerson was taken ill, he was appointed Lord Lieutenant of Rutland in 1613 on the death of his father, a position he held until his own death the following year. In August 1613 21-year-old Harington succeeded to his father's title and a heritage of debts, vainly attempted to retrieve the family fortunes by obtaining a royal patent on the minting of lead farthings from the mint under a scheme proposed by Gerard de Malynes on 10 April. After the farthings proved unpopular, the young Lord Harington of Exton died at Kew on 27 February 1614 and was buried at Exton. On 18 February he had sold the lordship of Exton to Sir Baptist Hicks, by his will, made at the same time, left the overplus of the estates, after the creditors had been paid, to his two sisters, two-thirds to Lucy, Countess of Bedford, one-third to Frances, Lady Chichester, whose kneeling effigy exists in Pilton Church in Devon, first wife of Sir Robert Chichester of Raleigh.

The Countess of Bedford sold the remaining family estates in Rutland. Harington's contemporaries wrote of him in the highest terms, his funeral sermon was preached by Richard Stock, pastor of All Hallows, Bread Street, published as "The Church's Lament for the Loss of the Godly", with a small woodprint portrait. Appended to this publication were an epitaph and elegies by F. Herring and Sir Thomas Roe. At the same time a poem entitled "Sorrows Lenitive, written upon occasion of the death of that hopeful and noble young gentleman, &c.", was written by Abraham Jackson, dedicated to Harington's mother and sister Lucy. John Donne took leave of poetry in a funeral ode on Harington, Thomas Gataker, in his "Discours Apologetical", London, 1654, p. 36, styles him a "mirror of nobility". A portrait is in Henry Holland's Herωologia. Attribution This article incorporates text from a publication now in the public domain: Bradley, Emily Tennyson. "Harington, John". In Stephen, Leslie. Dictionary of National Biography.

24. London: Smith, Elder & Co