In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a -polytope consist of k-polytopes that may have -polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope; some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, set-theoretic abstract polytopes. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli; the German term polytop was coined by the mathematician Reinhold Hoppe, was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a wide class of objects, different definitions are attested in mathematical literature.
Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes. They represent different approaches to generalizing the convex polytopes to include other objects with similar properties; the original approach broadly followed by Ludwig Schläfli, Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron in two and three dimensions. Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two.
However this definition does not allow star polytopes with interior structures, so is restricted to certain areas of mathematics. The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in p-space are equivalent to tilings of the -sphere, while others may be tilings of other elliptic, flat or toroidal -surfaces – see elliptic tiling and toroidal polyhedron. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra, so forth; the idea of constructing a higher polytope from those of lower dimension is sometimes extended downwards in dimension, with an seen as a 1-polytope bounded by a point pair, a point or vertex as a 0-polytope. This approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a polyhedron is the generic object in any dimension and polytope means a bounded polyhedron.
This terminology is confined to polytopes and polyhedra that are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and is defined by its vertices. Polytopes in lower numbers of dimensions have standard names: A polytope comprises elements of different dimensionality such as vertices, faces, cells and so on. Terminology for these is not consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically. Authors may use j-facet to indicate an element of j dimensions; some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element; the terms adopted in this article are given in the table below: An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope.
Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope, so on; these bounding sub-polytopes may be referred to as faces, or j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, consists of a single point. A 1-dimensional face is called an edge, consists of a line segment. A 2-dimensional face consists of a polygon, a 3-dimensional face, sometimes called a cell, consists of a polyhedron. A polytope may be convex; the convex polytopes are the simplest kind of polytopes, form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of half-spaces; this definition allows a polytope to be neither finite. Polytopes are defined in e.g. in linear programming. A polytope is bounded. A polytope is said to be pointed; every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set.
A polytope is finite if it is defin
The Proposed National Education Service is a'unified National Education Service for England to move towards cradle-to-grave learning, free at the point of use', proposed by the Labour Party in their manifesto for the 2017 general election. The National Education Service was again included in the Labour Party's manifesto for the 2019 General Election.... Labour will create a unified National Education Service for England to move towards cradle-to-grave learning, free at the point of use; the NES will be built on the principle that ‘Every Child – and Adult Matters’ and will incorporate all forms of education, from early years through to adult education. The National Education Service is being to ensure opportunities for life-long learning are available to meet both of the core aims of education: personal education and education for employment. While early Labour Party statements have suggested that the NES would be "free at the point of use", as the details of the service are developed it is to that this statement will be taken to apply to what is identified as entitlement for all up to the age of 18 and that a variety of funding strategies will apply post-18.
For example, where there are national priorities for retraining, such training could be free at the point of use by those who are retraining. Where post-18 education is for personal fulfilment rather than national skill shortages, a charge would be expected; the labour party believes this is an example of how scholarships for post-18 education could be used to signal to prospective students what the national skill shortages are. In England, it is believed by labour,that the current approach to the school system means that government has limited options to ensure the curriculum offered to students post 16 meets national priorities. In both university and school systems, it is believed by labour, an unforeseen consequence has been that the school and university leaders have managed to negotiate salaries which are considered high, which has led to publishing of salary league tables; the education sector is divided into these phases. In early years child care schools higher education workplace and vocational training adult educationAll phases are subject to quality assurance and monitoring through independent inspection through OFSTED or QAA, as they are for all phases with the exception of workplace training.
A team of 16 experts were appointed by Labour leader Jeremy Corbyn to the Lifelong Learning Commission in February 2019. This commission was intended to focus on expanding and transforming lifelong learning as a key component of the NES; the commission was led by Labour MP and Shadow Minister for Higher Education, Further Education and Skills, Gordon Marsden and Shadow Secretary of State for Education, Angela Rayner. The commission was launched in a speech given by Jeremy Corbyn to the Make UK/EEF annual conference in London's QEII centre on February 19th 2019. Membership of the commission comprised: Co-Chair – Estelle Morris, former education secretary Co-chair – Dave Ward, general secretary, Communication Workers Union Alison Fuller – Professor – Vocational Education and Work, UCL Seamus Nevin – Chief Economist, Make UK Ewart Keep – Director of Centre for Skills and Organisational Performance, Oxford University Mary Kellett – Vice Chancellor, Open University Graeme Atherton – Director of the National Education Opportunities Network Joyce Black – Assistant Director, R&D, Learning and Work Institute Amatey Doku – Vice President Higher Education, National Union of Students Kirstie Donnelly – Managing Director and Guilds Vicky Duckworth – Professor in Education, Edge Hill University David Latchman – Master of Birkbeck Dave Phoenix – Vice Chancellor, London South Bank University Carole Stott – Former Chair of the Board and Trust, Association of Colleges Matt Waddup – National Head Of Policy & Campaigns – University and College Union Tom Wilson – Chair of UFI, Former Head of Unionlearn.
The Commission's findings and proposals were announced during the launch of Labour's education manifesto for the 2019 UK General Election, in Blackpool on the 12th November 2019. Labour Party Manifesto proposal Jeremy Corbyn,'Education is a collective good – it’s time for a National Education Service' Angela Rayner outlines 10-point charter for National Education Service
Road Dreams is a series of six television programmes, each of 25 minutes duration, last shown on British terrestrial television in the early 1990s on Channel 4. The programmes were created by Elliott Bristow, being a compilation of Super 8 mm film footage filmed by Bristow during an extended stay in the United States; each programme opens with the following text: "In 1968 Elliott Bristow went to New York for a two week holiday. In 1980, after twelve years and 500,000 miles on the road in America he returned to England with three trunks containing 75 hours of silent Super 8 film - his diary of this'holiday'!" The programmes depict Bristow's journeys across the USA. Many pieces were filmed whilst driving; because of the source material, the technical quality of the programme suffers. The footage is accompanied by occasional explanatory comments from Bristow, readings by Mark Murphy and music tracks; the music is but not instrumental classical and new-age music. A DVD with 13 short films using the Road Dreams footage is available and four of the original six original episodes can be viewed on YouTube.
A book titled "Home in October: Twenty-one Years of Road Dreams" by Elliot Bristow was published in a hardcover edition of 192 pages by Mainstream Publishing in September 1991, ISBN 1-85158-334-3. According to Bristow, this book never made it past the proposal stage, was erroneously publicised by an over-enthusiastic publisher. An ebook "Road Dreams an American Adventure" for the iPad is on sale in the iTunes Bookstore from 31 July 2013 - ISBN 978-0-9575498-1-4. In early 2006 rumours circulated of a second series of Road Dreams being in production but these may have been a hoax. According to his website, Bristow is working on a new version of Road Dreams called Codachrome; the following is a listing of the items and music tracks featured in each programme. The artist names and track titles are as listed at the end of each programme; the names in brackets are those of commercially available albums. In many cases tracks many be available on other albums; some of the music used in Road Dreams does not appear to be available commercially.
The epilogue of each programme is the same and contains the following reading from Jack Kerouac’s novel On the Road. We arrived in Arizona at dawn. I woke up from a deep sleep to find everybody sleeping like lambs and the car parked God knows where, because I couldn't see out the steamy windows. I got out of the car. We were in the mountains: there was a heaven of sunrise, cool purple airs, red mountainsides, emerald pastures in valleys and transmuting clouds of gold, it was time for me to drive on. Programme One Items: Preview Arrival in New York Travels in New England "Snowbird" journey to Florida and Georgia EpilogueMusic Leo Kottke - Machine #2 Rick Loveridge - Breakfast Special Keith Thompson - East Side Hum Rick Wakeman - Dandelion Airs Penguin Cafe Orchestra - No's 1-4 Rick Loveridge - Suitable Favours Richard W Gilks - Road Dreams Theme Rick Loveridge - Person To PersonProgramme Two Items: Preview: The Dakotas Culver, Kansas Running down to Alamosa Burma-Shave On to Las Vegas EpilogueMusic: Leo Kottke - Machine #2 Rick Loveridge - Brimscombe Penguin Cafe Orchestra - Perpetuum Mobile Francois Godefroy - Dancers Rick Loveridge - Never Here Never Home Rick Loveridge - Leave A Message Leo Kottke - Pamela Brown Richard W Gilks - Road Dreams Theme Rick Loveridge - Person To PersonProgramme Three Items: Preview: Iowa Winter Driving Detroit to Washington DC Eastern suburbs and Jersey Shore EpilogueMusic: Leo Kottke - Machine #2 Rick Loveridge - Root And Branch Keith Thompson - Slipping South Rick Loveridge - Brimscombe St Saens - Carnival Of The Animals - The Aquarium Rick Wakeman - Waterfalls Penguin Cafe Orchestra - White Mischief Keith Thompson - East Side Hum Richard W Gilks - Eastway Richard W Gilks - Road Dreams Theme Rick Loveridge - Person To PersonProgramme Four Items: Preview: Rhode Island Boston: Fall in New England Cross-country to Oregon Stonehenge Down to San Francisco EpilogueMusic: Leo Kottke - Machine #2 Pat Metheny - Goin' Ahead Leo Kottke - Born To Be With You Rick Loveridge - Never Here Never Home Rick Loveridge - Leave A Message Rick Loveridge - Suitable Favours Richard W Gilks - Road Dreams Theme Rick Loveridge - Person To PersonProgramme Five Items: Preview: Los Angeles New York Cross-country Drive Los Angeles EpilogueMusic: Leo Kottke - Machine #2 Francois Godefroy - Cactus Lunch Tom Waits - Frank's Wild Years Francois Godefroy - Dancers Pat Metheny Group - Cross The Heartland Penguin Cafe Orchestra - Music For A Found Harmonium Richard W Gilks - Road Dreams Theme Rick Loveridge - Person To PersonProgramme Six Items: Preview Train ride to Oregon Leave L.
A. Final Cross-country trip Long Island and leave EpilogueMusic: Leo Kottke - Machine #2 Unknown Artist - Blue Valley Junction Rick Loveridge - Breakfast Special Simon Kitson - LA Freeway Philip Aaberg - Spring Creek Rick Loveridge - Home in October Rick Loveridge - Suitable Favours Francois Godefroy - Tracy's Big Moment Richard W Gilks -
In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution. If Y is a random variable with a normal distribution, P is the standard logistic function X = P has a logit-normal distribution, it is known as the logistic normal distribution, which refers to a multinomial logit version. A variable might be modeled as logit-normal if it is a proportion, bounded by zero and one, where values of zero and one never occur; the probability density function of a logit-normal distribution, for 0 ≤ x ≤ 1, is: f X = 1 σ 2 π 1 x e − 2 2 σ 2 where μ and σ are the mean and standard deviation of the variable’s logit. The density obtained by changing the sign of μ is symmetrical, in that it is equal to f, shifting the mode to the other side of 0.5. The moments of the logit-normal distribution have no analytic solution; the moments can be estimated by numerical integration, however numerical integration can be prohibitive when the values of μ, σ 2 are such that the density function diverges to infinity at the end points zero and one.
An alternative is to use the observation that the logit-normal is a transformation of a normal random variable. This allows us to approximate the moments via the following quasi Monte Carlo estimate E ≈ 1 K − 1 ∑ i = 1 K − 1 n, where P is the standard logistic function, Φ μ, σ 2 − 1 is the inverse cumulative distribution function of a normal distribution with mean and variance μ, σ 2; when the derivative of the density equals 0 the location of the mode x satisfies the following equation: logit = σ 2 + μ. For some values of the parameters there are two solutions, i.e. the distribution is bimodal. The logistic normal distribution is a generalization of the logit–normal distribution to D-dimensional probability vectors by taking a logistic transformation of a multivariate normal distribution; the probability density function is: f X = 1 | 2 π Σ | 1 2 1 ∏ i = 1 D x i e − 1 2 ⊤ Σ − 1, x ∈ S D, where x − D denot
Zacharias Wagenaer was a clerk, merchant, member of the Court of Justice, opperhoofd of Deshima and the only German governor of the Dutch Cape Colony. In 35 years he traveled over four continents. Zacharias was the son of a painter. In 1633 he traveled from Dresden via Hamburg to Amsterdam. There he worked for Willem Blaeu. Within a year he enlisted as a soldier in the armed forces of the Dutch West India Company to serve in "New Holland" in 1634. Three years he was hired as a writer by the newly arrived governor of the colony, Count John Maurice, Prince of Nassau-Siegen. In Recife he kept a sort of diary with 109 water-colour drawings of curious fish, strange birds and harmful animals, lovely tasty fruit and nasty, poisonous worms and big, brown or black people, published as "Thier-Buch". There are pictures of the Smooth Hammerhead, slender filefish and Cirripedia. In 1 April 1641, he left Dutch Brazil, arrived on Texel on 17 June, he traveled back to Dresden arriving 12 October. After four months, he left Dresden to return to the Netherlands arriving at Amsterdam on 29 March 1642, took a position with the Dutch East India Company.
On 29 September 1642 he sailed for the Indies as an apprentice officer. In the next year he became an assistant for the governors Antonie van Diemen and Cornelis van der Lijn. Three times he became a member of the Court of Justice at Batavia. In 1653 he went on a mission to Canton to open up again trade relations, which proved fruitless, due to a civil war after the Fall of the Ming Dynasty. In 1657 he rose to the rank of "opperhoofd" of the VOC at the small island in Nagasaki bay in the Japanese island of Kyushu, Dejima, he traveled to the capital Edo in a tributary mission and escaped from a burning city, which started on 2 March 1657. In 1659, as one of the first "opperhoofden", he ordered a dinner service. Wagener made the design of this Japanese porcelain, according to the European taste white and blue, with many flowers. In 1660 Wagner was involved in the peace negotiations with the sultan of Makassar; the port had about 2000 Portuguese traders and for years threatened the Dutch spice trade on the Moluccas.
The next year he was head of the Public Works in Batavia. In 1662 he went to Cape of Good Hope with five slaves and two horses, he followed Jan van Riebeeck as a governor on 6 May. Riebeeck left the next day. In December 1663 he asked Batavia to send him some pottery from Persia, he negotiated with the Hottentots about cattle for the Company. By abstaining from further expeditions Wagener could pursue his policy to refrain from an interference in tribal disputes, to keep neutral. After five years studying, the German student Georg Friedrich Wreede wrote a compendium of the Dutch and Hottentot language. Wagener appointed him in Mauritius. Wagener was one of the five people laying the foundation of the Castle of Good Hope, started in August 1665, he constructed a waterbasin, supplying the ships with fresh water, a hospital, a school and a church. In 1666 his wife Anna Auxbrebis, whom he had married in 1648, died. On 27 September 1666 he resigned and Wagener went back to Batavia with his stepdaughter.
He sold his slaves from Bengal. With presents he went to see the susuhunan of Mataram, who refused to trade with the VOC. While his knowledge of the Malay or Javanese language wasn't good, the mission turned out to be fruitless; the year after he sailed back to Amsterdam as a vice-admiral, in ill health. He was buried on 16 October 1668 in the Old Church. An excerpt of his diary was translated from the High Dutch into English and published in 1704 and 1732. A manuscript with fundamentally the same excerpt—but in German—has been a part of the Collection of Prints and Photographs of the Dresden State Art Collections since at least the beginning of the 18th century. Wagner served two terms as opperhoofd in alternation with Joan Boucheljon: Joan Bouchelion: 23 October 1655 – 1 November 1656 Zacharias Wagenaer: 1 November 1656 – 27 October 1657 Joan Bouchelion: 27 October 1657 – 23 October 1658 Zacharias Wagenaer: 22 October 1658 – 4 November 1659 Joan Bouchelion: 4 November 1659 – 26 October 1660 Wagenern, Zacharias.
Hlengiwe Buhle Mkhize MP is a former Minister of Higher Education and Training, having been appointed to the position by former President Jacob Zuma since 17 October 2017, after serving as Deputy Minister of Telecommunications and Postal Services in the government of South Africa. In February 2018, she was sacked from cabinet by President Cyril Ramaphosa. First elected to the National Assembly of South Africa in 2009 as part of the African National Congress. Professor Mkhize holds a Bachelor of Arts in Psychology, Social Work and Sociology from the University of Zululand. Mkhize is a founding member, trustee, of the Children and Violence Trust since 1995, had been a trustee of the Malibongwe Business Trust from 2005, she was a senior lecturer and researcher at Wits University from 1990 until 1995. Mkhize was a board member of the South African Prisoner's Organisation for Human Rights from 1994 to 1995. Prior to her appointment, Mkhize was ambassador to the Netherlands, had a short stint as Deputy Minister for Correctional Services and most served as the Deputy Minister of Telecommunications and Postal Services.
People by Hlengiwe Mkhize at People's Assembly