Split (1)

train · 40.2k rows

source stringlengths 31 168	text stringlengths 51 3k
https://en.wikipedia.org/wiki/Journal%20of%20Symbolic%20Logic	The Journal of Symbolic Logic is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by Mathematical Reviews, Zentralblatt MATH, and Scopus. Its 2009 MCQ was 0.28, and its 2009 impact factor was 0.631. External links Mathematics journals Academic journals established in 1936 Multilingual journals Quarterly journals Association for Symbolic Logic academic journals Logic journals Cambridge University Press academic journals
https://en.wikipedia.org/wiki/Hamudi%20Brick	Mohamad "Hamoudi" Brik (, ; born March 19, 1978) is a former Arab-Israeli footballer. External links Profile and statistics of Hamudi Brick on One.co.il 1978 births Living people Arab citizens of Israel Arab-Israeli footballers Israeli Muslims Israeli men's footballers Maccabi Kafr Kanna F.C. players Maccabi Netanya F.C. players Bnei Sakhnin F.C. players Hapoel Acre F.C. players Ahva Arraba F.C. players Liga Leumit players Israeli Premier League players Men's association football defenders
https://en.wikipedia.org/wiki/Probabilistic%20automaton	In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset. The number of stochastic languages is uncountable. The concept was introduced by Michael O. Rabin in 1963; a certain special case is sometimes known as the Rabin automaton (not to be confused with the subclass of ω-automata also referred to as Rabin automata). In recent years, a variant has been formulated in terms of quantum probabilities, the quantum finite automaton. Informal Description For a given initial state and input character, a deterministic finite automaton (DFA) has exactly one next state, and a nondeterministic finite automaton (NFA) has a set of next states. A probabilistic automaton (PA) instead has a weighted set (or vector) of next states, where the weights must sum to 1 and therefore can be interpreted as probabilities (making it a stochastic vector). The notions states and acceptance must also be modified to reflect the introduction of these weights. The state of the machine as a given step must now also be represented by a stochastic vector of states, and a state accepted if its total probability of being in an acceptance state exceeds some cut-off. A PA is in some sense a half-way step from deterministic to non-deterministic, as it allows a set of next states but with restrictions on their weights. However, this is somewhat misleading, as the PA utilizes the notion of the real numbers to define the weights, which is absent in the definition of both DFAs and NFAs. This additional freedom enables them to decide languages that are not regular, such as the p-adic languages with irrational parameters. As such, PAs are more powerful than both DFAs and NFAs (which are famously equally powerful). Formal Definition The probabilistic automaton may be defined as an extension of a nondeterministic finite automaton , together with two probabilities: the probability of a particular state transition taking place, and with the initial state replaced by a stochastic vector giving the probability of the automaton being in a given initial state. For the ordinary non-deterministic finite automaton, one has a finite set of states a finite set of input symbols a transition function a set of states distinguished as accepting (or final) states . Here, denotes the power set of . By use of currying, the transition function of a non-deterministic finite automaton can be written as a membership function so that if and otherwise. The curried transition function can be understood to be a matrix with matrix entries The matrix is then a square
https://en.wikipedia.org/wiki/Conceptual%20physics	Conceptual physics is an approach to teaching physics that focuses on the ideas of physics rather than the mathematics. It is believed that with a strong conceptual foundation in physics, students are better equipped to understand the equations and formulas of physics, and to make connections between the concepts of physics and their everyday life. Early versions used almost no equations or math-based problems. Paul G. Hewitt popularized this approach with his textbook Conceptual Physics: A New Introduction to your Environment in 1971. In his review at the time, Kenneth W. Ford noted the emphasis on logical reasoning and said "Hewitt's excellent book can be called physics without equations, or physics without computation, but not physics without mathematics." Hewitt's wasn't the first book to take this approach. Conceptual Physics: Matter in Motion by Jae R. Ballif and William E. Dibble was published in 1969. But Hewitt's book became very successful. As of 2022, it is in its 13th edition. In 1987 Hewitt wrote a version for high school students. The spread of the conceptual approach to teaching physics broadened the range of students taking physics in high school. Enrollment in conceptual physics courses in high school grew from 25,000 students in 1987 to over 400,000 in 2009. In 2009, 37% of students took high school physics, and 31% of them were in Physics First, conceptual physics courses, or regular physics courses using a conceptual textbook. This approach to teaching physics has also inspired books for science literacy courses, such as From Atoms to Galaxies: A Conceptual Physics Approach to Scientific Awareness by Sadri Hassani. References Further reading External links Conceptual Physics website Physics education
https://en.wikipedia.org/wiki/Andrzej%20Schinzel	Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory. Education Schinzel received an MSc in 1958 at Warsaw University, Ph.D. in 1960 from Institute of Mathematics of the Polish Academy of Sciences where he studied under Wacław Sierpiński, with a habilitation in 1962. He was a member of the Polish Academy of Sciences. Career Schinzel was a professor at the Institute of Mathematics of the Polish Academy of Sciences (IM PAN). His principal interest was the theory of polynomials. His 1958 conjecture on the prime values of polynomials, known as Schinzel's hypothesis H, both extends the Bunyakovsky conjecture and broadly generalizes the twin prime conjecture. He also proved Schinzel's theorem on the existence of circles through any given number of integer points. Schinzel was the author of over 200 research articles in various branches of number theory, including elementary, analytic and algebraic number theory. He was the editor of Acta Arithmetica for over four decades. Private life Andrzej Schinzel was the oldest brother of a Polish chess master Władysław Schinzel (born 1943). References External links Schinzel's page at IM PAN (list of publications) Andrzej Schinzel's picture 1937 births 2021 deaths 20th-century Polish mathematicians 21st-century Polish mathematicians Number theorists Members of the Polish Academy of Sciences University of Warsaw alumni People from Sandomierz
https://en.wikipedia.org/wiki/Scotland%20national%20football%20team%20records%20and%20statistics	This page details Scotland national football team records and statistics; the most capped players, the players with the most goals, and Scotland's match record by opponent and decade. Player records Most capped players Players in bold are still active with Scotland. Top goalscorers Players in bold are still active with Scotland. Hat-tricks Table Wartime internationals, not regarded as official matches, are not included in the list. Team records Head to head records Statistics include official FIFA recognised matches, five matches from a 1967 overseas tour that were reclassified as full internationals in 2021, and a match against a Hong Kong League XI played on 23 May 2002 that the Scottish Football Association includes in its statistical totals. By period Statistics include official FIFA recognised matches, five matches from a 1967 overseas tour that were reclassified as full internationals in 2021, and a match against a Hong Kong League XI played on 23 May 2002 that the Scottish Football Association includes in its statistical totals. Notes References National association football team records and statistics
https://en.wikipedia.org/wiki/List%20of%20Hull%20City%20A.F.C.%20records%20and%20statistics	Most league goals Most goals in a season Most league appearances Players Youngest Player Matt Edeson, 16 years & 63 days – Hull City vs Fulham – 10 October 1992 Oldest Player Steve Harper, 40 years & 71 days – Hull City vs Manchester United – 24 May 2015 Oldest goal scorer Dean Windass, 39 years & 235 days – Hull City v Portsmouth Results Biggest Victory 11–1 vs Carlisle United, Division 3, 14 January 1939 In Premier League (home): 6–0 vs Fulham, Premier League, 28 December 2013 In Premier League (away): 4–0 vs Cardiff City, Premier League, 22 February 2014 Biggest Defeat 0–8 vs Wolverhampton Wanderers, Division 2, 4 November 1911 0–8 vs Wigan Athletic, EFL Championship, 14 July 2020 Transfer fees Paid £13,000,000 – Ryan Mason from Tottenham Hotspur – 2016 £10,000,000 – Abel Hernández from Palermo – 2014 £8,000,000 (reportedly) – Jake Livermore from Tottenham Hotspur – 2014 £7,000,000 – Robert Snodgrass from Norwich City – 2014 £6,000,000 (reportedly) (rising to £7,000,000) – Shane Long from West Bromwich Albion – 2014 & Nikica Jelavić from Everton – 2014 £5,250,000 (reportedly) – Tom Huddlestone from Tottenham – 2013 £5,000,000 – Jimmy Bullard from Fulham – 2009 £3,500,000 – Stephen Hunt from Reading – 2009 (undisclosed fee reportedly in the region of £3.5 million) £3,000,000 – Seyi Olofinjana from Stoke City – 2009 £2,600,000 – Nick Proschwitz from Paderborn 07 – 2012 (€3.3 million) £2,500,000 – Anthony Gardner from Tottenham Hotspur – 2008 (originally a loan with the option to sign permanently for a fee in the region of £2.5 million) £2,500,000 – Kamil Zayatte from Young Boys – 2009 (undisclosed fee reported to equal the club's transfer record for Gardner) £2,000,000 – Péter Halmosi from Plymouth Argyle – 2008 (undisclosed fee reportedly in the region of £2 million) £1,800,000 – Steven Mouyokolo from Boulogne – 2009 (undisclosed fee reportedly in the region of £1.8 million) £1,700,000 – Kamel Ghilas from Celta Vigo – 2009 (undisclosed fee reportedly in the region of £1.7 million) £1,500,000 – Daniel Cousin from Rangers – 2008 £1,000,000 – Caleb Folan from Wigan Athletic – 2007 £500,000 – Dean Marney from Tottenham Hotspur – 2006 (undisclosed fee reported as £500,000 plus up to a further conditional £500,000) Received £22,000,000 – Jarrod Bowen to West Ham United – 2020 £17,000,000 – Harry Maguire to Leicester City – 2017 £12,500,000 – Shane Long to Southampton – 2014 £10,000,000 – Robert Snodgrass to West Ham – 2017 £10,000,000 – Jake Livermore to West Bromwich Albion – 2017 £8,000,000 – James Chester to West Bromwich Albion – 2015 £4,000,000 – Michael Turner to Sunderland – 2009 £2,000,000 – Sam Ricketts to Bolton Wanderers – 2009 (undisclosed fee reportedly in the region of £2–3 million) £1,250,000 – Leon Cort to Crystal Palace – 2006 £1,000,000 – Craig Fagan to Derby County – 2007 (£750,000 plus a further £250,000 due to promotion) £900,000 (reported) – Jack Hobbs to Nottingham For
https://en.wikipedia.org/wiki/Dynamic%20convex%20hull	The dynamic convex hull problem is a class of dynamic problems in computational geometry. The problem consists in the maintenance, i.e., keeping track, of the convex hull for input data undergoing a sequence of discrete changes, i.e., when input data elements may be inserted, deleted, or modified. It should be distinguished from the kinetic convex hull, which studies similar problems for continuously moving points. Dynamic convex hull problems may be distinguished by the types of the input data and the allowed types of modification of the input data. Planar point set It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore, if the convex hull is required to be reported in traditional way as a polygon, the lower bound for the worst-case computational complexity of the recomputation of the convex hull is , since this time is required for a mere reporting of the output. This lower bound is attainable, because several general-purpose convex hull algorithms run in linear time when input points are ordered in some way and logarithmic-time methods for dynamic maintenance of ordered data are well-known. This problem may be overcome by eliminating the restriction on the output representation. There are data structures that can maintain representations of the convex hull in an amount of time per update that is much smaller than linear. For many years the best algorithm of this type was that of Overmars and van Leeuwen (1981), which took time O(log2 n) per update, but it has since been improved by Timothy M. Chan and others. In a number of applications finding the convex hull is a step in an algorithm for the solution of the overall problem. The selected representation of the convex hull may influence on the computational complexity of further operations of the overall algorithm. For example, the point in polygon query for a convex polygon represented by the ordered set of its vertices may be answered in logarithmic time, which would be impossible for convex hulls reported by the set of it vertices without any additional information. Therefore, some research of dynamic convex hull algorithms involves the computational complexity of various geometric search problems with convex hulls stored in specific kinds of data structures. The mentioned approach of Overmars and van Leeuwen allows for logarithmic complexity of various common queries. References . Convex hull algorithms
https://en.wikipedia.org/wiki/Carath%C3%A9odory%20metric	In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory. Definition Let (X, \|\| \|\|) be a complex Banach space and let B be the open unit ball in X. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by (thus fixing the curvature to be −4). Then the Carathéodory metric d on B is defined by What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy. Properties For any point x in B, d can also be given by the following formula, which Carathéodory attributed to Erhard Schmidt: For all a and b in B, with equality if and only if either a = b or there exists a bounded linear functional ℓ ∈ X∗ such that \|\|ℓ\|\| = 1, ℓ(a + b) = 0 and Moreover, any ℓ satisfying these three conditions has \|ℓ(a − b)\| = \|\|a − b\|\|. Also, there is equality in (1) if \|\|a\|\| = \|\|b\|\| and \|\|a − b\|\| = \|\|a\|\| + \|\|b\|\|. One way to do this is to take b = −a. If there exists a unit vector u in X that is not an extreme point of the closed unit ball in X, then there exist points a and b in B such that there is equality in (1) but b ≠ ±a. Carathéodory length of a tangent vector There is an associated notion of Carathéodory length for tangent vectors to the ball B. Let x be a point of B and let v be a tangent vector to B at x; since B is the open unit ball in the vector space X, the tangent space TxB can be identified with X in a natural way, and v can be thought of as an element of X. Then the Carathéodory length of v at x, denoted α(x, v), is defined by One can show that α(x, v) ≥ \|\|v\|\|, with equality when x = 0. See also Earle–Hamilton fixed point theorem References Hyperbolic geometry Metric geometry
https://en.wikipedia.org/wiki/Utah%20Jazz%20all-time%20roster	The following is a list of players, both past and current, who have appeared at least in one game for the New Orleans/Utah Jazz NBA basketball franchise. Players Note: Statistics are correct through the end of the season. A to B \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Loyola Marymount \|\| align="center"\|1 \|\| align="center"\| \|\| 28 \|\| 613 \|\| 55 \|\| 69 \|\| 175 \|\| 21.9 \|\| 2.0 \|\| 2.5 \|\| 6.3 \|\| align=center\| \|- \|align="left" bgcolor="#CCFFCC"\|x \|\| align="center"\|G \|\| align="left"\|Kansas \|\| align="center"\|1 \|\| align="center"\| \|\| 59 \|\| 1,209 \|\| 121 \|\| 67 \|\| 467 \|\| 20.5 \|\| 2.1 \|\| 1.1 \|\| 7.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Missouri State \|\| align="center"\|1 \|\| align="center"\| \|\| 4 \|\| 30 \|\| 2 \|\| 1 \|\| 10 \|\| 7.5 \|\| 0.5 \|\| 0.3 \|\| 2.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Virginia Tech \|\| align="center"\|2 \|\| align="center"\|– \|\| 51 \|\| 677 \|\| 82 \|\| 93 \|\| 281 \|\| 13.3 \|\| 1.6 \|\| 1.8 \|\| 5.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Duke \|\| align="center"\|1 \|\| align="center"\| \|\| 38 \|\| 416 \|\| 23 \|\| 25 \|\| 211 \|\| 10.9 \|\| 0.6 \|\| 0.7 \|\| 5.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Rice \|\| align="center"\|2 \|\| align="center"\|– \|\| 34 \|\| 294 \|\| 38 \|\| 10 \|\| 105 \|\| 8.6 \|\| 1.1 \|\| 0.3 \|\| 3.1 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Penn State \|\| align="center"\|2 \|\| align="center"\|– \|\| 104 \|\| 1,060 \|\| 186 \|\| 50 \|\| 274 \|\| 10.2 \|\| 1.8 \|\| 0.5 \|\| 2.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|UNLV \|\| align="center"\|1 \|\| align="center"\| \|\| 1 \|\| 2 \|\| 0 \|\| 0 \|\| 0 \|\| 2.0 \|\| 0.0 \|\| 0.0 \|\| 0.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Bradley \|\| align="center"\|3 \|\| align="center"\|– \|\| 144 \|\| 1,922 \|\| 427 \|\| 109 \|\| 734 \|\| 13.3 \|\| 3.0 \|\| 0.8 \|\| 5.1 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G/F \|\| align="left"\|Georgia \|\| align="center"\|3 \|\| align="center"\|– \|\| 197 \|\| 3,740 \|\| 538 \|\| 194 \|\| 1,494 \|\| 19.0 \|\| 2.7 \|\| 1.0 \|\| 7.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|BYU \|\| align="center"\|1 \|\| align="center"\| \|\| 28 \|\| 248 \|\| 66 \|\| 10 \|\| 72 \|\| 8.9 \|\| 2.4 \|\| 0.4 \|\| 2.6 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|FIU \|\| align="center"\|3 \|\| align="center"\|– \|\| 145 \|\| 3,037 \|\| 257 \|\| 560 \|\| 1,264 \|\| 20.9 \|\| 1.8 \|\| 3.9 \|\| 8.7 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|Arizona State \|\| align="center"\|2 \|\| align="center"\|– \|\| 77 \|\| 418 \|\| 114 \|\| 11 \|\| 190 \|\| 5.4 \|\| 1.5 \|\| 0.1 \|\| 2.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Seton Hall \|\| align="center"\|1 \|\| align="center"\| \|\| 5 \|\| 44 \|\| 12 \|\| 1 \|\| 9 \|\| 8.8 \|\| 2.4 \|\| 0.2 \|\| 1.8 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|Kansas \|\| align="center"\|3 \|\| align="center"\|– \|\| 68 \|\| 611 \|\| 203 \|\| 11 \|\| 221 \|\| 9.0 \|\| 3.0 \|\| 0.2 \|\| 3.3 \|\| align=center\| \|- \|align="left"\| \|\| align="c
https://en.wikipedia.org/wiki/Censoring%20%28statistics%29	In statistics, censoring is a condition in which the value of a measurement or observation is only partially known. For example, suppose a study is conducted to measure the impact of a drug on mortality rate. In such a study, it may be known that an individual's age at death is at least 75 years (but may be more). Such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75. Censoring also occurs when a value occurs outside the range of a measuring instrument. For example, a bathroom scale might only measure up to 140 kg. If a 160-kg individual is weighed using the scale, the observer would only know that the individual's weight is at least 140 kg. The problem of censored data, in which the observed value of some variable is partially known, is related to the problem of missing data, where the observed value of some variable is unknown. Censoring should not be confused with the related idea truncation. With censoring, observations result either in knowing the exact value that applies, or in knowing that the value lies within an interval. With truncation, observations never result in values outside a given range: values in the population outside the range are never seen or never recorded if they are seen. Note that in statistics, truncation is not the same as rounding. Types Left censoring – a data point is below a certain value but it is unknown by how much. Interval censoring – a data point is somewhere on an interval between two values. Right censoring – a data point is above a certain value but it is unknown by how much. Type I censoring occurs if an experiment has a set number of subjects or items and stops the experiment at a predetermined time, at which point any subjects remaining are right-censored. Type II censoring occurs if an experiment has a set number of subjects or items and stops the experiment when a predetermined number are observed to have failed; the remaining subjects are then right-censored. Random (or non-informative) censoring is when each subject has a censoring time that is statistically independent of their failure time. The observed value is the minimum of the censoring and failure times; subjects whose failure time is greater than their censoring time are right-censored. Interval censoring can occur when observing a value requires follow-ups or inspections. Left and right censoring are special cases of interval censoring, with the beginning of the interval at zero or the end at infinity, respectively. Estimation methods for using left-censored data vary, and not all methods of estimation may be applicable to, or the most reliable, for all data sets. A common misconception with time interval data is to class as left censored intervals where the start time is unknown. In these cases we have a lower bound on the time interval, thus the data is right censored (despite the fact that the missing start point is to the left of t
https://en.wikipedia.org/wiki/Angus%20Macintyre	Angus John Macintyre FRS, FRSE (born 1941) is a British mathematician and logician who is a leading figure in model theory, logic, and their applications in algebra, algebraic geometry, and number theory. He is Emeritus Professor of Mathematics, at Queen Mary University of London. Education After undergraduate study at the University of Cambridge, he completed his PhD at Stanford University under the supervision of Dana Scott in 1968. Career and research From 1973 to 1985, he was Professor of Mathematics at Yale University. From 1985 to 1999, he was Professor of Mathematical Logic at Merton College at the University of Oxford. In 1999, Macintyre moved to the University of Edinburgh, where he was Professor of Mathematics until 2002, when he moved to Queen Mary College, University of London. Macintyre was the first Scientific Director of the International Centre for Mathematical Sciences (ICMS) in Edinburgh. Macintyre is known for many important results. These include classification of aleph-one categorical theories of groups and fields in 1971, which was very influential in the development of geometric stability theory. In 1976, he proved a result on quantifier elimination for p-adic fields from which a theory of semi-algebraic and subanalytic geometry for p-adic fields follows (in analogy with that for the real field) as shown by Jan Denef and Lou van den Dries and others. This quantifier elimination theorem was used by Jan Denef in 1984 to prove a conjecture of Jean-Pierre Serre on rationality of various p-adic Poincaré series, and subsequently these methods have been applied to prove rationality of a wide range of generating functions in group theory (e.g. subgroup growth) and number theory by various authors, notably Dan Segal and Marcus du Sautoy. Macintyre worked with Zoé Chatzidakis and Lou van den Dries on definable sets over finite fields generalising the estimates of Serge Lang and André Weil to definable sets an revisiting the work of James Ax on the logic of finite and pseudofinite fields. He initiated and proved results on the model theory of difference fields and of Frobenius automorphisms, where he proved extensions of Ax's work to this setting (including model-companions and decidability). Independently Ehud Hrushovski has proved model-theoretic results on Frobenius automorphisms. Macintyre developed a first-order model theory for intersection theory and showed connections to Alexander Grothendieck's standard conjectures on algebraic cycles. Macintyre has proved many results on the model theory of real and complex exponentiation. With Alex Wilkie he proved the decidability of real exponential fields (solving a problem of Alfred Tarski) modulo Schanuel's conjecture from transcendental number theory. With Lou van den Dries he initiated and studied the model theory of logarithmic-exponential series and Hardy fields. Together with David Marker and Lou van den Dries, he proved several results on the model theory of the real field e
https://en.wikipedia.org/wiki/Vrap%C4%8Di%C5%A1te	Vrapčište (, , ) is a village and seat of the municipality of Vrapčište, North Macedonia. History In statistics gathered by Vasil Kanchov in 1900, the village of Vrapčište was inhabited by 1300 Turks, 325 Orthodox Bulgarians, 165 Muslim Albanians and 50 Romani. A policy of Turkification of the Albanian population was employed by the Yugoslav authorities in cooperation with the Turkish government, stretching the period of 1948-1959. Starting in 1948, Turkish schools were opened in areas with large Albanian majorities, such as Vrapčište. Demographics As of the 2021 census, Vrapčište had 4,003 residents with the following ethnic composition: Turks 2,765 Albanians 912 Persons for whom data are taken from administrative sources 203 Macedonians 118 Others 5 According to the 2002 census, the village had a total of 4,874 inhabitants. Ethnic groups in the village include: Turks 2,899 Albanians 1,777 Macedonians 172 Others 26 Sports Local football club FK Vrapčište plays in the OFS Gostivar league. References External links Villages in Vrapčište Municipality Albanian communities in North Macedonia
https://en.wikipedia.org/wiki/Verner%20Emil%20Hoggatt%20Jr.	Verner Emil Hoggatt Jr. (June 26, 1921 – August 11, 1980) was an American mathematician, known mostly for his work in Fibonacci numbers and number theory. Hoggatt received a Ph.D. from Oregon State University in 1955 for his dissertation on The Inverse Weierstrass P-Function. Besides his contributions in Fibonacci numbers and number theory it is as the co-founder of the Fibonacci association and publisher of the associated journal Fibonacci Quarterly for which he is best remembered. Howard Eves commented, "During his long and outstanding tenure at San Jose State University, Vern directed an enormous number of master's theses, and put out an amazing number of attractive papers... He became the authority on Fibonacci and related numbers." See also Fibonacci numbers The Fibonacci Association Alfred Brousseau References Verner Hoggatt Biography Fibonacci mathematicians of the 20th century External links The Official website of the Fibonacci Association The Fibonacci Quarterly 1921 births 1980 deaths 20th-century American mathematicians Number theorists Oregon State University alumni San Jose State University faculty
https://en.wikipedia.org/wiki/Dimiter%20Skordev	Dimiter Skordev () (born 1936 in Sofia) is a professor in the Department of Mathematical Logic and Applications, Faculty of Mathematics and Computer Science at the University of Sofia. Chairman of the department in 1972-2000. Doyen and pioneer of mathematical logic research in Bulgaria who developed a Bulgarian school in the theory of computability, namely the algebraic (or axiomatic) recursion theory. He was the 1981 winner of Acad. Nikola Obreshkov Prize, the highest Bulgarian award in mathematics, bestowed for his monograph Combinatory Spaces and Recursiveness in Them. Skordev's field of scientific interests include computability and complexity in analysis, mathematical logic, generalized recursion theory, and theory of programs and computation. Skordev has more than 45 years of lecturing experience in calculus, mathematical logic, logic programming, discrete mathematics, and computer science. He has authored about 90 scientific publications including two monographs, and was one of the authors of the new Bulgarian phonetic keyboard layout proposed (but rejected) to become a state standard in 2006. Notes References Dimiter Skordev Historical notes on the development of mathematical logic in Sofia 1936 births Living people 20th-century Bulgarian mathematicians 21st-century Bulgarian mathematicians Mathematical logicians Bulgarian logicians Bulgarian philosophers Scientists from Sofia
https://en.wikipedia.org/wiki/Sergei%20Skorykh	Sergei Skorykh (born 25 May 1984) is a Kazakh football midfielder. Career statistics Club International Statistics accurate as of match played 1 December 2014 References External links 1984 births Living people Men's association football midfielders Kazakhstani men's footballers Kazakhstan men's international footballers Kazakhstan Premier League players FC Irtysh Pavlodar players FC Tobol players FC Zhetysu players FC Taraz players FC Kaisar players FC Shakhter Karagandy players Sportspeople from Petropavl
https://en.wikipedia.org/wiki/Murat%20Suyumagambetov	Murat Suyumagambetov (; born 14 October 1983 in Aktau) is a Kazakhstani football forward who last played for FC Kyran. He also plays for the Kazakhstan national football team. Career statistics International goals References 1983 births Living people Men's association football midfielders Kazakhstani men's footballers Kazakhstan men's international footballers Kazakhstan Premier League players FC Zhenis players FC Shakhter Karagandy players FC Tobol players FC Caspiy players FC Zhetysu players FC Ordabasy players FC Kairat players FC Vostok players FC Taraz players People from Aktau
https://en.wikipedia.org/wiki/Department%20of%20Mathematical%20Logic%20%28Bulgarian%20Academy%20of%20Sciences%29	The Department of Mathematical Logic at the Bulgarian Academy of Sciences was created by the Institute of Mathematics and Informatics in implementation of Government Decree N0. 236 of November 3, 1959. Its first chairman was Boyan Petkanchin (1907–87) who worked to promote and disseminate the knowledge of mathematical logic both in the professional mathematical community in Bulgaria and as popular science. Vladimir Sotirov and Radoslav Pavlov joined the department in 1970, followed by George Gargov, Anatoly Buda, Lyubomir Ivanov, Slavyan Radev and Solomon Passy in 1976-89. In 1996-2000 the department was joined by Dimiter Dobrev, Jordan Zashev and Dimitar Guelev. From 1971 to 1989 the department was merged with the corresponding division of the Faculty of Mathematics and Informatics at Sofia University, with Dimiter Skordev heading the integrated structure since 1971. In 1989 the institutional relationship with Sofia University was severed, and the department resumed as a division of the Institute of Mathematics and Informatics, headed since then by Lyubomir Ivanov. The logicians Bogdan Dyankov, Hristo Smolenov, Veselin Petrov and Marion Mircheva stayed with the department for various periods of time, all of them coming from the Institute of Philosophy at the Bulgarian Academy of Sciences once the latter was dissolved on account of the dissident activities of its members in 1989. The research of the department is mostly in the area of algebraic recursion theory, modal, temporal and other non-classical logics, as well as logic programming including the development of a version of the Prolog programming language. The department developed also the Streamlined System adopted as the official national system for the Romanization of Bulgarian, and eventually codified by the Bulgarian Law of Transliteration in 2009. A joint multi-institutional project led by the department has contributed to the development and introduction of a new Bulgarian phonetic keyboard layout for personal computers and mobile phones. Besides their research activities, members of the department have an extensive lecturing practice at various faculties of Sofia University as well as other Bulgarian universities. Some members of the department have earned public recognition for their non-academic activities. Sotirov, Ivanov, and Passy were returned MPs in the VII Grand National Assembly on the side of the Union of Democratic Forces, and co-authored the new Bulgarian Constitution. In 2011 the departments of Mathematical Logic and Algebra were merged to form the Department of Algebra and Logic at the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences. References Department of Mathematical Logic Skordev D. Historical notes on the development of mathematical logic in Sofia. Annuaire de l'Univ. de Sofia, Fac. de Math. et Inf. 96, 2004. pp. 11–21. Andreev A., I. Derzhanski eds. Bulgarian Academy of Sciences: Institute of Mathematics and Informatics,
https://en.wikipedia.org/wiki/Boyan%20Petkanchin	Boyan Petkanchin () (April 8, 1907 – March 3, 1987) was a prominent Bulgarian mathematician, working in geometry and foundation of mathematics. As a first chairman of the Department of Mathematical Logic at the Bulgarian Academy of Sciences he worked to promote and disseminate the knowledge of mathematical logic both in the professional mathematical community in Bulgaria and as popular science. External links Historical notes on the development of mathematical logic in Sofia 1907 births 1987 deaths Bulgarian logicians 20th-century Bulgarian mathematicians 20th-century Bulgarian philosophers
https://en.wikipedia.org/wiki/Noether%20Lecture	The Noether Lecture is a distinguished lecture series that honors women "who have made fundamental and sustained contributions to the mathematical sciences". The Association for Women in Mathematics (AWM) established the annual lectures in 1980 as the Emmy Noether Lectures, in honor of one of the leading mathematicians of her time. In 2013 it was renamed the AWM-AMS Noether Lecture and since 2015 is sponsored jointly with the American Mathematical Society (AMS). The recipient delivers the lecture at the yearly American Joint Mathematics Meetings held in January. The ICM Emmy Noether Lecture is an additional lecture series, sponsored by the International Mathematical Union. Beginning in 1994 this lecture was delivered at the International Congress of Mathematicians, held every four years. In 2010 the lecture series was made permanent. The 2021 Noether Lecture was supposed to have been given by Andrea Bertozzi of UCLA, but it was cancelled due to Bertozzi's connections to policing. The cancellation was made during the George Floyd protests: "This decision comes as many of this nation rise up in protest over racial discrimination and brutality by police". Noether Lecturer ICM Emmy Noether Lecturers See also Falconer Lecture Kovalevsky Lecture List of mathematics awards List of things named after Emmy Noether References External links Lists of women scientists Women in mathematics Science awards honoring women Awards and prizes of the Association for Women in Mathematics Lecture series American Mathematical Society Quadrennial events Awards established in 1980 Awards established in 1994 1980 establishments in the United States 1994 establishments in the United States International Congress of Mathematicians
https://en.wikipedia.org/wiki/Complex%20geodesic	In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. Definition Let (X, \|\| \|\|) be a complex Banach space and let B be the open unit ball in X. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by and denote the corresponding Carathéodory metric on B by d. Then a holomorphic function f : Δ → B is said to be a complex geodesic if for all points w and z in Δ. Properties and examples of complex geodesics Given u ∈ X with \|\|u\|\| = 1, the map f : Δ → B given by f(z) = zu is a complex geodesic. Geodesics can be reparametrized: if f is a complex geodesic and g ∈ Aut(Δ) is a bi-holomorphic automorphism of the disc Δ, then f o g is also a complex geodesic. In fact, any complex geodesic f1 with the same image as f (i.e., f1(Δ) = f(Δ)) arises as such a reparametrization of f. If for some z ≠ 0, then f is a complex geodesic. If where α denotes the Caratheodory length of a tangent vector, then f is a complex geodesic. References Hyperbolic geometry Geodesic (mathematics)
https://en.wikipedia.org/wiki/List%20of%20career%20achievements%20by%20Tiger%20Woods	This page details statistics, records, and other achievements pertaining to Tiger Woods. Career records and statistics Woods has won 82 official PGA Tour events, tied with Sam Snead also 82, and nine ahead of Jack Nicklaus's 73 wins. (See List of golfers with most PGA Tour wins.) Woods has won 15 majors, second all time behind Jack Nicklaus' 18. Woods is 14–1 when going into the final round of a major with at least a share of the lead. Woods scoring average in 2000 is the lowest in PGA Tour history, both adjusted, 67.79, and unadjusted, 68.17. Woods has the lowest career scoring average in PGA Tour history. Woods has amassed the most career earnings of any player in PGA Tour history (even after inflation is considered). Woods is one of five players (along with Gene Sarazen, Ben Hogan, Gary Player, and Jack Nicklaus) to have won all four professional major championships in his career, known as the Career Grand Slam, and was the youngest to do so. Woods is the only player to have won all four professional major championships in a row, accomplishing the feat in the 2000–2001 seasons. This feat became known as the "Tiger Slam". Woods set the all-time PGA Tour record for most consecutive cuts made, with 142. The streak started in 1998, he set the record at the 2003 Tour Championship with 114 (passing Byron Nelson's previous record of 113 and Jack Nicklaus at 105) and extended this mark to 142 before it ended on May 13, 2005 at the EDS Byron Nelson Championship. Many consider this to be one of the most remarkable golf accomplishments of all time, given the margin by which he broke the old record and given that during the streak, the next longest streak by any other player was usually only in the 10s or 20s. When Byron Nelson played far fewer players made the cut in a given event. Woods has won a record 22.8% (82 out of 359) of his professional starts on the PGA Tour. Woods is the only golfer to have won the U.S. Amateur three consecutive times (1994–1996). Awards records Woods has been the PGA Player of the Year a record eleven times. Woods has been the PGA Tour Player of the Year a record eleven times. Woods has been the PGA Tour Money Leader a record ten times. Woods has been the Vardon Trophy winner a record nine times. Woods has been the recipient of the Byron Nelson Award a record nine times. Miscellaneous Woods owns a 55–4 record when holding at least a share of the lead after 54 holes, and 44–2 record when holding the outright lead. Woods has only lost once when leading by more than one shot after 54 holes. Yang Yong-eun began the final round of the 2009 PGA Championship two strokes behind Woods and defeated him by three strokes. Woods has a 39–11 record when leading after 36 holes in Tour events, including an 8–3 record in majors. Woods has won 14 tournaments wire-to-wire, including seven times while holding the lead outright after each round: 2000 U.S. Open, 2000 PGA Championship (tied after 1st and 4th rounds), 2000 WG
https://en.wikipedia.org/wiki/AMS-55	AMS-55 may refer to: USS Seagull (AMS-55), battleship NBS AMS 55 aka Abramowitz and Stegun, a mathematics textbook
https://en.wikipedia.org/wiki/Encyclopedia%20of%20the%20History%20of%20Arabic%20Science	The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a marked influence on the Middle Ages in Europe. It is written by internationally recognized experts in the field and edited by Roshdi Rashed in collaboration with Régis Morelon. Volume one covers "Astronomy—Theoretical and applied". Volume two covers "Mathematics and the Physical Sciences". Volume three covers "Technology, Alchemy, and the Life Sciences". Editions French edition: ":fr:Histoire des sciences arabes", 3 vol., Le Seuil, Paris, 1997, (). Arabic edition: "Mawsu‘a Tarikh al-‘ulum al-‘arabiyya", 3 vol., Markaz Dirasat al-Wahda al-‘arabiyya, Beirut, 1997, (, 978-9953-450-73-5). Contributors A partial list of contributors include: Volume 1 R. Morelon and George Saliba (Arabic astronomy) David A. King (astronomy in Islamic society) Edward Stewart Kennedy (mathematical geography) J. Vernet and J. Samsó (Arabic science in Andalusia) H. Grosset-Grange (Arabic nautical sciences) Volume 2 A. S. Saidan (numeration and arithmetic) Boris A. Rosenfeld and A. P. Yushkevich (geometry) J.-C. Chabrier and M. Rozhanskaya (music and statics) M.-Th. Debarnot (trigonometry, algebra) Roshdi Rashed (geometrical optics) G. Russell (physiological optics) Volume 3 Donald Routledge Hill (engineering) A. Miquel (geography) Toufic Fahd (botany and agriculture) G. Anawati (Arabic alchemy) E. Savage-Smith (medicine) F. Micheau (scientific institutions in the medieval Near East) J. Jolivet (classifications of the sciences) M. Mahdi (historiography) B. Goldstein (heritage of Arabic science in Hebrew) H. Hugonnard-Roche, A. Allard, D. Lindberg, R. Halleux, and D. Jacquart (Western reception of various Arabic sciences) Notes References J. L. Berggren. "Reviewed work(s): Encyclopedia of the History of Arabic Science by Roshdi Rashed". Journal of the American Oriental Society. Vol. 120, No. 2 (Apr. - Jun., 2000), pp. 282-283. Sonja Brentjes. "Reviewed work(s): Encyclopedia of the History of Arabic Science by Roshdi Rashed". Technology and Culture. Vol. 40, No. 2 (Apr., 1999), pp. 399-401. Charles Burnett. "Reviewed work(s): Encyclopedia of the History of Arabic Science by Roshdi Rashed; Régis Morelon". The British Journal for the History of Science. Vol. 31, No. 1 (Mar., 1998), pp. 72–73. External links Volume 1 Volume 2 History Arabic Science History Arabic Science History Arabic Science Science in the medieval Islamic world Works about the history of mathematics 20th-century encyclopedias
https://en.wikipedia.org/wiki/Simen%20Brenne	Simen Brenne (born 17 March 1981) is a Norwegian footballer who plays for Råde. Brenne has played in 15 games for Norway, and scored one goal since his debut in 2007. Career statistics International goals Honours Club Fredrikstad Norwegian Football Cup (1): 2006 Lillestrøm Norwegian Football Cup (1): 2007 Strømsgodset Tippeligaen (1): 2013 References External links Odd Grenland profile 1981 births Living people Norwegian men's footballers Norway men's international footballers Norway men's under-21 international footballers Moss FK players Fredrikstad FK players Lillestrøm SK players Odds BK players Strømsgodset Toppfotball players Sarpsborg 08 FF players Eliteserien players Norwegian First Division players Footballers from Fredrikstad Men's association football midfielders
https://en.wikipedia.org/wiki/Jordan%E2%80%93Chevalley%20decomposition	In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. The decomposition is easy to describe when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normal form. Analogues of the Jordan-Chevalley decomposition exist for elements of linear algebraic groups, Lie algebras, and Lie groups, and the decomposition is an important tool in the study of these objects. Decomposition of a linear operator Consider linear operators on a finite-dimensional vector space over a field. An operator is semisimple if every T-invariant subspace has a complementary T-invariant subspace (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xm of it is the zero operator. An operator x is unipotent if x − 1 is nilpotent. Now, let x be any operator. A Jordan–Chevalley decomposition of x is an expression of it as a sum x = xs + xn, where xs is semisimple, xn is nilpotent, and xs and xn commute. Over a perfect field, such a decomposition exists (cf. #Proof of uniqueness and existence), the decomposition is unique, and the xs and xn are polynomials in x with no constant terms. In particular, for any such decomposition over a perfect field, an operator that commutes with x also commutes with xs and xn. If x is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses x as a product x = xs · xu, where xs is semisimple, xu is unipotent, and xs and xu commute. Again, over a perfect field, such a decomposition exists, the decomposition is unique, and xs and xu are polynomials in x. The multiplicative version of the decomposition follows from the additive one since, as is easily seen to be invertible, and is unipotent. (Conversely, by the same type of argument, one can deduce the additive version from the multiplicative one.) If x is written in Jordan normal form (with respect to some basis) then xs is the endomorphism whose matrix contains just the diagonal terms of x, and xn is the endomorphism whose matrix contains just the off-diagonal terms; xu is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element. Proof of uniqueness and existence The uniqueness follows from the fact are polynomial in x: if is another decomposition such that and commute, then , and both commute with x, hence with since they are polynomials in . The sum of commuting nilpotent endomorphisms is nilpotent, and over a perfect field the sum of commuting semisimple endomorphisms is again semisimple. Since the only operator which is both se
https://en.wikipedia.org/wiki/Glivenko%27s%20theorem	Glivenko's theorem may refer to: Glivenko's theorem (probability theory) Glivenko's theorem or Glivenko's translation, a double-negation translation for propositional logic See also Glivenko–Cantelli theorem Glivenko–Stone theorem
https://en.wikipedia.org/wiki/David%20Avis	David Michael Avis (born March 20, 1951) is a Canadian and British computer scientist known for his contributions to geometric computations. Avis is a professor in computational geometry and applied mathematics in the School of Computer Science, McGill University, in Montreal. Since 2010, he belongs to Department of Communications and Computer Engineering, School of Informatics, Kyoto University. Avis received his Ph.D. in 1977 from Stanford University. He has published more than 70 journal papers and articles. Writing with Komei Fukuda, Avis proposed a reverse-search algorithm for the vertex enumeration problem; their algorithm generates all of the vertices of a convex polytope. Selected publications References External links School of Computer Science(McGill Univ.) David Avis’ homepage(McGill Univ.) David Avis' homepage(Kyoto Univ.) http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/a/Avis:David.html 1951 births Living people Researchers in geometric algorithms Stanford University School of Humanities and Sciences alumni Academic staff of McGill University 20th-century British mathematicians 21st-century British mathematicians Anglophone Quebec people Stanford University School of Engineering alumni
https://en.wikipedia.org/wiki/Izvestiya%3A%20Mathematics	Izvestiya: Mathematics is the English translation of the Russian mathematical journal Izvestiya Rossiiskoi Akademii Nauk, Seriya Matematicheskaya () which was founded in 1937. Since 1995, the journal has been published jointly by Turpion, the Russian Academy of Sciences, and the London Mathematical Society. The journal covers all fields of mathematics but pays special attention to: algebra, algebraic geometry, mathematical logic, number theory, mathematical analysis, geometry, topology, and differential equations. Since 2008 electronic access to the content back to the first English translation volume has been hosted by IOP Publishing. The Editor in Chief is V. V. Kozlov, Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow, Russia. Indexing and abstracting The journal is indexed in the following bibliographic databases: Web of Science (SCI-E) MathSciNet Scopus NASA Astrophysical Data System INIS External links Izvestiya Mathematics on IOPscience (IOP Publishing) London Mathematical Society Russian Academy of Sciences Turpion References Mathematics journals Academic journals established in 1937 Russian Academy of Sciences academic journals London Mathematical Society
https://en.wikipedia.org/wiki/CGAL	The Computational Geometry Algorithms Library (CGAL) is an open source software library of computational geometry algorithms. While primarily written in C++, Scilab bindings and bindings generated with SWIG (supporting Python and Java for now) are also available. The software is available under dual licensing scheme. When used for other open source software, it is available under open source licenses (LGPL or GPL depending on the component). In other cases commercial license may be purchased, under different options for academic/research and industrial customers. History The CGAL project was founded in 1996, as a consortium of eight research institutions in Europe and Israel: Utrecht University, ETH Zurich, Free University of Berlin, INRIA Sophia Antipolis, Martin-Luther-University Halle-Wittenberg, Max Planck Institute for Informatics Saarbrücken, Johannes Kepler University Linz, and Tel-Aviv University. The original funding for the project came from the ESPRIT project of the European Union. Originally, its licensing terms allowed its software to be used freely for academic purposes, with commercial licenses available for other uses. CGAL Releases 3.x were distributed under the QPL license. Starting with CGAL 4.0, released in 2012, CGAL is distributed under the GPL version 3. it is managed by a thirteen-member editorial board, with an additional 30 developers and reviewers. The project started in 1996 as the pooling of the previous efforts of several project participants:PlaGeo and SpaGeo from Utrecht University, LEDA of the Max-Planck-Institute for Informatics and C++GAL of INRIA Sophia Antipolis. The LEDA library encompasses a broader range of algorithms. A comparison of the two libraries is provided by Kettner and Näher. Three CGAL User workshops held in 2002, 2004, and 2008 highlighted research results related to CGAL, and many additional papers related to CGAL have appeared in other conferences, workshops, and journals. In 2023 the project won the SoCG Test of Time Award Scope The library covers the following topics: Geometry kernels - basic geometric operations on geometric primitives Arithmetic and algebra Convex hull algorithms Polygons and polyhedra Polygon and polyhedron operations Arrangements Point set triangulations Delaunay triangulations Voronoi diagrams Mesh generation Geometry processing Search structures (k-d tree) Shape analysis, fitting, and distances Interpolation Kinetic data structures Platforms The library is supported on a number of platforms: Microsoft Windows (GNU G++, Microsoft Visual C++, Intel C++ Compiler) GNU g++ (Solaris, Linux, Mac OS) Clang The CGAL library depends on the Boost libraries, and several CGAL packages on the Eigen C++ library. See also OPEN CASCADE OpenSCAD (uses CGAL) References External links CGAL Homepage Geometric algorithms C++ libraries Python (programming language) libraries Free computer libraries Max Planck Institute for Informatics
https://en.wikipedia.org/wiki/Marcel%20L%C3%A9gaut	Marcel Légaut (27 April 1900 – 6 November 1990) was a French Christian philosopher and mathematician. Biography Marcel Légaut was born in Paris, where he received his Ph.D. in Mathematics from the École Normale Supérieure in 1925. He taught in various faculties (among them Rennes and Lyon) until 1943. Under the impact of the Second World War and the rapid French defeat in 1940, Légaut acknowledged the lack of certain fundamental aspects in his life as well as in the lives of other university professors and civil servants. That is why he tried to alternate teaching with farm work. After three years his project was no longer accepted and he left the University to live as a shepherd in the Pré-Alpes (Haut-Diois). Légaut had married in 1940, and between 1945 and 1953 he became the father of six children. Twenty years after this renunciation of his social status (and the consequent rooting in common life), Légaut, almost sixty years old, started a deep and personal reflection around man's condition and existence, which was captured in the two volumes of his main work: Human Accomplishment. Monsieur Portal, his mentor during his youth, had helped him discover that honesty and intellectual independence are essential to a vigorous, spiritual life. This wasn't usual in the Catholic education of the time, where obedience and adherence to the prevailing doctrine took priority. M. Portal had learned something from the modernist crisis period. Encouraged in this secular direction by M. Portal, Légaut ruled out the usual lifestyle in the Catholic system of the time (priesthood and religious life), and instead thoroughly followed the scientific path. As a result of these youthful decisions, but also by the way he lived the full engagement, Légaut became a leader of the Catholic students in the Normale Écoles during the interwar period up to the French defeat. He overcame the temptation of absolutizing his function as celibate leader in the group, and therefore Légaut didn't consider himself as a philosopher, or a theologian of formation or occupation. His only degree was his tenacity and freedom of thought (other names for his responsibility). Writing as an older man, Légaut recovered his dream as a youth: to express spiritual life just as it is, in a sincere and honest way, avoiding utopian speculation. Légaut died in Avignon. Publications Marcel Légaut is the author of twenty books, sixteen of which have been published from 1970. His main work, published in two volumes in 1970 and 1971, is L’accomplissement humain (Human Accomplishment). (Vol. 1: L’homme à la recherche de son humanité; Vol.2: Introduction à l'intelligence du passé et de l'avenir du christianisme. Prior to these, there is a small book called: Travail de la foi (1962). L’homme à la recherche de son humanité was published in English under the title "True humanity" (1982). After 1971: Mutation de l’Église et conversion personnelle (1975), Priéres d’homme (1974, 1978, 1984), Débat sur la f
https://en.wikipedia.org/wiki/List%20of%20metropolitan%20areas%20in%20Japan	This is a list of in Japan by population as defined by the Statistics Bureau of Japan (SBJ) and the Center for Spatial Information Service of the University of Tokyo. The region containing most of the people in Japan between Tokyo and Fukuoka is often called the Taiheiyō Belt. Population Census The Statistics Bureau of Japan (SBJ) defines a metropolitan area as one or more central cities and its associated outlying municipalities. To qualify as an outlying municipality, the municipality must have at least 1.5% of its resident population aged 15 and above commuting to school or work into one of the central cities. To qualify as a central city, a city must either be a designated city of any population or a non-designated city with a city proper population of at least 500,000. Metropolitan areas of designated cities are defined as "major metropolitan areas" (大都市圏) while those of non-designated cities are simply "metropolitan areas" (都市圏). If multiple central cities are close enough such that their outlying cities overlap, they are combined and a single metropolitan area is defined rather than independently. 2015 Population Census The metropolitan areas written in bold are the 11 major metropolitan areas of Japan. 2015 MMA: Major Metropolitan Area MA: Metropolitan Area Source: Statistics Bureau of Japan 2010 Population Census The metropolitan areas written in bold are the 11 major metropolitan areas of Japan. 2010 MMA: Major Metropolitan Area MA: Metropolitan Area Source: Statistics Bureau of Japan Changes from 2005 census The following changes to metropolitan area definitions were made in the 2010 Census report. New central cities in Kantō and Keihanshin major metropolitan areas Sagamihara in the Kantō MMA and Sakai in the Keihanshin MMA have become designated cities in 2010 and 2006 respectively. These cities are already well within their MMAs and should not greatly alter their formation. Niigata and Okayama major metropolitan areas Niigata became a designated city in 2007 and Okayama became a designated city in 2009. These cities therefore formed major metropolitan areas in the 2010 census. Shizuoka, Hamamatsu major metropolitan area Hamamatsu also became a designated city in 2007. As its outlying areas overlap with Shizuoka, the two cities formed a single major metropolitan area in the 2010 census. Utsunomiya metropolitan area Utsunomiya qualified as a central city for the 2010 census, resulting from mergers with neighboring municipalities and subsequent population growth. 2005 Population Census The metropolitan areas written in bold are the 8 major metropolitan areas of Japan. October 1, 2005 MMA: Major Metropolitan Area MA: Metropolitan Area Source: Statistics Bureau of Japan Urban Employment Area Urban Employment Area is another definition of metropolitan areas, defined by the Center for Spatial Information Service, the University of Tokyo. 2015 The Center for Spatial Information Service, the University of Tokyo has defined
https://en.wikipedia.org/wiki/Ukrainian%20Physics%20and%20Mathematics%20Lyceum	The Ukrainian Physics and Mathematics Lyceum (UPML) is a boarding high school and one of the few science magnet schools in Ukraine. It is located in Kyiv and affiliated with Taras Shevchenko National University of Kyiv. History The lyceum was established in 1963 and was known as the Republican Specialized Physics and Mathematics Boarding School until 1992. In total 5,324 students graduated from UPML by 2013. Many of the alumni are winners of National Ukrainian and International Science Olympiads: only between 1963 and 2005 as many as 65 UPML students were awarded medals at International Olympiads in Physics (IPhO), Mathematics (IMO), Chemistry (IChO) and Informatics (IOI), 148 – at all-Soviet Olympiads, 696 – at Ukrainian National Olympiads. The lyceum is currently publicly funded in its entirety and suffers from regular budget shortfalls. Nevertheless, tuition, board and lodging are free of charge for all admitted students. In 2007 UPML became the first Ukrainian school with its name visible from outer space. Alumni Students of the lyceum study from 8th through 11th grades. Due to the organizational affiliation students of the graduating class (11th grade) have the right to be admitted to engineering and natural sciences departments of Taras Shevchenko National University of Kyiv through a preferential admission process. Between 2000 and 2005 all lyceum graduates were admitted to top universities in Ukraine and overseas: about 90% of them went on to study at Taras Shevchenko National University of Kyiv, 6–7% at Moscow Institute of Physics and Technology. A growing number of the lyceum alumni continue their studies overseas: in the United States, Canada, Europe and Asia. Many alumni work in the IT industry and build their careers in the R&D sector. See also Lviv Physics and Mathematics Lyceum Kyiv Natural Science Lyceum No. 145 External links Official website References Schools in Kyiv Educational institutions established in 1963 1963 establishments in Ukraine Secondary schools in Ukraine
https://en.wikipedia.org/wiki/Schur%20functor	In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior powers and symmetric powers of a vector space. Schur functors are indexed by Young diagrams in such a way that the horizontal diagram with n cells corresponds to the nth symmetric power functor, and the vertical diagram with n cells corresponds to the nth exterior power functor. If a vector space V is a representation of a group G, then also has a natural action of G for any Schur functor . Definition Schur functors are indexed by partitions and are described as follows. Let R be a commutative ring, E an R-module and λ a partition of a positive integer n. Let T be a Young tableau of shape λ, thus indexing the factors of the n-fold direct product, E × E × ... × E, with the boxes of T. Consider those maps of R-modules satisfying the following conditions (1) is multilinear, (2) is alternating in the entries indexed by each column of T, (3) satisfies an exchange condition stating that if are numbers from column i of T then where the sum is over n-tuples x' obtained from x by exchanging the elements indexed by I with any elements indexed by the numbers in column (in order). The universal R-module that extends to a mapping of R-modules is the image of E under the Schur functor indexed by λ. For an example of the condition (3) placed on suppose that λ is the partition and the tableau T is numbered such that its entries are 1, 2, 3, 4, 5 when read top-to-bottom (left-to-right). Taking (i.e., the numbers in the second column of T) we have while if then Examples Fix a vector space V over a field of characteristic zero. We identify partitions and the corresponding Young diagrams. The following descriptions hold: For a partition λ = (n) the Schur functor Sλ(V) = Symn(V). For a partition λ = (1, ..., 1) (repeated n times) the Schur functor Sλ(V) = Λn(V). For a partition λ = (2, 1) the Schur functor Sλ(V) is the cokernel of the comultiplication map of exterior powers Λ3(V) → Λ2(V) ⊗ V. For a partition λ = (2, 2) the Schur functor Sλ(V) is the quotient of Λ2(V) ⊗ Λ2(V) by the images of two maps. One is the composition Λ3(V) ⊗ V → Λ2(V) ⊗ V ⊗ V → Λ2(V) ⊗ Λ2(V), where the first map is the comultiplication along the first coordinate. The other map is a comultiplication Λ4(V) → Λ2(V) ⊗ Λ2(V). For a partition λ = (n, 1, ..., 1), with 1 repeated m times, the Schur functor Sλ(V) is the quotient of Λn(V) ⊗ Symm(V) by the image of the composition of the comultiplication in exterior powers and the multiplication in symmetric powers: Applications Let V be a complex vector space of dimension k. It's a tautological representation of its automorphism group GL(V). If λ is a diagram where each row has no more than k cells, then Sλ(V) is an irreducible GL(V)-representation of highest weight λ. In
https://en.wikipedia.org/wiki/Chicago%20Bulls%20all-time%20roster	The following is a list of players, both past and current, who appeared at least in one game for the Chicago Bulls NBA franchise. Players Note: Statistics are correct through the end of the season. A to B \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Loyola Marymount \|\| align="center"\|2 \|\| align="center"\|– \|\| 67 \|\| 958 \|\| 95 \|\| 91 \|\| 296 \|\| 14.3 \|\| 1.4 \|\| 1.4 \|\| 4.4 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|West Virginia \|\| align="center"\|1 \|\| align="center"\| \|\| 8 \|\| 29 \|\| 5 \|\| 2 \|\| 4 \|\| 3.6 \|\| 0.6 \|\| 0.3 \|\| 0.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Arizona \|\| align="center"\|1 \|\| align="center"\| \|\| 10 \|\| 120 \|\| 26 \|\| 13 \|\| 37 \|\| 12.0 \|\| 2.6 \|\| 1.3 \|\| 3.7 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Villanova \|\| align="center"\|2 \|\| align="center"\|– \|\| 114 \|\| 1,339 \|\| 259 \|\| 36 \|\| 508 \|\| 11.7 \|\| 2.3 \|\| 0.3 \|\| 4.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|Wake Forest \|\| align="center"\|1 \|\| align="center"\| \|\| 6 \|\| 67 \|\| 19 \|\| 2 \|\| 9 \|\| 11.2 \|\| 3.2 \|\| 0.3 \|\| 1.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|UNLV \|\| align="center"\|2 \|\| align="center"\|– \|\| 2 \|\| 3 \|\| 1 \|\| 0 \|\| 0 \|\| 1.5 \|\| 0.5 \|\| 0.0 \|\| 0.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|Australia \|\| align="center"\|1 \|\| align="center"\| \|\| 73 \|\| 1,007 \|\| 280 \|\| 65 \|\| 439 \|\| 13.8 \|\| 3.8 \|\| 0.9 \|\| 6.0 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|UNLV \|\| align="center"\|1 \|\| align="center"\| \|\| 36 \|\| 961 \|\| 88 \|\| 203 \|\| 302 \|\| 26.7 \|\| 2.4 \|\| 5.6 \|\| 8.4 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Villanova \|\| align="center"\|4 \|\| align="center"\|– \|\| 207 \|\| 3,645 \|\| 421 \|\| 456 \|\| 989 \|\| 17.6 \|\| 2.0 \|\| 2.2 \|\| 4.8 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F/C \|\| align="left"\|Cincinnati \|\| align="center"\|1 \|\| align="center"\| \|\| 14 \|\| 116 \|\| 32 \|\| 7 \|\| 18 \|\| 8.3 \|\| 2.3 \|\| 0.5 \|\| 1.3 \|\| align=center\| \|- \|align="left" bgcolor="#FFCC00"\|+ \|\| align="center"\|G \|\| align="left"\|Iowa \|\| align="center"\|7 \|\| align="center"\|– \|\| 518 \|\| 13,319 \|\| 948 \|\| 1,741 \|\| 5,553 \|\| 25.7 \|\| 1.8 \|\| 3.4 \|\| 10.7 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Duke \|\| align="center"\|2 \|\| align="center"\|– \|\| 92 \|\| 975 \|\| 88 \|\| 105 \|\| 350 \|\| 10.6 \|\| 1.0 \|\| 1.1 \|\| 3.8 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|F \|\| align="left"\|St. John's \|\| align="center"\|3 \|\| align="center"\|– \|\| 175 \|\| 5,424 \|\| 733 \|\| 507 \|\| 2,194 \|\| 31.0 \|\| 4.2 \|\| 2.9 \|\| 12.5 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C \|\| align="left"\|Turkey \|\| align="center"\|3 \|\| align="center"\|– \|\| 152 \|\| 2,021 \|\| 666 \|\| 65 \|\| 438 \|\| 13.3 \|\| 4.4 \|\| 0.4 \|\| 2.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|G \|\| align="left"\|Texas \|\| align="center"\|1 \|\| align="center"\| \|\| 61 \|\| 1,857 \|\| 126 \|\| 303 \|\| 909 \|\| 30.4 \|\| 2.1 \|\| 5.0 \|\| 14.9 \|\| align=center\| \|- \|align="left"\| \|\| align="center"\|C
https://en.wikipedia.org/wiki/Units%20of%20measurement%20in%20transportation	The units of measurement in transportation describes the unit of measurement used to express various transportation quantities, as used in statistics, planning, and their related applications. Transportation quantity The currently popular units are: Length of journey kilometre (km) or kilometer is a metric unit used, outside the US, to measure the length of a journey; the international statute mile (mi) is used in the US; 1 mi = 1.609344 km nautical mile is rarely used to derive units of transportation quantity. Traffic flow vehicle-kilometre (vkm) as a measure of traffic flow, determined by multiplying the number of vehicles on a given road or traffic network by the average length of their trips measured in kilometres. vehicle-mile (, or VMT) same as before but measures the trip expressed in miles. Passenger Payload quantity Passenger; Person (often abbreviated as either "pax" or "p.") Passenger-distance Passenger-distance is the distance (km or miles) travelled by passengers on transit vehicles; determined by multiplying the number of unlinked passenger trips by the average length of their trips. passenger-kilometre or pkm internationally; passenger-mile (or pmi ?) sometimes in the US; 1 pmi = 1.609344 pkm Passengers per hour per direction Passengers per hour per direction (pphpd) measures the maximum route capacity of a transport system. Passengers per bus hour A system may carry a high number of passengers per distance (km or mile) but a relatively low number of passengers per bus hour if vehicles operate in congested areas and thus travel at slower speed. Passengers per bus distance A transit system serving a community with a widely dispersed population must operate circuitous routes that tend to carry fewer passengers per distance (km or mile). A higher number is more favorable. Freight Freight is measured in mass-distance. A simple unit of freight is the kilogram-kilometre (kgkm), the service of moving one kilogram of payload a distance of one kilometre. Payload quantity kilogram (kg), the standard SI unit of mass. tonne (t), a non-SI but an accepted metric unit, defined as 1,000 kilograms. "short ton" is used in the US; 1 short ton = 2,000 pounds = 0.907 tonnes. 1 t = (1/0.907) short tons = 1.102 short tons. Payload-distance kilogram-kilometre (kg⋅km), moving 1 kg of cargo a distance of 1 km; tonne-kilometre or kilometre-tonne (t⋅km or km⋅t, also tkm or kmt), the transportation of one tonne over one kilometre; 1 tkm = 1,000 kgkm. ton-mile in the US: 1 ton-mile * ( 0.907185 t / short ton) * ( 1.609344 km / mile ) = 1.460 tkm Usage The metric units (pkm and tkm) are used internationally. (In aviation where United States customary units are widely used, the International Air Transport Association (IATA) releases its statistics in the metric units.) In the US, sometimes United States customary units are used. Derivation The dimension of the measure is the product of the payload mass and the distance tr
https://en.wikipedia.org/wiki/Sirens%20%28Dizzee%20Rascal%20song%29	"Sirens" is the seventh single release from British rapper Dizzee Rascal, and the lead single from his third studio album Maths + English. The single was playlisted on BBC Radio 1's 1-Upfront list and the music video for the single made the top thirty of that chart. Despite mediocre amounts of airplay on both television and radio, the single was a success for Dizzee Rascal, returning him to the top twenty of the UK Singles Chart after previous single "Off 2 Work" / "Graftin'" was his first to miss the top forty. The song became his fourth top twenty hit and was his first single to be released on 7" vinyl. This song features a sample from the song "Here to Stay" by Korn. Jorja Smith samples the chorus in her song "Blue Lights". Track listing CD "Sirens" "Dean" 7" Vinyl "Sirens" "Like Me" 12" Vinyl "Sirens" "Sirens" (a cappella) "Sirens" (Chase & Status remix) "Sirens" (Chase & Status remix instrumental) Music video The video premiered in 2007. It shows him being chased by foxhunters on horses chasing him through his house, then onto the streets and towards the end of the video, he is trapped down an alleyway and is seemingly killed, and his blood is smeared on the hunters' faces. This suggests that Dizzee is effectively game for the toffs, where 'game' represents the lower classes, deliberately held down by the rich, while the police are just one of the riches tools to do so. Charts References 2007 singles Dizzee Rascal songs XL Recordings singles Songs written by Dizzee Rascal 2007 songs
https://en.wikipedia.org/wiki/Product%20integral	A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral (Type I below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the geometric integral (Type II below), the bigeometric integral (Type III below), and some other integrals of non-Newtonian calculus. Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The geometric integral, together with the geometric derivative, is useful in image analysis and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay). The bigeometric integral, together with the bigeometric derivative, is useful in some applications of fractals, and in the theory of elasticity in economics. This article adopts the "product" notation for product integration instead of the "integral" (usually modified by a superimposed "times" symbol or letter P) favoured by Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field. Basic definitions The classical Riemann integral of a function can be defined by the relation where the limit is taken over all partitions of the interval whose norms approach zero. Roughly speaking, product integrals are similar, but take the limit of a product instead of the limit of a sum. They can be thought of as "continuous" versions of "discrete" products. The most popular product integrals are the following: Type I: Volterra integral The type I product integral corresponds to Volterra's original definition. The following relationship exists for scalar functions : which is not a multiplicative operator. (So the concepts of product integral and multiplicative integral are not the same). The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra, where the last equality is no longer true (see the references below). When applied to scalars belonging to a non-commutative field, to matrixes, and to operators, i.e. to mathematical objects that don't commute, the Volterra integral splits in two definitions Left Product integral With the notation of left products (i.e. normal products applied from left) Right Product Integral With the notation of right products (i.e. applied from right) Where is the identity matrix and D is a partition of the interval [a,b] in the Riemann sense, i.e. the limit is over the maximum interval in the partition. Note how in this case time ordering comes evident in the definitions. For scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus. Type II: geometric in
https://en.wikipedia.org/wiki/Isaak%20Yaglom	Isaak Moiseevich Yaglom (; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom. Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan. As the author of several books, translated into English, that have become academic standards of reference, he has an international stature. His attention to the necessities of learning (pedagogy) make his books pleasing experiences for students. The seven authors of his Russian obituary recount "…the breadth of his interests was truly extraordinary: he was seriously interested in history and philosophy, passionately loved and had a good knowledge of literature and art, often came forward with reports and lectures on the most diverse topics (for example, on Alexander Blok, Anna Akhmatova, and the Dutch painter M. C. Escher), actively took part in the work of the cinema club in Yaroslavl and the music club at the House of Composers in Moscow, and was a continual participant of conferences on mathematical linguistics and on semiotics." University life Yaglom started his higher education at Moscow State University in 1938. During World War II he volunteered, but due to myopia he was deferred from military service. In the evacuation of Moscow he went with his family to Sverdlovsk in the Ural Mountains. He studied at Sverdlovsk State University, graduated in 1942, and when the usual Moscow faculty assembled in Sverdlovsk during the war, he took up graduate study. Under the geometer Veniamin Kagan he developed his Ph.D. thesis which he defended in Moscow in 1945. It is reported that this thesis "was devoted to projective metrics on a plane and their connections with different types of complex numbers (where , or , or else )." Institutes and titles During his career, Yaglom was affiliated with these institutions: Moscow Energy Institute (1946) – lecturer in mathematics Moscow State University (1946 – 49) – lecturer, department of analysis and differential geometry Orekhovo-Zuevo Pedagogical Institute (1949–56) – lecturer in mathematics Lenin State Pedagogical Institute (Moscow) (1956–68) – obtained D.Sc. 1965 Moscow Evening Metallurgical Institute (1968–74) – professor of mathematics Yaroslavl State University (1974–83) – professor of mathematics Academy of Pedagogical Sciences (1984–88) – technical consultant Affine geometry In 1962 Yaglom and Vladimir G. Ashkinuse published Ideas and Methods of Affine and Projective Geometry, in Russian. The text is limited to affine geometry since projective geometry was put off to a second volume that did not appear. The concept of hyperbolic angle is developed through area of hyperbolic sectors. A treatment of Routh's theorem is given at page 193. This textbook, published by the Ministry of Education, includes 234 exercises with hints and solutions in an appendix. English translations Isaac Yaglom wrote over 40 books and many articles. Several were transla
https://en.wikipedia.org/wiki/2003%20Cricket%20World%20Cup%20statistics	2003 Cricket World Cup statistics lists all the major statistics and records for the 2003 Cricket World Cup held in South Africa, Zimbabwe and Kenya from 9 February to 24 March 2003. Talha Jubair became the youngest player to participate in Cricket World Cup. Sri Lanka's clinical demolition of Canada for 36 runs created a new World Cup record for the lowest innings score, a dubious distinction that was, at the time, the lowest score in ODI history. Records tumbled when defending champions Australia took on minnows Namibia, with Glenn McGrath claiming the World Cup's best bowling figures (7/15), a performance that helped Australia defeat Namibia by 256 runs. Team-mate Adam Gilchrist created a new wicket-keeping dismissal record in the same match, with 6. Against Namibia, Indian players Sachin Tendulkar and Sourav Ganguly recorded the second highest partnership in World Cup cricket (244 runs). India and Australia clashed in a one-sided battle in the final, with Australia creating multiple records (highest World Cup final score, highest score by a captain in a World Cup final – Ricky Ponting, most sixes by a batsman – Ponting) in a match; with Australia winning by 125 runs. Tendulkar's 673 runs, the most runs scored in a single World Cup history to date, was the consolation for India as he won the 2003 Cricket World Cup Man of the Series award. The World Cup also saw fielding records in an innings (Mohammad Kaif) and tournament (Ponting). The World Cup broke the record for most sixes in the tournament (with 266), but this was easily surpassed in the 2007 edition (with 373). Records Team totals Highest team totals The highest score of the 2003 Cricket World Cup came in the finals when Australia scored 359 runs against India in 50 overs. This represents the highest score made in the finals of Cricket World Cup. Note: Only scores of 310 or higher are listed. Lowest team totals Canada were bowled out for the lowest ever total in World Cup history against Sri Lanka; which was also, at the time, the lowest ever total in ODI history. Note: Only scores of 100 or lower are listed. Bowling Most wickets in the tournament Vaas's haul of 23 wickets in the tournament was, at the time, the record in World Cup history. His record was equalled or bettered by three bowlers in the 2007 edition of the World Cup (Glenn McGrath, Muttiah Muralitharan and Shaun Tait). Note: Only top 10 players shown. Sorted by wickets then bowling average. Best bowling Note: Only top ten performances listed. Batting Most runs in the tournament The 2003 Cricket World Cup had four cricketers scoring over 400 runs in the tournament (two Indians and two Australians), a record that has been bettered when ten cricketers scored more than 400 runs in the 2007 Cricket World Cup. Sachin's 673 runs in the 2003 Cricket World Cup is the current record for most runs scored in a single edition in World Cup history. Note: Only top 10 players shown. Sorted by total. Highest individual scores
https://en.wikipedia.org/wiki/Euclidean%20relation	In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other." Definition A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c. To write this in predicate logic: Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b is related to c: Properties Due to the commutativity of ∧ in the definition's antecedent, aRb ∧ aRc even implies bRc ∧ cRb when R is right Euclidean. Similarly, bRa ∧ cRa implies bRc ∧ cRb when R is left Euclidean. The property of being Euclidean is different from transitivity. For example, ≤ is transitive, but not right Euclidean, while xRy defined by 0 ≤ x ≤ y + 1 ≤ 2 is not transitive, but right Euclidean on natural numbers. For symmetric relations, transitivity, right Euclideanness, and left Euclideanness all coincide. However, a non-symmetric relation can also be both transitive and right Euclidean, for example, xRy defined by y=0. A relation that is both right Euclidean and reflexive is also symmetric and therefore an equivalence relation. Similarly, each left Euclidean and reflexive relation is an equivalence. The range of a right Euclidean relation is always a subset of its domain. The restriction of a right Euclidean relation to its range is always reflexive, and therefore an equivalence. Similarly, the domain of a left Euclidean relation is a subset of its range, and the restriction of a left Euclidean relation to its domain is an equivalence. Therefore, a right Euclidean relation on X that is also right total (respectively a left Euclidean relation on X that is also left total) is an equivalence, since its range (respectively its domain) is X. A relation R is both left and right Euclidean, if, and only if, the domain and the range set of R agree, and R is an equivalence relation on that set. A right Euclidean relation is always quasitransitive, as is a left Euclidean relation. A connected right Euclidean relation is always transitive; and so is a connected left Euclidean relation. If X has at least 3 elements, a connected right Euclidean relation R on X cannot be antisymmetric, and neither can a connected left Euclidean relation on X. On the 2-element set X = { 0, 1 }, e.g. the relation xRy defined by y=1 is connected, right Euclidean, and antisymmetric, and xRy defined by x=1 is connected, left Euclidean, and antisymmetric. A relation R on a set X is right Euclidean if, and only if, the restriction R := Rran(R) is an equivalence and for each x in X\ran(R), all elements to which x is related under R are equivalent under R. Similarly, R on X is left Euclidean if, and only if, R := Rdom(R) is an equivalence and for each x in X\dom(R), all elements that are related to x under R are equivalent under
https://en.wikipedia.org/wiki/Unfold	Unfold may refer to: Science Unfoldable cardinal, in mathematics Unfold (higher-order function), in computer science a family of anamorphism functions Unfoldment (disambiguation), in spirituality and physics Unfolded protein response, in biochemistry Equilibrium unfolding, in biochemistry Unfolded state (denatured protein), in biochemistry Maximum variance unfolding (semidefinite embedding), in computer science Music Unfold (Marié Digby album), 2008 Unfold (John O'Callaghan album), 2011 Unfold (The Necks album), 2017 "Unfold" (Porter Robinson song), 2021 "Unfold", a song by De La Soul from the 2016 album And the Anonymous Nobody... See also Fold (disambiguation)
https://en.wikipedia.org/wiki/Burt%20Totaro	Burt James Totaro, FRS (b. 1967), is an American mathematician, currently a professor at the University of California, Los Angeles, specializing in algebraic geometry and algebraic topology. Education and early life Totaro participated in the Study of Mathematically Precocious Youth while in grade school and enrolled at Princeton University at the age of thirteen, becoming the youngest freshman in its history. He scored a perfect 800 on the math portion and a 690 on the verbal portion of the SAT-I exam at the age of 12. He graduated in 1984 and went on to graduate school at the University of California, Berkeley, receiving his Ph.D. in 1989. Career and research Since 2009, he has been one of three managing editors of the journal Compositio Mathematica; he is also on the editorial boards of Forum of Mathematics, Pi and Sigma, the Journal of the American Mathematical Society, and the Bulletin of the American Mathematical Society. In 2012, he became a Professor in the UCLA Department of Mathematics. Totaro's work is influenced by the Hodge conjecture, and is based on the connections and application of topology to algebraic geometry. His work has applications in a number of diverse areas of mathematics, from representation theory to Lie theory and group cohomology. Selected works Recognition In 2000, he was elected Lowndean Professor of Astronomy and Geometry at the University of Cambridge. In the same year, he was awarded the Whitehead Prize by the London Mathematical Society. In 2009, Totaro was elected Fellow of the Royal Society. He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to algebraic geometry, Lie theory and cohomology and their connections and for service to the profession". References 20th-century American mathematicians 21st-century American mathematicians Algebraic geometers Fellows of the Royal Society Fellows of the American Mathematical Society Whitehead Prize winners University of California, Los Angeles faculty Cambridge mathematicians Lowndean Professors of Astronomy and Geometry Princeton University alumni University of California, Berkeley alumni Living people 1967 births Topologists
https://en.wikipedia.org/wiki/Profinite	In mathematics, the term profinite is used for profinite groups, topological groups profinite sets, also known as "profinite spaces" or "Stone spaces"
https://en.wikipedia.org/wiki/Almost%20symplectic%20manifold	In differential geometry, an almost symplectic structure on a differentiable manifold is a two-form on that is everywhere non-singular. If in addition is closed then it is a symplectic form. An almost symplectic manifold is an Sp-structure; requiring to be closed is an integrability condition. References Further reading Smooth manifolds Symplectic geometry
https://en.wikipedia.org/wiki/Torsion-free%20abelian%20group	In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case. Definitions An abelian group is said to be torsion-free if no element other than the identity is of finite order. Explicitly, for any , the only element for which is . A natural example of a torsion-free group is , as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group is torsion-free for any . An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a . A non-finitely generated countable example is given by the additive group of the polynomial ring (the free abelian group of countable rank). More complicated examples are the additive group of the rational field , or its subgroups such as (rational numbers whose denominator is a power of ). Yet more involved examples are given by groups of higher rank. Groups of rank 1 Rank The rank of an abelian group is the dimension of the -vector space . Equivalently it is the maximal cardinality of a linearly independent (over ) subset of . If is torsion-free then it injects into . Thus, torsion-free abelian groups of rank 1 are exactly subgroups of the additive group . Classification Torsion-free abelian groups of rank 1 have been completely classified. To do so one associates to a group a subset of the prime numbers, as follows: pick any , for a prime we say that if and only if for every . This does not depend on the choice of since for another there exists such that . Baer proved that is a complete isomorphism invariant for rank-1 torsion free abelian groups. Classification problem in general The hardness of a classification problem for a certain type of structures on a countable set can be quantified using model theory and descriptive set theory. In this sense it has been proved that the classification problem for countable torsion-free abelian groups is as hard as possible. Notes References . . Algebraic structures Abelian group theory Properties of groups
https://en.wikipedia.org/wiki/Flat%20neighborhood%20network	Flat Neighborhood Network (FNN) is a topology for distributed computing and other computer networks. Each node connects to two or more switches which, ideally, entirely cover the node collection, so that each node can connect to any other node in two "hops" (jump up to one switch and down to the other node). This contrasts to topologies with fewer cables per node which communicate with remote nodes via intermediate nodes, as in Hypercube (see The Connection Machine). See also Thinking Machines Corporation built the Connection Machine employing hypercube topology for its compute nodes. Kentucky's Linux/Athlon Testbed KLAT2 is an archetypal implementation. External links The Aggregate (at the University of Kentucky) defines FNN and includes a bibliography. Supercomputers
https://en.wikipedia.org/wiki/Locality-sensitive%20hashing	In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. (The number of buckets is much smaller than the universe of possible input items.) Since similar items end up in the same buckets, this technique can be used for data clustering and nearest neighbor search. It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items. Hashing-based approximate nearest-neighbor search algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as locality-preserving hashing (LPH). Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion. Definitions An LSH family is defined for a metric space , a threshold , an approximation factor , and probabilities and . This family is a set of functions that map elements of the metric space to buckets . An LSH family must satisfy the following conditions for any two points and any hash function chosen uniformly at random from : if , then (i.e., and collide) with probability at least , if , then with probability at most . A family is interesting when . Such a family is called -sensitive. Alternatively it is defined with respect to a universe of items that have a similarity function . An LSH scheme is a family of hash functions coupled with a probability distribution over the functions such that a function chosen according to satisfies the property that for any . Locality-preserving hashing A locality-preserving hash is a hash function that maps points in a metric space to a scalar value such that for any three points . In other words, these are hash functions where the relative distance between the input values is preserved in the relative distance between the output hash values; input values that are closer to each other will produce output hash values that are closer to each other. This is in contrast to cryptographic hash functions and checksums, which are designed to have random output difference between adjacent inputs. Locality preserving hashes are related to space-filling curves. Amplification Given a -sensitive family , we can construct new families by either the AND-construction or OR-construction of . To create an AND-construction, we define a new family of hash functions , where each function is constructed from random functions from . We then say that for a hash function , if and only if all
https://en.wikipedia.org/wiki/Sequoia%20Hall	Sequoia Hall is the home of the Statistics Department on the campus of Stanford University in Stanford, California. History In 1891, the original building opened as Roble Hall, a three-story women's dormitory. Roble Hall housed the first women admitted to Stanford. In 1917, a new women's dormitory also called Roble Hall was constructed on another part of campus and the earlier building was renamed Sequoia Hall and renovated as a men's dormitory. During World War I, Sequoia Hall was used by the Army for officers attending the War Department civilian defense school. In the 1930s and 1940s, Sequoia Hall fell into disrepair and was vacant by 1945. In 1957, the building was deemed an earthquake hazard. The top two stories of the building were demolished and the bottom floor was renovated. The renovated building became home to the Statistics Department. In the late 1980s, Stanford University began planning a $120 million Science and Engineering Quad (SEQ) Project, scheduled to be completed by 1999. Part of this project included the construction of a new building for Statistics. On August 22, 1996, the original Sequoia Hall was demolished to make way for the new facility. The new Sequoia Hall opened January 17, 1998 on an adjacent site. The facility is current home to the Statistics Department. Further reading External links Official website of the Statistics Department at Stanford University Residential buildings completed in 1917 School buildings completed in 1998 Stanford University buildings and structures
https://en.wikipedia.org/wiki/Hyperparameter	In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. For example, if one is using a beta distribution to model the distribution of the parameter p of a Bernoulli distribution, then: p is a parameter of the underlying system (Bernoulli distribution), and α and β are parameters of the prior distribution (beta distribution), hence hyperparameters. One may take a single value for a given hyperparameter, or one can iterate and take a probability distribution on the hyperparameter itself, called a hyperprior. Purpose One often uses a prior which comes from a parametric family of probability distributions – this is done partly for explicitness (so one can write down a distribution, and choose the form by varying the hyperparameter, rather than trying to produce an arbitrary function), and partly so that one can vary the hyperparameter, particularly in the method of conjugate priors, or for sensitivity analysis. Conjugate priors When using a conjugate prior, the posterior distribution will be from the same family, but will have different hyperparameters, which reflect the added information from the data: in subjective terms, one's beliefs have been updated. For a general prior distribution, this is computationally very involved, and the posterior may have an unusual or hard to describe form, but with a conjugate prior, there is generally a simple formula relating the values of the hyperparameters of the posterior to those of the prior, and thus the computation of the posterior distribution is very easy. Sensitivity analysis A key concern of users of Bayesian statistics, and criticism by critics, is the dependence of the posterior distribution on one's prior. Hyperparameters address this by allowing one to easily vary them and see how the posterior distribution (and various statistics of it, such as credible intervals) vary: one can see how sensitive one's conclusions are to one's prior assumptions, and the process is called sensitivity analysis. Similarly, one may use a prior distribution with a range for a hyperparameter, perhaps reflecting uncertainty in the correct prior to take, and reflect this in a range for final uncertainty. Hyperpriors Instead of using a single value for a given hyperparameter, one can instead consider a probability distribution of the hyperparameter itself; this is called a "hyperprior." In principle, one may iterate this, calling parameters of a hyperprior "hyperhyperparameters," and so forth. See also Empirical Bayes method References Further reading Bayesian statistics Sensitivity analysis
https://en.wikipedia.org/wiki/Chinese%20people%20in%20Italy	The community of Chinese people in Italy has grown rapidly in the past ten years. Official statistics indicate there are at least 330,495 Chinese citizens in Italy, although these figures do not account for former Chinese citizens who have acquired Italian nationality or Italian-born people of Chinese descent. Demographics Prato, Tuscany has the largest concentration of Chinese people in Italy and all of Europe. It has the second largest population of Chinese people overall in Italy after Milan. Religion In total, approximately one quarter of the Chinese community was classified as belonging to the Chinese (folk) religion. The surveyors weren't able to determine a precise Taoist identity; only 1.1% of the surveyed people identified as such, and the analysts preferred to consider Taoism as an "affluent" of the Chinese religion. The survey found that 39.9% of the Chinese had a thoroughly atheist identity, not believing in any god, nor belonging to any religious organisation, nor practicing any religious activity. The study also analysed the Chinese Christian community, finding it comprised 8% of the total population (of which 3.6% were Catholics, 3.3% Protestants and 1.1% Jehovah's Witnesses). The Christian community was small, but larger than that of the province of origin, especially for the Catholics and the Jehovah's Witnesses, the latter being an illegal religion in China. Protestants were found to be basically nondenominational and largely (70%) women. In the years 2011 and 2012 the ISTAT made a survey regarding the religious affiliation among the immigrants in Italy, the religion of the Chinese people in Italy were as follows: Non religious: 44.5% Buddhists: 44.4% Christians: 7.3% Other religions: 3.8% Community relations In 2007, several dozen protesters took to the streets in Milan over alleged discrimination. The northern Italian town of Treviso also ordered Chinese-run businesses to take down their lanterns because they looked "too oriental". Cities with significant Chinese communities Based on Demo ISTAT statistics. Milan 18,918 (1.43% on total resident population) Rome 12,013 Prato 11,882 (6.32%) Turin 5,437 Florence 3,890 (1.05%) Campi Bisenzio 3,018 (6.87%) Reggio Emilia 2,925 (1.72%) Bologna 2,654 Naples 2,456 Brescia 2,394 (1.23%) Venice 2,163 Empoli 1,759 (3.67%) Genoa 1,637 Forlì 1,607 (1.36%) Padua 1,571 Fucecchio 1,502 (6.39%) The city of Prato has the second largest Chinese immigrant population in Italy (after Milan with Italy's largest Chinatown). Legal Chinese residents in Prato on 31 December 2008 were 9,927. Local authorities estimate the number of Chinese citizens living in Prato to be around 45,000, illegal immigrants included. Most overseas Chinese come from the city of Wenzhou in the province of Zhejiang, some of them having moved from the Chinatown in Paris. In 2021 there were 33871 (2,466%) Chinese in Milan and 33649 (16,764%) in Prato Notable people References Further reading (Archive) Luigi Berz
https://en.wikipedia.org/wiki/Cadabra	Cadabra might refer to: Cadabra (computer program), a computer algebra system for field theory problems Cadabra Design Automation, a former EDA company purchased by Numerical Technologies Cadabra, Inc., now Amazon.com, Inc. See also Abracadabra (disambiguation) Kadabra (disambiguation)
https://en.wikipedia.org/wiki/Spacetime%20diagram	A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity. Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations. The history of an object's location through time traces out a line or curve on a spacetime diagram, referred to as the object's world line. Each point in a spacetime diagram represents a unique position in space and time and is referred to as an event. The most well-known class of spacetime diagrams are known as Minkowski diagrams, developed by Hermann Minkowski in 1908. Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes. Introduction to kinetic diagrams Position versus time graphs In the study of 1-dimensional kinematics, position vs. time graphs (called x-t graphs for short) provide a useful means to describe motion. Kinematic features besides the object's position are visible by the slope and shape of the lines. In Fig 1-1, the plotted object moves away from the origin at a positive constant velocity (1.66 m/s) for 6 seconds, halts for 5 seconds, then returns to the origin over a period of 7 seconds at a non-constant speed (but negative velocity). At its most basic level, a spacetime diagram is merely a time vs position graph, with the directions of the axes in a usual p-t graph exchanged; that is, the vertical axis refers to temporal and the horizontal axis to spatial coordinate values. Especially when used in special relativity (SR), the temporal axes of a spacetime diagram are often scaled with the speed of light , and thus are often labeled by This changes the dimension of the addressed physical quantity from <Time> to <Length>, in accordance with the dimension associated with the spatial axis, which is frequently labeled Standard configuration of reference frames To ease insight into how spacetime coordinates, measured by observers in different reference frames, compare with each other, it is useful to standardize and simplify the setup. Two Galilean reference frames (i.e., conventional 3-space frames), S and S′ (pronounced "S prime"), each with observers O and O′ at rest in their respective frames, but measuring the other as moving with speeds ±v are said to be in standard configuration, when: The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′. The origins of frames S and S′ coincide at time t = 0 in frame S and also at t′ = 0 in frame S′. Frame S′ moves in the x-direction of frame S with velocity v as measured in frame S. T
https://en.wikipedia.org/wiki/Daniel%20Rajna	Daniel Rajna born 1968, is a South African ballet dancer. After gaining a BSc in applied mathematics at UCT, he trained at the UCT Ballet school, Cape Town. He joined the former CAPAB Ballet in 1990, before leaving in 1997 to join PACT Ballet in Pretoria. He returned to Cape Town in 1999 and was a principal dancer at the Cape Town City Ballet. He is known for his interpretation of dramatic ballets and his partnership with friend, Tracy Li. He and his wife Leanne Voysey, a former principal dancer with Cape Town City Ballet, have a son Finn (born July 2007). He is the son of composer Thomas Rajna. He has performed as a guest artist in Hong Kong, Zimbabwe, The United States, South Africa and Taiwan. They were also both invited to the 2004 International Ballet Festival of Miami. He retired in August 2007 after several performances of Camille. After retirement Rajna made a radical career change and after studying for three years, joined a civil consultancy as a dam designer. He continued his association with ballet and was asked back on occasion to coach members of the ballet company. In 2015 Rajna and Li were invited to come out of retirement to give two performances of Veronica Paeper's ballet, "Carmen" accompanied by the Cape Town Philharmonic Orchestra. Awards FNB Vita Award 1999 Balletomanes Award for best male dancer, 1996, 2000, 2002 and 2006 Daphne Levy Award for his partnership with Tracy Li, 2001 Notable roles Pluto in Orpheus in the Underworld Albrecht in Giselle Basilio in Don Quixote Crassus in Spartacus Armand in Camille Don Jose in Carmen Puck in A Midsummer Night's Dream James in La Sylphide Romeo in Romeo and Juliet Nutcracker Prince in The Nutcracker Florimund in The Sleeping Beauty Prince Siegfried in Swan Lake References External links Daniel Rajna at the Dance Directory Cape Town City Ballet Company Pas de Deux Couple Tracy Li & Daniel Rajna Dancers exit stage for new future Tracy Li and Daniel Rajna to reunite for Cape Town City Ballet Carmen gets retired dancers back on stage Final Pas de deux from the ballet "Camille" 1968 births Living people South African male ballet dancers South African ballet dancers Alumni of Rondebosch Boys' High School University of Cape Town alumni
https://en.wikipedia.org/wiki/Prevalent%20and%20shy%20sets	In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces. The term "shy" was suggested by the American mathematician John Milnor. Definitions Prevalence and shyness Let be a real topological vector space and let be a Borel-measurable subset of is said to be prevalent if there exists a finite-dimensional subspace of called the probe set, such that for all we have for -almost all where denotes the -dimensional Lebesgue measure on Put another way, for every Lebesgue-almost every point of the hyperplane lies in A non-Borel subset of is said to be prevalent if it contains a prevalent Borel subset. A Borel subset of is said to be shy if its complement is prevalent; a non-Borel subset of is said to be shy if it is contained within a shy Borel subset. An alternative, and slightly more general, definition is to define a set to be shy if there exists a transverse measure for (other than the trivial measure). Local prevalence and shyness A subset of is said to be locally shy if every point has a neighbourhood whose intersection with is a shy set. is said to be locally prevalent if its complement is locally shy. Theorems involving prevalence and shyness If is shy, then so is every subset of and every translate of Every shy Borel set admits a transverse measure that is finite and has compact support. Furthermore, this measure can be chosen so that its support has arbitrarily small diameter. Any finite or countable union of shy sets is also shy. Analogously, countable intersection of prevalent sets is prevalent. Any shy set is also locally shy. If is a separable space, then every locally shy subset of is also shy. A subset of -dimensional Euclidean space is shy if and only if it has Lebesgue measure zero. Any prevalent subset of is dense in If is infinite-dimensional, then every compact subset of is shy. In the following, "almost every" is taken to mean that the stated property holds of a prevalent subset of the space in question. Almost every continuous function from the interval into the real line is nowhere differentiable; here the space is with the topology induced by the supremum norm. Almost every function in the space has the property that Clearly, the same property holds for the spaces of -times differentiable functions For almost every sequence has the property that the series diverges. Prevalence version of the Whitney embedding theorem: Let be a compact manifold of class and dimension contained in For almost every function is an embedding of If is a compact subset of with Hausdorff dimension and then, for almost every function also has Hausdorff dimension For almost every function has the property that all of its periodic points are hyperbolic. In
https://en.wikipedia.org/wiki/Donald%20J.%20Wheeler	Donald J. Wheeler is an American author, statistician and expert in quality control. Wheeler graduated from the University of Texas in 1966 and holds M.S. and Ph.D. degrees in statistics from Southern Methodist University. From 1970 to 1982 he taught in the Statistics Department at the University of Tennessee, where he was an associate professor. Since 1982 he has worked as a consultant. He is the author of 22 textbooks. His books have been translated into five languages and are in use in over 40 countries. He has been invited to contribute to two state-of-the-art anthologies, and has had articles published in 16 refereed journals. He is a fellow of both the American Statistical Association and American Society for Quality. He was awarded the 2010 Deming Medal by the American Society for Quality. Wheeler has been a monthly columnist for both Quality Digest and Quality magazine. He has conducted over 1000 seminars for over 250 companies and organizations in 17 countries on five continents, and has had students come from 30 countries to attend his seminars in the United States. References External links http://www.spcpress.com/ American statisticians Southern Methodist University alumni Fellows of the American Statistical Association Living people Year of birth missing (living people)
https://en.wikipedia.org/wiki/Transverse%20measure	In mathematics, a measure on a real vector space is said to be transverse to a given set if it assigns measure zero to every translate of that set, while assigning finite and positive (i.e. non-zero) measure to some compact set. Definition Let V be a real vector space together with a metric space structure with respect to which it is complete. A Borel measure μ is said to be transverse to a Borel-measurable subset S of V if there exists a compact subset K of V with 0 < μ(K) < +∞; and μ(v + S) = 0 for all v ∈ V, where is the translate of S by v. The first requirement ensures that, for example, the trivial measure is not considered to be a transverse measure. Example As an example, take V to be the Euclidean plane R2 with its usual Euclidean norm/metric structure. Define a measure μ on R2 by setting μ(E) to be the one-dimensional Lebesgue measure of the intersection of E with the first coordinate axis: An example of a compact set K with positive and finite μ-measure is K = B1(0), the closed unit ball about the origin, which has μ(K) = 2. Now take the set S to be the second coordinate axis. Any translate (v1, v2) + S of S will meet the first coordinate axis in precisely one point, (v1, 0). Since a single point has Lebesgue measure zero, μ((v1, v2) + S) = 0, and so μ is transverse to S. See also Prevalent and shy sets References Measures (measure theory)
https://en.wikipedia.org/wiki/Geoff%20Bascand	Geoff Bascand was the Deputy Governor and Head of Operations at the Reserve Bank of New Zealand. He was Government Statistician and the Chief Executive of Statistics New Zealand until May 2013. Bascand is a graduate of the University of Otago and the Australian National University with a BA (Honours) degree in Geography and a master's degree in Economics. Career Bascand has worked for the New Zealand Treasury, the International Monetary Fund in Washington, and the New Zealand Department of Labour. He was appointed one of three Deputy Government Statisticians for Statistics New Zealand in July 2004 and was responsible for Macro-Economic, Environment, Regional and Geography Statistics. He was appointed Government Statistician and Chief Executive of Statistics New Zealand on 22 May 2007. He started his career in 1981 at the Treasury as an economic analyst and later became Director of Forecasting. From 1998 until 2004, Bascand was the General Manager of the Labour Market Policy Group at the Department of Labour. As well as holding senior policy and management positions at the Treasury and the Department of Labour, Bascand has been a Research Fellow at the Centre of Policy Studies at Monash University in Australia, and from 1996 until 1997 he was a staff economist at the International Monetary Fund in Washington DC. In February 2005, he was a recipient of a Leadership Development Centre Fellowship award. On 12 February 2013 Bascand announced his resignation at Statistics New Zealand, finishing there on 24 May 2013. He has accepted a position at the Reserve Bank as Deputy Governor and Head of Operations. References Australian National University alumni Living people New Zealand mathematicians Government Statisticians of New Zealand University of Otago alumni New Zealand statisticians 21st-century New Zealand public servants Year of birth missing (living people) People educated at Taieri College
https://en.wikipedia.org/wiki/Graeme%20Ruxton	Graeme Ruxton is a zoologist known for his research into behavioural ecology and evolutionary ecology. Life and work Ruxton received his PhD in Statistics and Modelling Science in 1992 from the University of Strathclyde. His studies focus on the evolutionary pressures on aggregation by animals, and predator-prey aspects of sensory ecology. He researched visual communication in animals at the University of Glasgow, where he was professor of theoretical ecology. In 2013 he became professor at the University of St Andrews, Scotland. Publications Ruxton has published numerous papers on antipredator adaptations, along with contributions to textbooks. His book Living in Groups has been cited over 2300 times. His textbook Avoiding Attack. The Evolutionary Ecology of Crypsis, Warning Signals and Mimicry has been cited over 1150 times. His paper "Collective memory and spatial sorting in animal groups" has been cited over 1300 times, while his paper on the use of statistics in behavioural ecology, "The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test", has been cited over 850 times. Honours and awards In 2012 Ruxton was elected a Fellow of the Royal Society of Edinburgh. References External links University of Glasgow bio of Ruxton British ecologists Evolutionary biologists Living people Mathematical ecologists Year of birth missing (living people) Camouflage researchers
https://en.wikipedia.org/wiki/Tangential%20quadrilateral	In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons. Other less frequently used names for this class of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral. Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral or inscribed quadrilateral, it is preferable not to use any of the last five names. All triangles can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to be able to have an incircle. Special cases Examples of tangential quadrilaterals are the kites, which include the rhombi, which in turn include the squares. The kites are exactly the tangential quadrilaterals that are also orthodiagonal. A right kite is a kite with a circumcircle. If a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral, and if it is both tangential and a trapezoid, it is called a tangential trapezoid. Characterizations In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter. According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral: Conversely a convex quadrilateral in which a + c = b + d must be tangential. If opposite sides in a convex quadrilateral ABCD (that is not a trapezoid) intersect at E and F, then it is tangential if and only if either of or Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if and only if the incircles in the two triangles ABC and ADC are tangent to each other. A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if Further, a convex quadrilateral with successive sides a, b, c, d is tangential if and only if where Ra, Rb, Rc, Rd are the radii in the circles externally tan
https://en.wikipedia.org/wiki/Quotient%20of%20subspace%20theorem	In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. Let (X, \|\|·\|\|) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. The induced norm \|\| · \|\| on E, defined by is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that for with K > 1 a universal constant. The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N. In fact, the constant c can be made arbitrarily close to 1, at the expense of the constant K becoming large. The original proof allowed Notes References Banach spaces Asymptotic geometric analysis Theorems in functional analysis
https://en.wikipedia.org/wiki/Complex%20conjugate%20root%20theorem	In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. It follows from this (and the fundamental theorem of algebra) that, if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proved by using the intermediate value theorem. Examples and consequences The polynomial x2 + 1 = 0 has roots ±&hairsp;i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial has roots and thus can be factored as In computing the product of the last two factors, the imaginary parts cancel, and we get The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors. If the roots are and , they form a quadratic . If the third root is , this becomes . Corollary on odd-degree polynomials It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots; Therefore some of them must be real. This requires some care in the presence of multiple roots; but a complex root and its conjugate do have the same multiplicity (and this lemma is not hard to prove). It can also be worked around by considering only irreducible polynomials; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above. This corollary can also be proved directly by using the intermediate value theorem. Proof One proof of the theorem is as follows: Consider the polynomial where all ar are real. Suppose some complex number ζ is a root of P, that is . It needs to be shown that as well. If P(ζ&hairsp;&hairsp;) = 0, then which can be put as Now and given the properties of complex conjugation, Since it follows that That is, Note that this works only because the ar are real, that is, . If any of the coefficients were non-real, the roots would not necessarily come in conjugate pairs. Notes Theorems in complex analysis Theorems about polynomials Articles containing proofs
https://en.wikipedia.org/wiki/M.%20Riesz%20extension%20theorem	The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments. Formulation Let be a real vector space, be a vector subspace, and be a convex cone. A linear functional is called -positive, if it takes only non-negative values on the cone : A linear functional is called a -positive extension of , if it is identical to in the domain of , and also returns a value of at least 0 for all points in the cone : In general, a -positive linear functional on cannot be extended to a -positive linear functional on . Already in two dimensions one obtains a counterexample. Let and be the -axis. The positive functional can not be extended to a positive functional on . However, the extension exists under the additional assumption that namely for every there exists an such that Proof The proof is similar to the proof of the Hahn–Banach theorem (see also below). By transfinite induction or Zorn's lemma it is sufficient to consider the case dim . Choose any . Set We will prove below that . For now, choose any satisfying , and set , , and then extend to all of by linearity. We need to show that is -positive. Suppose . Then either , or or for some and . If , then . In the first remaining case , and so by definition. Thus In the second case, , and so similarly by definition and so In all cases, , and so is -positive. We now prove that . Notice by assumption there exists at least one for which , and so . However, it may be the case that there are no for which , in which case and the inequality is trivial (in this case notice that the third case above cannot happen). Therefore, we may assume that and there is at least one for which . To prove the inequality, it suffices to show that whenever and , and and , then . Indeed, since is a convex cone, and so since is -positive. Corollary: Krein's extension theorem Let E be a real linear space, and let K ⊂ E be a convex cone. Let x ∈ E\(−K) be such that R x + K = E. Then there exists a K-positive linear functional φ: E → R such that φ(x) > 0. Connection to the Hahn–Banach theorem The Hahn–Banach theorem can be deduced from the M. Riesz extension theorem. Let V be a linear space, and let N be a sublinear function on V. Let φ be a functional on a subspace U ⊂ V that is dominated by N: The Hahn–Banach theorem asserts that φ can be extended to a linear functional on V that is dominated by N. To derive this from the M. Riesz extension theorem, define a convex cone K ⊂ R×V by Define a functional φ1 on R×U by One can see that φ1 is K-positive, and that K + (R × U) = R × V. Therefore φ1 can be extended to a K-positive functional ψ1 on R×V. Then is the desired extension of φ. Indeed, if ψ(x) > N(x), we have: (N(x), x) ∈ K, whereas leading to a contradiction. Notes References Theorems in convex geometry Theorems in functional analysis
https://en.wikipedia.org/wiki/McDiarmid%27s%20inequality	In probability theory and theoretical computer science, McDiarmid's inequality is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain functions when they are evaluated on independent random variables. McDiarmid's inequality applies to functions that satisfy a bounded differences property, meaning that replacing a single argument to the function while leaving all other arguments unchanged cannot cause too large of a change in the value of the function. Statement A function satisfies the bounded differences property if substituting the value of the th coordinate changes the value of by at most . More formally, if there are constants such that for all , and all , Extensions Unbalanced distributions A stronger bound may be given when the arguments to the function are sampled from unbalanced distributions, such that resampling a single argument rarely causes a large change to the function value. This may be used to characterize, for example, the value of a function on graphs when evaluated on sparse random graphs and hypergraphs, since in a sparse random graph, it is much more likely for any particular edge to be missing than to be present. Differences bounded with high probability McDiarmid's inequality may be extended to the case where the function being analyzed does not strictly satisfy the bounded differences property, but large differences remain very rare. There exist stronger refinements to this analysis in some distribution-dependent scenarios, such as those that arise in learning theory. Sub-Gaussian and sub-exponential norms Let the th centered conditional version of a function be so that is a random variable depending on random values of . Bennett and Bernstein forms Refinements to McDiarmid's inequality in the style of Bennett's inequality and Bernstein inequalities are made possible by defining a variance term for each function argument. Let Proof The following proof of McDiarmid's inequality constructs the Doob martingale tracking the conditional expected value of the function as more and more of its arguments are sampled and conditioned on, and then applies a martingale concentration inequality (Azuma's inequality). An alternate argument avoiding the use of martingales also exists, taking advantage of the independence of the function arguments to provide a Chernoff-bound-like argument. For better readability, we will introduce a notational shorthand: will denote for any and integers , so that, for example, Pick any . Then, for any , by triangle inequality, and thus is bounded. Since is bounded, define the Doob martingale (each being a random variable depending on the random values of ) as for all and , so that . Now define the random variables for each Since are independent of each other, conditioning on does not affect the probabilities of the other variables, so these are equal to the expressions Note that . In addition, Then
https://en.wikipedia.org/wiki/Boudghene%20Ben%20Ali%20Lotfi%20Airport	Boudghene Ben Ali Lotfi Airport () is an airport located 5 km north of Béchar, a city in the Béchar Province of Algeria. Airlines and destinations Statistics See also List of airports in Algeria Béchar Ouakda aerodrome Benali Boudghene References External links Airports in Algeria Buildings and structures in Béchar Province
https://en.wikipedia.org/wiki/Commandant%20Ferradj%20Airport	Commandant Ferradj Airport is an airport in Tindouf, Algeria . Airlines and destinations Statistics References OurAirports - Tindouf Airports in Algeria Buildings and structures in Tindouf Province
https://en.wikipedia.org/wiki/Timimoun%20Airport	Timimoun Airport is an airport serving Timimoun, a town in the Adrar Province of Algeria . The airport is in the desert southeast of the town. Airlines and destinations Statistics See also Transport in Algeria List of airports in Algeria References External links Airports in Algeria Buildings and structures in Adrar Province
https://en.wikipedia.org/wiki/Galust%20Petrosyan	Galust Petrosyan (, born on 5 September 1981) is a retired Armenian football forward. He was a member of the Armenia national football team, with 7 caps and 1 goal scored. National team statistics References External links Living people 1981 births Footballers from Yerevan Armenian men's footballers Armenia men's international footballers Armenian expatriate men's footballers Expatriate men's footballers in Moldova Expatriate men's footballers in Belarus Expatriate men's footballers in Iran Armenian expatriate sportspeople in Moldova Armenian Premier League players FC Ararat Yerevan players FC Pyunik players FC Zimbru Chișinău players FC Smorgon players Sanati Kaveh Tehran F.C. players Mes Sarcheshme players Ulisses FC players Men's association football forwards
https://en.wikipedia.org/wiki/Central%20carrier	In the context of von Neumann algebras, the central carrier of a projection E is the smallest central projection, in the von Neumann algebra, that dominates E. It is also called the central support or central cover. Definition Let L(H) denote the bounded operators on a Hilbert space H, M ⊂ L(H) be a von Neumann algebra, and M' the commutant of M. The center of M is Z(M) = M' ∩ M = {T ∈ M \| TM = MT for all M ∈ M}. The central carrier C(E) of a projection E in M is defined as follows: C(E) = ∧ {F ∈ Z(M) \| F is a projection and F ≥ E}. The symbol ∧ denotes the lattice operation on the projections in Z(M): F1 ∧ F2 is the projection onto the closed subspace Ran(F1) ∩ Ran(F2). The abelian algebra Z(M), being the intersection of two von Neumann algebras, is also a von Neumann algebra. Therefore, C(E) lies in Z(M). If one thinks of M as a direct sum (or more accurately, a direct integral) of its factors, then the central projections are the projections that are direct sums (direct integrals) of identity operators of (measurable sets of) the factors. If E is confined to a single factor, then C(E) is the identity operator in that factor. Informally, one would expect C(E) to be the direct sum of identity operators I where I is in a factor and I · E ≠ 0. An explicit description The projection C(E) can be described more explicitly. It can be shown that Ran C(E) is the closed subspace generated by MRan(E). If N is a von Neumann algebra, and E a projection that does not necessarily belong to N and has range K = Ran(E). The smallest central projection in N that dominates E is precisely the projection onto the closed subspace [N' K] generated by N' K. In symbols, if F' = ∧ {F ∈ N \| F is a projection and F ≥ E} then Ran(F' ) = [N' K]. That [N' K] ⊂ Ran(F' ) follows from the definition of commutant. On the other hand, [N' K] is invariant under every unitary U in N' . Therefore the projection onto [N' K] lies in (N')' = N. Minimality of F' then yields Ran(F' ) ⊂ [N' K]. Now if E is a projection in M, applying the above to the von Neumann algebra Z(M) gives Ran C(E) = [ Z(M)' Ran(E) ] = [ (M' ∩ M)' Ran(E) ] = [MRan(E)]. Related results One can deduce some simple consequences from the above description. Suppose E and F are projections in a von Neumann algebra M. Proposition ETF = 0 for all T in M if and only if C(E) and C(F) are orthogonal, i.e. C(E)C(F) = 0. Proof: ETF = 0 for all T in M. ⇔ [M Ran(F)] ⊂ Ker(E). ⇔ C(F) ≤ 1 - E, by the discussion in the preceding section, where 1 is the unit in M. ⇔ E ≤ 1 - C(F). ⇔ C(E) ≤ 1 - C(F), since 1 - C(F) is a central projection that dominates E. This proves the claim. In turn, the following is true: Corollary Two projections E and F in a von Neumann algebra M contain two nonzero sub-projections that are Murray-von Neumann equivalent if C(E)C(F) ≠ 0. Proof: C(E)C(F) ≠ 0. ⇒ ETF ≠ 0 for some T in M. ⇒ ETF has polar decomposition UH for some partial isometry U and positive operator H in M. ⇒ Ra
https://en.wikipedia.org/wiki/TerraNova%20%28test%29	TerraNova is a series of standardized achievement tests used in the United States designed to assess K-12 student achievement in reading, language arts, mathematics, science, social studies, vocabulary, spelling, and other areas. The test series is published by CTB/McGraw-Hill. On June 30, 2015 McGraw-Hill Education announced that Data Recognition Corporation (DRC) had agreed to acquire "key assets" of the CTB/McGraw-Hill assessment business. TerraNova was created with an update in 1996 CTB to the California Achievement Tests and the California Tests of Basic Skills. TerraNova are used by many Department of Defense Dependents Schools. The state of California used the test as part of the CAT/6 or California Achievement Tests, 6th edition as part of the statewide STAR testing program, though only in certain grades. The CAT series of tests have been available for quite some time and before many US states began developing their own standards-based tests as part of an overall testing movement in the United States, which began in the early 2000s. The CAT were also widely used outside of California to assess student achievement. TerraNova are used widely throughout the United States. Tests The tests are usually multiple choice and answered with bubble sheets. Many sections take fifteen minutes to a few hours, and the tests sometimes extend to over one day. Fifth grade and above include short answers. The results are nationally norm-referenced, meaning that students' scores reflect their achievement in comparison to all students who took the test nationally. Typically this is expressed as a raw score that is then converted into a percentile ranking. This is different from criterion-referenced tests, which measure student performance based on mastery of the material. In 2015, CTB released TerraNova 3 Online. The online administered test includes a full battery of constructed-response and selected-response test items provides detailed diagnostic information on students' basic and applied skills. References External links DRC \| CTB - TerraNova publishers TerraNova - TerraNova Product Page Page about TerraNova Standardized tests in the United States Student assessment and evaluation Achievement tests
https://en.wikipedia.org/wiki/N%C3%A9ron%20model	In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the "push-forward" of AK from Spec(K) to Spec(R), in other words the "best possible" group scheme AR defined over R corresponding to AK. They were introduced by for abelian varieties over the quotient field of a Dedekind domain R with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains. Definition Suppose that R is a Dedekind domain with field of fractions K, and suppose that AK is a smooth separated scheme over K (such as an abelian variety). Then a Néron model of AK is defined to be a smooth separated scheme AR over R with fiber AK that is universal in the following sense. If X is a smooth separated scheme over R then any K-morphism from XK to AK can be extended to a unique R-morphism from X to AR (Néron mapping property). In particular, the canonical map is an isomorphism. If a Néron model exists then it is unique up to unique isomorphism. In terms of sheaves, any scheme A over Spec(K) represents a sheaf on the category of schemes smooth over Spec(K) with the smooth Grothendieck topology, and this has a pushforward by the injection map from Spec(K) to Spec(R), which is a sheaf over Spec(R). If this pushforward is representable by a scheme, then this scheme is the Néron model of A. In general the scheme AK need not have any Néron model. For abelian varieties AK Néron models exist and are unique (up to unique isomorphism) and are commutative quasi-projective group schemes over R. The fiber of a Néron model over a closed point of Spec(R) is a smooth commutative algebraic group, but need not be an abelian variety: for example, it may be disconnected or a torus. Néron models exist as well for certain commutative groups other than abelian varieties such as tori, but these are only locally of finite type. Néron models do not exist for the additive group. Properties The formation of Néron models commutes with products. The formation of Néron models commutes with étale base change. An Abelian scheme AR is the Néron model of its generic fibre. The Néron model of an elliptic curve The Néron model of an elliptic curve AK over K can be constructed as follows. First form the minimal model over R in the sense of algebraic (or arithmetic) surfaces. This is a regular proper surface over R but is not in general smooth over R or a group scheme over R. Its subscheme of smooth points over R is the Néron model, which is a smooth group scheme over R but not necessarily proper over R. The fibers in general may have several irreducible components, and to form the Néron model one discards all multiple components, all points where two components intersect, and all singular points of the components. Tate's algorithm calculates the special fiber of the Néron model of an elliptic curve, or more precisely the fibers of the mi
https://en.wikipedia.org/wiki/Skip%20counting	Skip counting is a mathematics technique taught as a kind of multiplication in reform mathematics textbooks such as TERC. In older textbooks, this technique is called counting by twos (threes, fours, etc.). In skip counting by twos, a person can count to 10 by only naming every other even number: 2, 4, 6, 8, 10. Combining the base (two, in this example) with the number of groups (five, in this example) produces the standard multiplication equation: two multiplied by five equals ten. References Mathematics education Multiplication
https://en.wikipedia.org/wiki/Countryman%20line	In mathematics, a Countryman line (named after Roger Simmons Countryman Jr.) is an uncountable linear ordering whose square is the union of countably many chains. The existence of Countryman lines was first proven by Shelah. Shelah also conjectured that, assuming PFA, every Aronszajn line contains a Countryman line. This conjecture, which remained open for three decades, was proven by Justin Moore. References Roger S. Countryman, Jr. Spaces having a -monotone base. Preprint, 1970. Order theory
https://en.wikipedia.org/wiki/Area%20of%20a%20triangle	In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya (section 2.6). Although simple, this formula is only useful if the height can be readily found, which is not always the case. For example, the land surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. Other frequently used formulas for the area of a triangle use trigonometry, side lengths (Heron's formula), vectors, coordinates, line integrals, Pick's theorem, or other properties. History Heron of Alexandria found what is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica, written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry. In 499 Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, expressed the area of a triangle as one-half the base times the height in the Aryabhatiya (section 2.6). A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in Shushu Jiuzhang ("Mathematical Treatise in Nine Sections"), written by Qin Jiushao. Using trigonometry The height of a triangle can be found through the application of trigonometry. Knowing SAS (side-angle-side) Using the labels in the image on the right, the altitude is . Substituting this in the formula derived above, the area of the triangle can be expressed as: (where α is the interior angle at A, β is the interior angle at B, is the interior angle at C and c is the line AB). Furthermore, since sin α = sin (π − α) = sin (β + ), and similarly for the other two angles: Knowing AAS (angle-angle-side) and analogously if the known side is a or c. Knowing ASA (angle-side-angle) and analogously if the known side is b or c. Using side lengths (Heron's formula) The shape of the triangle is determined by the lengths of the sides. Ther
https://en.wikipedia.org/wiki/Sazonov%27s%20theorem	In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (), is a theorem in functional analysis. It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is a Hilbert–Schmidt operator. The result is also important in the study of stochastic processes and the Malliavin calculus, since results concerning probability measures on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not γ-radonifying. Statement of the theorem Let G and H be two Hilbert spaces and let T : G → H be a bounded operator from G to H. Recall that T is said to be γ-radonifying if the push forward of the canonical Gaussian cylinder set measure on G is a bona fide measure on H. Recall also that T is said to be a Hilbert–Schmidt operator if there is an orthonormal basis } of G such that Then Sazonov's theorem is that T is γ-radonifying if it is a Hilbert–Schmidt operator. The proof uses Prokhorov's theorem. Remarks The canonical Gaussian cylinder set measure on an infinite-dimensional Hilbert space can never be a bona fide measure; equivalently, the identity function on such a space cannot be γ-radonifying. See also References Stochastic processes Theorems in functional analysis Theorems in measure theory
https://en.wikipedia.org/wiki/List%20of%20Dundee%20United%20F.C.%20records%20and%20statistics	This page details Dundee United records. Player records Appearances Most international appearances: Maurice Malpas (55 for Scotland) Most League appearances: Maurice Malpas (617, 1981–2000) Youngest player: Chris Mochrie, aged 16 years and 27 days (against Greenock Morton in the Scottish Championship on 4 May 2019) Oldest player: Jimmy Brownlie, aged 40 years and eight months (against Hearts at Tynecastle in February 1926, as an emergency goalkeeper) All-time appearances As of 1 January 2007 (Competitive matches only, includes appearances as substitute): Goalscorers Most League goals: Peter McKay (158 during 1947–1954) Most League goals in one season: Johnny Coyle (43 in 1955–56) Youngest scorer: David Goodwillie, aged 16 years and 11 months (against Hibernian) All-time goalscorers As of 1 January 2007 (Competitive matches only, includes appearances as substitute): Club records Scores Biggest win: 14–0 v Nithsdale Wanderers, Scottish Cup 1st Round, 17 January 1931 Biggest league win: 12–1 v East Stirlingshire, Scottish Football League Division Two, 13 April 1936 Worst defeat: 1–12 v Motherwell, Scottish Football League Division Two, 23 January 1954 Goals Most league goals: 108 during 1935–36 in Division Two (3.2 per match) Fewest league goals: 21 during 1911–12 in Division Two (0.95 per match) Fastest goals: Finn Dossing, after 14 seconds into the Division One match against Hamilton Academical at Tannadice on 16 October 1965 and Johnny Russell, also after 14 seconds in a Scottish Cup tie against Rangers at Tannadice on 2 February 2013. Most goals in one match: Albert Juliussen, with six of United's seven in a North Eastern League Cup match against St Bernard's on 15 November 1941 Several players have scored five: Collie Martin in a Scottish Football League Division Two match in December 1914 Tim Williamson in a Scottish Cup match in January 1931 Willie Ouchterlonie in Division Two matches in November and December 1933 Willie Black in the same competition in March 1939 Paul Sturrock scored all five in a 5–0 Premier Division win against Morton in November 1984. Wins Most consecutive wins: 11 during the last 6 matches of 1982–83 and the first five of the 1983–84 Most league wins in a season: 24 from 36 games (1928–29 and 1982–83) Attendances Highest home attendance: 28,000 v CF Barcelona, European Fairs Cup 2nd Round 2nd Leg, 16 November 1966 Largest crowd involving Dundee United: 100,000+ against Selangor for the formal opening of the Shah Alam Stadium, Selangor, Malaysia, in July 1994 Transfers Record transfers as of 1 August 2008 (figures based on press reports). Where some records have been equalled, only the first instance is shown. Bought The current record signing is former Scotland international Steven Pressley, who was signed from Coventry City in July 1995 for £750,000. Source: https://web.archive.org/web/20081208045249/http://www.dundeeunitedfc.co.uk/index.asp?pg=302 Sold The current record sale is former Scotland interna
https://en.wikipedia.org/wiki/John%20Smith%20%28Canadian%20poet%29	John William Smith (10 August 1927 – 16 March 2018) was a Canadian poet. Early years Born in Toronto, Ontario to English immigrant parents, Smith earned a degree in mathematics and physics from the University of Toronto. He then studied philosophy in London, and later returned to Toronto to earn an MA in English. Career After earning his Master's degree in English, Smith remained in Toronto and taught high school English for seven years. Later, he moved to Prince Edward Island to teach at Prince of Wales College. He served a term as Dean of Arts at the University of Prince Edward Island, taught there for many years, and is currently Professor Emeritus. As the author of several volumes of verse, Smith's work has appeared in a number of anthologies, including The New Poets of Prince Edward Island (1991), Landmarks (2001), and Coastlines: Poetry of Atlantic Canada (2002). An interview with Smith is included in Meetings with Maritime Poets: Interviews (2006) by Anne Compton. In 2002, Smith was the first to be appointed poet laureate of Prince Edward Island, and held the position until 2004. Smith died on 16 March 2018, aged 90. Bibliography Winter in Paradise (Charlottetown: Square Deal, 1972) Of the Swimmer Among the Coral and of the Monk in the Mountains (Charlottetown: Square Deal, 1976) Sucking Stones (Dunvegan, ON: Quadrant,1982) Midnight Found You Dancing (Charlottetown: Ragweed,1986) Strands the Length of the Wind (Charlottetown: Ragweed,1993) Fireflies in the Magnolia Grove (Charlottetown: Acorn, 2004) Maps of Invariance (Toronto: Fitzhenry & Whiteside, 2005) References External links John Smith biography 1927 births 2018 deaths Poets from Toronto 20th-century Canadian poets Canadian male poets 20th-century Canadian male writers Poets Laureate of places in Canada
https://en.wikipedia.org/wiki/GURPS%20Bestiary	GURPS Bestiary is a source book for the GURPS role-playing game system containing information and statistics of animals. It also contains information animal player character templates, and tips for fitting animals into adventures. The first edition was published in 1988. Contents The GURPS Bestiary contains over 200 creatures to populate the various worlds of the GURPS universe. The book classifies creatures by terrain type, and deals with normal animals, legendary beasts, and otherworld creatures. The book also contains GM commentaries on handling animal encounters, hunting and trapping, animals as companions, and how to create new animals. This supplement describes several hundred animals and monsters, mostly organized by habitat (e.g., arctic, desert, forest, jungle, swamp and subterranean) plus dinosaurs, domestic animals, otherworldly creatures, and "loathsome crawlers"; the book also includes rules and guidelines for game-mastering animals, animal companions, and hunting. GURPS Bestiary is a universal sourcebook for GURPS that is intended to be usable in many different settings. Publication history The 1st edition of the GURPS Bestiary was written by Steffan O'Sullivan, with a cover by Ken Kelly and illustrations by Dan Carroll, and was first published by Steve Jackson Games in 1988 as a 112-page book. The 2nd edition of GURPS Bestiary was updated by Chris McCubbin and Bob Schroeck, and had rules for were-creatures that wound up in GURPS Shapeshifters (2003). McCubbin and Schroeck wrote an article in a 1993 issue of Pyramid (a magazine published by Steve Jackson Games), offering extra material for the book. The current version of GURPS Bestiary is the 3rd edition, which was revised by Hunter Johnson and again featured Ken Kelly's cover art. Released in 2000, this expanded edition is 128 pages in length. Johnson published extra material for the 3rd edition in a 2000 issue of Pyramid. As of August 2014, the 3rd edition of GURPS Bestiary was out of print, and a 4th edition had not yet been released for the 4th edition GURPS RPG system. All three editions featured the same cover image by Ken Kelly. Reception Jim Bambra reviewed GURPS Bestiary for Dragon magazine #140 (December 1988). He comments on the book: "The art is fairly poor — about the same standard as that in the Monster Manual — so don't expect to be thrilled by the illustrations. The GURPS Bestiary has much to offer GURPS GMs, but it is of limited use to anyone looking for new critters to add to his favorite game system." See also List of GURPS books References Sources Bestiary Website GURPS Bestiary by Stephan O'Sullivan Bestiary Role-playing game supplements introduced in 1988
https://en.wikipedia.org/wiki/Gaussian%20free%20field	In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in d-dimensional Euclidean space. The continuum version is defined on Rd or on a bounded subdomain of Rd. It can be thought of as a natural generalization of one-dimensional Brownian motion to d time (but still one space) dimensions: it is a random (generalized) function from Rd to R. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval. In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways to the Schramm–Loewner evolution, see and . Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of uniform random planar domino tilings, see . The planar GFF is also the limit of the fluctuations of the characteristic polynomial of a random matrix model, the Ginibre ensemble, see . The structure of the discrete GFF on any graph is closely related to the behaviour of the simple random walk on the graph. For instance, the discrete GFF plays a key role in the proof by of several conjectures about the cover time of graphs (the expected number of steps it takes for the random walk to visit all the vertices). Definition of the discrete GFF Let P(x, y) be the transition kernel of the Markov chain given by a random walk on a finite graph G(V, E). Let U be a fixed non-empty subset of the vertices V, and take the set of all real-valued functions with some prescribed values on U. We then define a Hamiltonian by Then, the random function with probability density proportional to with respect to the Lebesgue measure on is called the discrete GFF with boundary U. It is not hard to show that the expected value is the discrete harmonic extension of the boundary values from U (harmonic with respect to the transition kernel P), and the covariances are equal to the discrete Green's function G(x, y). So, in one sentence, the discrete GFF is the Gaussian random field on V with covariance structure given by the Green's function associated to the transition kernel P. The continuum field The definition of the continuum field necessarily uses some abstract machinery, since it does not exist as a random height function. Instead, it is a random generalized function, or in other words, a probability distribution on distr
https://en.wikipedia.org/wiki/Richard%20Crandall	Richard E. Crandall (December 29, 1947 – December 20, 2012) was an American physicist and computer scientist who made contributions to computational number theory. Background Crandall was born in Ann Arbor, Michigan, and spent two years at Caltech before transferring to Reed College in Portland, Oregon, where he graduated in physics and wrote his undergraduate thesis on randomness. He earned his Ph.D in theoretical physics from Massachusetts Institute of Technology. Career In 1978, he became a physics professor at Reed College, where he taught courses in experimental physics and computational physics for many years, ultimately becoming Vollum Professor of Science and director of the Center for Advanced Computation. He was also, at various times, Chief Scientist at NeXT, Inc., Chief Cryptographer and Distinguished Scientist at Apple, and head of Apple's Advanced Computation Group. He was a pioneer in experimental mathematics. He developed the irrational base discrete weighted transform, a method of finding very large primes. He wrote several books and many scholarly papers on scientific programming and computation. Crandall was awarded numerous patents for his work in the field of cryptography and wrote a poker program that could bluff. He also owned and operated PSI Press, an online publishing company. Personal life Crandall was part Cherokee and proud of his Native heritage. He fronted a band called the Chameleons in 1981. He was working on an intellectual biography of Steve Jobs when he collapsed at his home in Portland, Oregon, from acute leukemia. He died 10 days later, on December 20, 2012, at the age of 64. Books Pascal Applications for the Sciences. John Wiley & Sons, New York 1983. with M. M. Colgrove: Scientific Programming with Macintosh Pascal. John Wiley & Sons, New York 1986. Mathematica for the Sciences, Addison-Wesley, Reading, Mass, 1991. Projects in Scientific Computation. Springer 1994. Topics in Advanced Scientific Computation. Springer 1996. with M. Levich: A Network Orange. Springer 1997. with C. Pomerance: Prime numbers: A Computational Perspective.'' Springer 2001. References External links Professor Richard E. Crandall; many of Crandall's papers can be found here Nicholas Wheeler, Remembering Prof. Crandall Stephen Wolfram, Remembering Richard Crandall (1947-2012) David Bailey and Jonathan Borwein, Mathematician/physicist/inventor Richard Crandall dies at 64 David Broadhurst, A prime puzzle in honor of Richard Crandall 1947 births 2012 deaths Scientists from Ann Arbor, Michigan Scientists from Portland, Oregon 20th-century American inventors 21st-century American inventors American atheists American computer scientists Apple Inc. employees Computational physicists Deaths from cancer in Oregon Deaths from acute leukemia Reed College faculty Reed College alumni
https://en.wikipedia.org/wiki/Rhind%20Mathematical%20Papyrus	The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1550 BC. The British Museum, where the majority of the papyrus is now kept, acquired it in 1865 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind. There are a few small fragments held by the Brooklyn Museum in New York City and an central section is missing. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. It was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost text from the reign of king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is tall and consists of multiple parts which in total make it over long. The papyrus began to be transliterated and mathematically translated in the late 19th century. The mathematical translation aspect remains incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from the period ("Year 11") of his successor, Khamudi. In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues with: This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out. The Rhind Papyrus was published in 1923 by Peet and contains a discussion of the text that followed Griffith's Book I, II and III outline. Chace published a compendium in 1927–29 which included photographs of the text. A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. Book I – Arithmetic and Algebra The first part of the Rhind papyrus consists of reference tables and a collection of 21 arithmetic and 20 algebraic problems. The problems start out with simple fractional expressions, followed by completion (sekem) problems and more involved linear equations (aha problems). The first part of the papyrus is taken up by the 2/n table. The fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. For example, . T
https://en.wikipedia.org/wiki/Shape%20parameter	In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributions that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the shape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does). For example, "peakedness" refers to how round the main peak is. Estimation Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite. Estimators of shape often involve higher-order statistics (non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as the L-moments. Maximum likelihood estimation can also be used. Examples The following continuous probability distributions have a shape parameter: Beta distribution Burr distribution Dagum distribution Erlang distribution ExGaussian distribution Exponential power distribution Fréchet distribution Gamma distribution Generalized extreme value distribution Log-logistic distribution Log-t distribution Inverse-gamma distribution Inverse Gaussian distribution Pareto distribution Pearson distribution Skew normal distribution Lognormal distribution Student's t-distribution Tukey lambda distribution Weibull distribution By contrast, the following continuous distributions do not have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that (where they exist) the skewness and kurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters. Exponential distribution Cauchy distribution Logistic distribution Normal distribution Raised cosine distribution Uniform distribution Wigner semicircle distribution See also Skewness Kurtosis Location parameter References Statistical parameters
https://en.wikipedia.org/wiki/Gyula%20O.%20H.%20Katona	Gyula O. H. Katona (born 16 March 1941 in Budapest) is a Hungarian mathematician known for his work in combinatorial set theory, and especially for the Kruskal–Katona theorem and his beautiful and elegant proof of the Erdős–Ko–Rado theorem in which he discovered a new method, now called Katona's cycle method. Since then, this method has become a powerful tool in proving many interesting results in extremal set theory. He is affiliated with the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences. Katona was secretary-general of the János Bolyai Mathematical Society from 1990 to 1996. In 1966 and 1968 he won the Grünwald Prize, awarded by the Bolyai Society to outstanding young mathematicians, he was awarded the Alfréd Rényi Prize of the Hungarian Academy of Sciences in 1975, and the same academy awarded him the Prize of the Academy in 1989. In 2011 the Alfréd Rényi Institute, the János Bolyai Society and the Hungarian Academy of Sciences organized a conference in honor of Katona's 70th birthday. Gyula O.H. Katona is the father of Gyula Y. Katona, another Hungarian mathematician with similar research interests to those of his father. References External links Katona's web site Katona on IMDB, appearing as himself in N is a Number Members of the Hungarian Academy of Sciences 20th-century Hungarian mathematicians Combinatorialists 1941 births Living people
https://en.wikipedia.org/wiki/Aronszajn%20line	In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality which contains no subset order-isomorphic to with the usual ordering the reverse of an uncountable subset of the Real numbers with the usual ordering. Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree. References Order theory
https://en.wikipedia.org/wiki/Field%20of%20definition	In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V. The issue of field of definition is of concern in diophantine geometry. Notation Throughout this article, k denotes a field. The algebraic closure of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field containing p elements. Affine n-space over a field F is denoted by An(F). Definitions for affine and projective varieties Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective space of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous. A k-algebraic set is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, ..., xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k. One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, ..., xn] is the zero-locus of a single element of k[x1, ..., xn] if k is not algebraically closed. A k-variety is called a variety if it is absolutely irreducible, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg[x1, ..., xn] that vanishes on V is the linear combination (over kalg) of polynomials in k[x1, ..., xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L. Equivalently, a k-variety V is a variety defined over k if and only if the function field k(V) of V is a regular extension of k, in the sense of Weil. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint. André Weil proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition. Examples The zero-locus of x12+ x22 is both a Q-variety and a Qalg-algebraic set but neither a variety nor a Qalg-variety, since it is the union of the Qalg-varieties defined by the polynomial
https://en.wikipedia.org/wiki/Straight%20Lines	Straight Lines may refer to: Straight line, in mathematical geometry Straight Lines (band), a Canadian pop-rock band Straight Lines (album), by Ken Vandermark Straight Lines (EP), by Junip "Straight Lines" (song), by Silverchair "Straight Lines", a song by New Musik from From A to B "Straight Lines", a song by Hayden Thorpe from Diviner See also No Straight Lines, an anthology by Justin Hall
https://en.wikipedia.org/wiki/Kaspar%20Gottfried%20Schweizer	Kaspar Gottfried Schweizer (16 February 1816 – 6 July 1873) was a Swiss astronomer who travelled to Moscow in 1845 to become Professor of Mathematics and Astronomy at the Survey Institute, and later director of the Moscow University Observatory. Schweizer was born in 1816 as the son of a pastor at Wila, Switzerland. In 1839, he went to Königsberg to assist Friedrich Wilhelm Bessel. From 1841 to 1845 he worked at Pulkovo Observatory under Friedrich Georg Wilhelm von Struve. Schweizer discovered five comets, and found one NGC object, NGC 7804, on 11 November 1864. References External links Kaspar Gottfried Schweizer, Historische Lexikon der Schweiz Kaspar Gottfried Schweizer, obituary Astronomers from the Russian Empire 19th-century Swiss astronomers Discoverers of comets 1816 births 1873 deaths
https://en.wikipedia.org/wiki/Hat%20notation	A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses. Estimated value In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value. For example, in the context of errors and residuals, the "hat" over the letter indicates an observable estimate (the residuals) of an unobservable quantity called (the statistical errors). Another example of the hat operator denoting an estimator occurs in simple linear regression. Assuming a model of , with observations of independent variable data and dependent variable data , the estimated model is of the form where is commonly minimized via least squares by finding optimal values of and for the observed data. Hat matrix In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ: Cross product In screw theory, one use of the hat operator is to represent the cross product operation. Since the cross product is a linear transformation, it can be represented as a matrix. The hat operator takes a vector and transforms it into its equivalent matrix. For example, in three dimensions, Unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in (pronounced "v-hat"). Fourier transform The Fourier transform of a function is traditionally denoted by . See also Exterior algebra Top-hat filter Circumflex, noting that precomposed glyphs [letter-with-circumflex] do not exist for all letters. References Mathematical notation
https://en.wikipedia.org/wiki/Norman%20Tome	Norman Tome (born 20 March 1973) is an Australian soccer player who represented Australia at the 1996 Atlanta Olympics. External links Career Statistics at OzFootball 1973 births Living people Australian men's soccer players Footballers at the 1996 Summer Olympics Olympic soccer players for Australia Bonnyrigg White Eagles FC players Sydney Olympic FC players Marconi Stallions FC players Men's association football forwards Place of birth missing (living people)
https://en.wikipedia.org/wiki/Evan%20Siegel	Evan Siegel is a professor of Mathematics and Computer Science. Biography Evan Siegel received his PhD in Mathematics from the CUNY Graduate Center in 2000, his MSc in Mathematics from New York University, and his BSc in Mathematics from MIT. He is currently an Associate Professor of Mathematics at New Jersey City University. In addition to his interest in Mathematics, Siegel is interested in the history of the Middle East and has numerous publications on this topic. Siegel does research in sources in Persian, French, Arabic, Turkish, Russian, German, and Georgian. In 1998-2000 he was an Editorial Board Member of the Journal of Azerbaijani Studies, and in 1994-2002 a Corresponding Secretary of the International Society for Azerbaijani Studies. Books An annotated translation of Ahmad Kasravi's History of the Iranian Constitutional Revolution. (2007) "Akinchi and Azerbaijani Self-Definition" in Michael Ursinus, Christoph Herzog, & Raoul Motika (ed.), Heidelberger Studien zur Geschichte und Kultur des modernen Vorderen Orients, vol. 27 (Frankfurt am Main, etc.: Peter Lang, 2001) "An Azerbaijani Poets' Duel over Iranian Constitutionalism" in Michael Ursinus, Raoul Motika, & Christoph Herzog (eds.), Presse und Öffentlichkeit im Nahen Östen (Istanbul: ISIS Yayinlari, 2000) "The Politics of Shahid-e Javid" in Werner Ende and Rainer Brunner (eds.), The Twelver Shia in Modern Times: Religious Culture and Political History (Leiden: Brill (Social, Economic and Political Studies of the Middle East, vol. 72), 2000) "Negahiye Kutahi be Bargozidehayi az Mollah Nasr od-Din/Montakhebiye az Nashriyeye Molla Nasr od-Din" in Janet Afary et al., Negareshi bar Zan va Jensiat dar Dawran-e Mashrute (Chicago: Historical Society of Iranian Women, 2000) "The Turkish Language in Iran by Ahmad Kasravi" (translated from Arabic), Journal of Azerbaijani Studies (vol. 1, no. 2, 1998) "A Woman's Letters to Molla Nasr od-Din(Tbilisi)" in Christopher Herzog et al. (eds.), Presse und Öffentlichkeit im Nahen Östen (Heidelberg: Heidelberg Orientverlag, 1995) "Chand Maqale az Mulla Nasr ud-Din", Nimeye Digar no. 17 (Winter 1993) Articles, papers, etc "A risala by Sheikh Fazlollah Nuri denouncing the Iranian constitutional movement," Images, Representations and Perceptions in the Shia World, Geneva, Switzerland, (2002) "Debates in Georgian Historiography on the Iranian Constitutional Revolution", Society for Iranian Studies Conference (2001) "The Uses of Classical Iranian Literature in Early Modern Azerbaijani Satire", International Conference on "The Middle Eastern Press as a Forum for Literature, Bamberg, Germany (October 2001) "Negahiye Kutahi be Bargozidehayi az Mollah Nasr od-Din/Montakhebiye az Nashriyeye Molla Nasr od-Din" in Janet Afary et al., Negareshi bar Zan va Jensiat dar Dawran-e Mashrute (Chicago: Historical Society of Iranian Women, 2000) "The Mullah and the Commissar: Mirza Jalil Muhammadquluzada in the Land of the Soviets", Middle East Studies Association (2000) "S
https://en.wikipedia.org/wiki/Quadratic%20growth	In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta notation, . This can be defined both continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural number variable). Examples Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such as the triangular numbers. The th triangular number has value , approximately . For a real function of a real variable, quadratic growth is equivalent to the second derivative being constant (i.e., the third derivative being zero), and thus functions with quadratic growth are exactly the quadratic polynomials, as these are the kernel of the third derivative operator . Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second finite difference being constant (the third finite difference being zero), and thus a sequence with quadratic growth is also a quadratic polynomial. Indeed, an integer-valued sequence with quadratic growth is a polynomial in the zeroth, first, and second binomial coefficient with integer values. The coefficients can be determined by taking the Taylor polynomial (if continuous) or Newton polynomial (if discrete). Algorithmic examples include: The amount of time taken in the worst case by certain algorithms, such as insertion sort, as a function of the input length. The numbers of live cells in space-filling cellular automaton patterns such as the breeder, as a function of the number of time steps for which the pattern is simulated. Metcalfe's law stating that the value of a communications network grows quadratically as a function of its number of users. See also Exponential growth References Asymptotic analysis
https://en.wikipedia.org/wiki/Mathematical%20operators	Mathematical operator can refer to: Operator (mathematics), a concept in mapping vector spaces Operation (mathematics), the basic symbols for addition, multiplication etc. Mathematical Operators (Unicode block), containing characters for mathematical, logical, and set notation
https://en.wikipedia.org/wiki/Eric%20Moe%20%28ice%20hockey%29	Eric Moe (born March 6, 1988, in Timrå, Sweden) is a defenceman playing for Leksands IF hockey team in the Swedish second league, HockeyAllsvenskan. Career statistics International play Played for Sweden in: 2006 World U18 Championships 2008 World Junior Championships (silver medal) International statistics External links References 1988 births Leksands IF players Living people Swedish ice hockey defencemen
https://en.wikipedia.org/wiki/Zyad%20Chaabo	Zyad Barakat Chaabo (; born 1 January 1979) is a Syrian former footballer who played as a striker. Career statistics International Scores and results list Syria's goal tally first. Honours Hutteen Syrian Cup: 2000–01 Al-Jaish Syrian Premier League: 2001–02, 2002–03 Syrian Cup: 2001–02, 2003–04 AFC Cup: 2004 Al-Karamah Syrian Premier League: 2007–08 Syrian Cup: 2007–08 Syria Nehru Cup runner-up: 2007 West Asian Games runner-up: 2005 Individual Best Syrian Footballer: 2003 Syrian Premier League top scorer: 2002–03 Nehru Cup top scorer: 2007 References External links Profile at syrialivesport.com 1979 births Living people People from Latakia Men's association football forwards Syrian men's footballers Syria men's international footballers Taliya SC players Al-Karamah SC players Persepolis F.C. players Al-Jaish SC (Syria) players Hutteen SC players Syrian expatriate men's footballers Expatriate men's footballers in Iran Syrian expatriate sportspeople in Iran Syrian Premier League players
https://en.wikipedia.org/wiki/Mika%20Niskanen	Mika Niskanen (born July 24, 1973 in Helsinki, Finland) is a professional ice hockey defenceman, currently with Ilves in the Finnish elite league SM-liiga. Career statistics Awards Elitserien playoff winner with HV71 in 2004. References External links 1973 births Espoo Blues players Finnish ice hockey defencemen HIFK (ice hockey) players HV71 players Ilves players KalPa players Living people Lahti Pelicans players Timrå IK players Ice hockey people from Helsinki
https://en.wikipedia.org/wiki/Kimmo%20Lotvonen	Kimmo Lotvonen (born January 11, 1976 in Oulu, Finland) is a defenceman for the Leksands IF hockey team in the Swedish HockeyAllsvenskan league. Career statistics References External links 1976 births Finnish ice hockey defencemen Oulun Kärpät players Living people Leksands IF players Lukko players Timrå IK players Ice hockey people from Oulu
https://en.wikipedia.org/wiki/Petri%20Kokko%20%28ice%20hockey%29	Petri Kokko (born February 1, 1975 in Oulu, Finland) is a professional ice hockey defenceman playing for the HC Energie Karlovy Vary hockey team. Career statistics References External links 1975 births Living people Ice hockey people from Oulu Finnish ice hockey defencemen Ilves players SaiPa players Timrå IK players
https://en.wikipedia.org/wiki/Geometrical%20properties%20of%20polynomial%20roots	In mathematics, a univariate polynomial of degree with real or complex coefficients has complex roots, if counted with their multiplicities. They form a multiset of points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plane that can be deduced from the degree and the coefficients of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree with real coefficients, which is less than for sufficiently large. In this article, a polynomial that is considered is always denoted where are real or complex numbers and ; thus is the degree of the polynomial. Continuous dependence on coefficients The roots of a polynomial of degree depend continuously on the coefficients. For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial). A consequence is that, for classical numeric root-finding algorithms, the problem of approximating the roots given the coefficients is ill-conditioned for many inputs. Conjugation The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form . It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be interpreted geometrically. Bounds on all roots Upper bounds on the absolute values of polynomial roots are widely used for root-finding algorithms, either for limiting the regions where roots should be searched, or for the computation of the computational complexity of these algorithms. Many such bounds have been given, and the sharper one depends generally on the specific sequence of coefficient that are considered. Most bounds are greater or equal to one, and are thus not sharp for a polynomial which have only roots of absolute values lower than one. However, such polynomials are very rare, as shown below. Any upper bound on the absolute values of roots provides a corresponding lower bound. In fact, if and is an upper bound of the
https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Tate%20height	In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate. Definition and properties Néron defined the Néron–Tate height as a sum of local heights. Although the global Néron–Tate height is quadratic, the constituent local heights are not quite quadratic. Tate (unpublished) defined it globally by observing that the logarithmic height associated to a symmetric invertible sheaf on an abelian variety is “almost quadratic,” and used this to show that the limit exists, defines a quadratic form on the Mordell–Weil group of rational points, and satisfies where the implied constant is independent of . If is anti-symmetric, that is , then the analogous limit converges and satisfies , but in this case is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes as a product of a symmetric sheaf and an anti-symmetric sheaf, and then is the unique quadratic function satisfying The Néron–Tate height depends on the choice of an invertible sheaf on the abelian variety, although the associated bilinear form depends only on the image of in the Néron–Severi group of . If the abelian variety is defined over a number field K and the invertible sheaf is symmetric and ample, then the Néron–Tate height is positive definite in the sense that it vanishes only on torsion elements of the Mordell–Weil group . More generally, induces a positive definite quadratic form on the real vector space . On an elliptic curve, the Néron–Severi group is of rank one and has a unique ample generator, so this generator is often used to define the Néron–Tate height, which is denoted without reference to a particular line bundle. (However, the height that naturally appears in the statement of the Birch and Swinnerton-Dyer conjecture is twice this height.) On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on , the product of with its dual. The elliptic and abelian regulators The bilinear form associated to the canonical height on an elliptic curve E is The elliptic regulator of E/K is where P1,...,Pr is a basis for the Mordell–Weil group E(K) modulo torsion (cf. Gram determinant). The elliptic regulator does not depend on the choice of basis. More generally, let A/K be an abelian variety, let B ≅ Pic0(A) be the dual abelian variety to A, and let P be the Poincaré line bundle on A × B. Then the abelian regulator of A/K is defined by choosing a basis Q1,...,Qr for the Mordell–Weil group A(K) modulo torsion and a basis η1,...,ηr for the Mordell–Weil group B(K) modulo torsion and setting (The definitions of elliptic and abelian regulato
https://en.wikipedia.org/wiki/Singapore%20math	Singapore math (or Singapore maths in British English) is a teaching method based on the national mathematics curriculum used for first through sixth grade in Singaporean schools. The term was coined in the United States to describe an approach originally developed in Singapore to teach students to learn and master fewer mathematical concepts at greater detail as well as having them learn these concepts using a three-step learning process: concrete, pictorial, and abstract. In the concrete step, students engage in hands-on learning experiences using physical objects which can be everyday items such as paper clips, toy blocks or math manipulates such as counting bears, link cubes and fraction discs. This is followed by drawing pictorial representations of mathematical concepts. Students then solve mathematical problems in an abstract way by using numbers and symbols. The development of Singapore math began in the 1980s when Singapore's Ministry of Education developed its own mathematics textbooks that focused on problem solving and developing thinking skills. Outside Singapore, these textbooks were adopted by several schools in the United States and in other countries such as Canada, Israel, the Netherlands, Indonesia, Chile, Jordan, India, Pakistan, Thailand, Malaysia, Japan, South Korea, the Philippines and the United Kingdom. Early adopters of these textbooks in the U.S. included parents interested in homeschooling as well as a limited number of schools. These textbooks became more popular since the release of scores from international education surveys such as Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA), which showed Singapore at the top three of the world since 1995. U.S. editions of these textbooks have since been adopted by a large number of school districts as well as charter and private schools. History Before the development of its own mathematics textbooks in the 1980s, Singapore imported its mathematics textbooks from other countries. In 1981, the Curriculum Development Institute of Singapore (CDIS) (currently the Curriculum Planning and Development Division) began to develop its own mathematics textbooks and curriculum. The CDIS developed and distributed a textbook series for elementary schools in Singapore called Primary Mathematics, which was first published in 1982 and subsequently revised in 1992 to emphasize problem solving. In the late 1990s, the country's Ministry of Education opened the elementary school textbook market to private companies, and Marshall Cavendish, a local and private publisher of educational materials, began to publish and market the Primary Mathematics textbooks. Following Singapore's curricular and instructional initiatives, dramatic improvements in math proficiency among Singaporean students on international assessments were observed. TIMSS, an international assessment for math and science among fourth and eighth graders, ran

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.