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https://en.wikipedia.org/wiki/Jordi%20Carchano
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Jordi Carchano (born 2 July 1984 in Sant Quirze del Vallès, Catalonia Spain) is a motorcycle road racer. He raced in the 125cc and 250cc World Championships from to .
Career statistics
By season
Races by year
(key) (Races in bold indicate pole position)
References
1984 births
Living people
Motorcycle racers from Catalonia
Spanish motorcycle racers
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https://en.wikipedia.org/wiki/Isaac%20Milner
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Isaac Milner (11 January 1750 – 1 April 1820) was a mathematician, an inventor, the President of Queens' College, Cambridge and Lucasian Professor of Mathematics.
He was instrumental in the 1785 religious conversion of William Wilberforce and helped him through many trials and was a great supporter of the abolitionists' campaign against the slave trade, steeling Wilberforce with his assurance before the 1789 Parliamentary debate:
He was also a natural philosopher and the Dean of Carlisle.
Biography
Milner was born on 11 January 1750 in Mabgate, Leeds. He began his education at a grammar school in Leeds in 1756, but this ended in 1760 with the death of his father. He was apprenticed as a weaver, reading the classics when time permitted, until his elder brother, Joseph Milner, provided him with an opportunity. Joseph was offered the mastership at Hull's grammar school and invited Isaac to become the institution's usher.
Through the patronage of his brother, Milner was subsequently freed from his duties in Hull and entered Queens' College, Cambridge, as a sizar in 1770. He graduated with a BA degree as senior wrangler in 1774, winning the first Smith's prize.
Shortly after he took his bachelor's degree he was ordained as deacon; in 1776 Queens' offered him a fellowship; in the following year he became a priest and college tutor; and in 1778 he was presented with the rectory of St Botolph's Church, Cambridge. However, he was a northerner at heart and thus was sent to reform the management of the Deanery of Carlisle. Taking a scientific approach to the Church of England's most northerly parishes he achieved success for the chapter and diocese. But Milner remained ambitious and seeking promotion he desired a return to Cambridge.
During these years his career as a natural philosopher began to take off. In 1776 Nevil Maskelyne hired him as a computer for the board of longitude, and two of his mathematical papers were presented to the Royal Society, of which he was elected fellow in 1780. In these papers Milner displayed three things: proficiency in mathematics, suspicion of French philosophy, and adherence to English Newtonian mechanics. In 1782 the Jacksonian professorship of natural philosophy was established and the syndicate selected Milner as the inaugural professor, a position he retained until 1792.
Besides lecturing, Milner also developed an important process to fabricate nitrous acid, a key ingredient in the production of gunpowder. His paper describing this process was published in the Royal Society's Philosophical Transactions in 1789 alongside an article of Joseph Priestley's, and the two corresponded on the subject. In later years Milner transferred his elaborate collection of chemical apparatus into the president's lodge at Queens' and performed experiments with E. D. Clarke, William Whewell, and the Wollaston brothers; he also collaborated with Humphry Davy and Joseph Banks in an attempt to cure gout.
Over the span of his forty-f
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https://en.wikipedia.org/wiki/Joshua%20King
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Joshua King (16 January 1798 – 1 September 1857) was the Lucasian Professor of Mathematics at the University of Cambridge from 1839 to 1849. He was also the President of Queens' College, Cambridge, from 1832 until his death and Vice-Chancellor of Cambridge University from 1833–4.
Education
Educated at Hawkshead Grammar School, Joshua King went first to Trinity College, Cambridge in 1815 but moved to Queens' College in February 1816 as a sizar (i.e. a student receiving some financial assistance), and graduated Senior Wrangler in 1819.
Career
He was elected a Fellow of the Queens’ in 1820, and served as its President from 1832 to his death – the first person not in holy orders to be so elected. In the University, he was Lucasian Professor of Mathematics from 1839, resigning because of ill-health in 1849 having given no lectures and published only one paper. His interests seemingly shifted from mathematics to law and politics, although he declined to stand as Tory candidate for Parliament for either the town or the University. He served on many committees, and was Vice-Chancellor in 1833/34. He died on 1 September 1857 aged 59, and was buried in the antechapel of the College.
"Joshua King came to Cambridge from Hawkshead Grammar School. It was soon evident that the school had produced someone of importance. He became Senior Wrangler, and his reputation in Cambridge was immense. It was believed that nothing less than a second Newton had appeared. They expected his work as a mathematician to make an epoch in the science. At an early age he became president of Queens’; later, he was Lucasian Professor. He published nothing; in fact, he did no mathematical work. But as long as he kept his health, he was an active and prominent figure in Cambridge, and he maintained his enormous reputation. When he died, it was felt that the memory of such an extraordinary man should not be permitted to die out, and his papers should be published. So his papers were examined, and nothing whatever worth publishing was found."
References
Senior Wranglers
19th-century English mathematicians
1798 births
1857 deaths
People from Ulverston
People educated at Hawkshead Grammar School
Alumni of Queens' College, Cambridge
Fellows of Queens' College, Cambridge
Presidents of Queens' College, Cambridge
Vice-Chancellors of the University of Cambridge
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https://en.wikipedia.org/wiki/Clement%20John%20Tranter
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Clement John Tranter, (16 August 1909 – 27 October 1991) was a British mathematics professor, researcher and the author of several key academic textbooks. Born in 1909 into a family of scientists, he served as a captain in the Second World War, before receiving his doctorate from the University of Oxford and later becoming professor of mathematical physics at the Royal Military College of Science in Shrivenham. His published works became popular in schools during the 1970s and were the standard textbooks used by A-level students for several years; they are still used in Far Eastern schools today.
He was made Commander of the Order of the British Empire and died of a sudden heart attack at his home in Highworth, close to Swindon. He was survived by his wife Joan, who later died on 6 December 2008.
Published works
Advanced Level Pure Mathematics, 1953.
Techniques of Mathematical Analysis, 1957.
Integral Transforms in Mathematical Physics, 1959. (translated to Spanish)
Differential Equations for Engineers and Scientists, 1961.
Mathematics For Sixth Form Scientists, 1964.
Bessel Functions with some Physical Applications, 1969.
References
L.W. Longdon & D.C. Stocks (1994) "Clement John Tranter", Bulletin of the London Mathematical Society 26(5):497–502.
1909 births
1991 deaths
People educated at Cirencester Grammar School
British non-fiction writers
20th-century British mathematicians
British physicists
People from Highworth
Commanders of the Order of the British Empire
Royal Artillery officers
Alumni of The Queen's College, Oxford
British male writers
20th-century non-fiction writers
Male non-fiction writers
Military personnel from Wiltshire
British Army personnel of World War II
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https://en.wikipedia.org/wiki/Toronto%20Raptors%20accomplishments%20and%20records
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This page details the all-time statistics, records, and other achievements pertaining to the Toronto Raptors of the National Basketball Association.
Individual accomplishments
All-NBA Team
All-NBA Defensive Team
All-Stars
All-Star Rookie Game
All-Star Rising Stars Challenge Game (formerly known as All-Star Rookie/Sophomore Challenge Game)
Slam Dunk champion
Three-Point Shootout champion
NBA Most Valuable Player
None
NBA Finals Most Valuable Player
NBA Defensive Player of the Year
• None
NBA Most Improved Player
Retired jerseys
None
Coach of the Year
Executive of the Year
Rookie of the Year
Sixth Man of the Year
All-Rookie
Conference Player of the Week*
Conference Player of the Month*
Conference Rookie of the Month*
Conference Coach of the Month*
In a rookie season
Most points scored by a rookie in one game with 48 – Charlie Villanueva, vs. Milwaukee Bucks, 26 March 2006
Most points in rookie season – Damon Stoudamire with 1,331 points
Most blocks in rookie season – Marcus Camby with 130 blocks
Most assists in rookie season – Damon Stoudamire with 653 assists
Most assists by a rookie in one game with 19 – Damon Stoudamire, vs. Houston Rockets, 2 February 1996
Most rebounds in rookie season – Chris Bosh with 557 rebounds
Eurobasket
EuroBasket 2007:
José Calderón (silver)
Jorge Garbajosa (silver)
EuroBasket 2011:
José Calderón (gold)
EuroBasket 2013:
Jonas Valančiūnas (silver)
EuroBasket 2015:
Jonas Valančiūnas (silver)
Eurobasket 2022:
Juancho Hernangomez (gold)
World Championship
2002 FIBA World Championship:
Antonio Davis
2003 FIBA Americas Championship:
Vince Carter
2006 FIBA World Championship:
Chris Bosh (bronze)
José Calderón (gold)
Jorge Garbajosa (gold)
2010 FIBA World Championship:
David Andersen
Leandro Barbosa
Linas Kleiza (bronze)
2014 FIBA World Championship:
DeMar DeRozan (gold)
Jonas Valančiūnas (fourth)
2019 FIBA Basketball World Cup:
Marc Gasol (gold)
Olympics
1996 Summer Olympics:
Žan Tabak
2000 Summer Olympics:
Vince Carter (gold)
2008 Summer Olympics:
Chris Bosh (gold)
José Calderón (silver)
Roko Ukić
2012 Summer Olympics:
José Calderón (silver)
Linas Kleiza
Jonas Valančiūnas
2016 Summer Olympics:
DeMar DeRozan (gold)
Kyle Lowry (gold)
Jonas Valančiūnas
2020 Summer Olympics:
Aron Baynes
Yuta Watanabe
Note: Beginning with the season the NBA began selecting a Player of the Week, Player of the Month and Rookie of the Month in both the Eastern and Western Conference. Beginning with the season the NBA began selecting a Coach of the Month in both the Eastern and Western Conference. Prior to selecting a winner in each conference a single winner for the entire league was selected for each of the aforementioned awards.
Team records
Regular season
Most points in a game – 144 vs. Sacramento Kings, 8 January 2021 (W 144–123)
Most points in a non-OT game – 144 vs. Sacramento Kings, 8 January 2021 (W 144–123)
Most assists in a game – 40 vs. Charlotte Hornets, 19
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https://en.wikipedia.org/wiki/Kamui%20Fujiwara
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is a Japanese character designer and manga artist. Fujiwara's father was a soldier in the Imperial Japanese Army during World War II. He excelled in mathematics and computer science when in grade school. He graduated from the Kuwasawa Design School. Fujiwara won an honorable mention in 1979 for his debut manga titled Itsu mo no Asa ni in the 18th Tezuka Award along with Toshio Nobe (also an honorable mention) and Tsukasa Hojo, who won the top prize awarded. He was heavily influenced by Katsuhiro Otomo, and a defining feature of his work is the fine attention to detail. His pen name "Kamui" has its origins in the name of the Ainu god of creation, Kamuy, and he has used it since high school. He has had stories published in the manga anthology series Petit Apple Pie.
Works
Manga
Buyo Buyo
"Chameko" (published in Manga Burikko)
Chocolate Panic
Clip
Color Mail
Deja Vu
Dragon Quest: Warriors of Eden
Dragon Quest Retsuden: Roto no Monshō
Dragon Quest Retsuden: Emblem of Roto Returns
Dragon Quest Retsuden: Emblem of Roto: Monshō o Tsugumono-tachi e
Drop
Fukugami Chōkidan
H2O (published in Manga Burikko)
Hot Ai-Q
Hyōi
Kanata e
Kenrō Densetsu: Kerberos Panzer Cops (written by Mamoru Oshii)
Oine (published in Manga Burikko, originally created by Kentarō Takekuma)
Old Testament: Genesis Books I & II (initially published by Core, republished by Tokuma Shoten)
Raika (created by Yū Terashima)
Saiyūki
Shifuku Sennen
Sōseiki
St. Michaela Gakuen Hyōryūki (created by Ei Takatori)
Teito Monogatari
Ultra Q
Unlucky Young Men
Yūtopia
Video games
Bōken Shōnen Kurabu ga Hou (character designer)
Grandia Xtreme (character designer)
World Neverland (character designer)
Gēmu Nihonshi Tenkabito: Odanobunaga (character designer)
Gēmu Nihonshi Tenkabito: Hidekichi to Ieyasu (character designer)
Terranigma (art director)
46 Okunen Monogatari: Harukanaru Eden E (Japanese version cover art)
References
External links
Kamui's Note (official site)
1959 births
Japanese illustrators
Japanese video game designers
Living people
People from Arakawa, Tokyo
Manga artists from Tokyo
Video game artists
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https://en.wikipedia.org/wiki/Divisor%20summatory%20function
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In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems.
Definition
The divisor summatory function is defined as
where
is the divisor function. The divisor function counts the number of ways that the integer n can be written as a product of two integers. More generally, one defines
where dk(n) counts the number of ways that n can be written as a product of k numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in k dimensions. Thus, for k=2, D(x) = D2(x) counts the number of points on a square lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis, and to the upper-right by the hyperbola jk = x. Roughly, this shape may be envisioned as a hyperbolic simplex. This allows us to provide an alternative expression for D(x), and a simple way to compute it in time:
, where
If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is known as the Gauss circle problem.
Sequence of D(n):
0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, ...
Dirichlet's divisor problem
Finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations. The leading behavior of the series is given by
where is the Euler–Mascheroni constant, and the error term is
Here, denotes Big-O notation. This estimate can be proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849. The Dirichlet divisor problem, precisely stated, is to improve this error bound by finding the smallest value of for which
holds true for all . As of today, this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for Gauss's circle problem, another lattice-point counting problem. Section F1 of Unsolved Problems in Number Theory
surveys what is known and not known about these problems.
In 1904, G. Voronoi proved that the error term can be improved to
In 1916, G. H. Hardy showed that . In particular, he demonstrated that for some constant , there exist values of x for which and values of x for which .
In 1922, J. van der Corput improved Dirichlet's bound to .
In 1928, J. van der Corput proved that .
In 1950, Chih Tsung-tao and independently in 1953 H. E. Richert proved that .
In 1969, Grigori Kolesnik demonstrated that .
In 1973, Grigori Kolesnik demonstrated that .
In 1982, Grigori Kolesnik demonstrated that .
In 1988, H. Iwaniec and C. J. Mozzochi proved that .
In 2003, M.N. Huxley improved this to show that .
So, lies somewhere between 1/4 and 131/416 (approx. 0.3149); it is widely
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https://en.wikipedia.org/wiki/Welch%E2%80%93Satterthwaite%20equation
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In statistics and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample variances, also known as the pooled degrees of freedom, corresponding to the pooled variance.
For sample variances , each respectively having degrees of freedom, often one computes the linear combination.
where is a real positive number, typically . In general, the probability distribution of {{math|χ}} cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution, whose effective degrees of freedom are given by the Welch–Satterthwaite equation'''
There is no assumption that the underlying population variances are equal. This is known as the Behrens–Fisher problem.
The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t-test.
See also
Pooled variance
References
Further reading
Michael Allwood (2008) "The Satterthwaite Formula for Degrees of Freedom in the Two-Sample t-Test", AP Statistics'', Advanced Placement Program, The College Board.
Theorems in statistics
Equations
Statistical approximations
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https://en.wikipedia.org/wiki/Mock%20modular%20form
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In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.
History
Ramanujan's 12 January 1920 letter to Hardy listed 17 examples of functions that he called mock theta functions, and his lost notebook contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.) Ramanujan pointed out that they have an asymptotic expansion at the cusps, similar to that of modular forms of weight , possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions. He called functions with similar properties "mock theta functions". Zwegers later discovered the connection of the mock theta function with weak Maass forms.
Ramanujan associated an order to his mock theta functions, which was not clearly defined. Before the work of Zwegers, the orders of known mock theta functions included
3, 5, 6, 7, 8, 10.
Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.
In the next few decades, Ramanujan's mock theta functions were studied by Watson, Andrews, Selberg, Hickerson, Choi, McIntosh, and others, who proved Ramanujan's statements about them and found several more examples and identities. (Most of the "new" identities and examples were already known to Ramanujan and reappeared in his lost notebook.) In 1936, Watson found that under the action of elements of the modular group, the order 3 mock theta functions almost transform like modular forms of weight (multiplied by suitable powers of q), except that there are "error terms" in the functional equations, usually given as explicit integrals. However, for many years there was no good definition of a mock theta function. This changed in 2001 when Zwegers discovered the relation with non-holomorphic modular forms, Lerch sums, and indefinite theta series. Zwegers showed, using the previous work of Watson and Andrews, that the mock theta functions of orders 3, 5, and 7 can be written as the sum of a weak Maass form of weight and a function that is bounded along geodesics ending at cusps. The weak Maass form has eigenvalue under the hyperbolic Laplacian (the same value as holomorphic modular forms of weight ); however, it increases exponentially fast near cusps, so it does not satisfy the usual growth condition for Maass wave forms. Zwegers proved this result in three different ways, by relating the mock theta functions to Hecke's theta functions of indefinite lattices of dimension 2, and to Appell–Lerc
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https://en.wikipedia.org/wiki/Bourbaki%20dangerous%20bend%20symbol
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The dangerous bend or caution symbol ☡ () was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group. It resembles a road sign that indicates a "dangerous bend" in the road ahead, and is used to mark passages tricky on a first reading or with an especially difficult argument.
Variations
Others have used variations of the symbol in their books. The computer scientist Donald Knuth introduced an American-style road-sign depiction in his Metafont and TeX systems, with a pair of adjacent signs indicating doubly dangerous passages.
Typography
In the LaTeX typesetting system, Knuth's dangerous bend symbol can be produced by
first loading the font manfnt (a font with extra symbols used in Knuth's TeX manual) with
\usepackage{manfnt}
and then typing
\dbend
There are several variations given by \lhdbend, \reversedvideodbend, \textdbend, \textlhdbend, and \textreversedvideodbend.
See also
Halmos box
References
External links
Knuth's use of the dangerous bend sign. Public domain GIF files.
Latex style file to provide a "danger" environment marked by a dangerous bend sign, based on Knuth's book.
Mathematical symbols
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https://en.wikipedia.org/wiki/Full
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Full may refer to:
People with the surname Full, including:
Mr. Full (given name unknown), acting Governor of German Cameroon, 1913 to 1914
A property in the mathematical field of topology; see Full set
A property of functors in the mathematical field of category theory; see Full and faithful functors
Satiety, the absence of hunger
A standard bed size, see Bed
Fulling, also known as tucking or walking ("waulking" in Scotland), term for a step in woollen clothmaking (verb: to full)
Full-Reuenthal, a municipality in the district of Zurzach in the canton of Aargau in Switzerland
See also
"Fullest", a song by the rapper Cupcakke
Ful (disambiguation)
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https://en.wikipedia.org/wiki/Indeterminate%20system
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In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable), but that property does not extend to nonlinear systems (e.g., the system with the equation ).
An indeterminate system by definition is consistent, in the sense of having at least one solution. For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.
Examples
The following examples of indeterminate systems of equations have respectively, fewer equations than, as many equations as, and more equations than unknowns:
Conditions giving rise to indeterminacy
In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point), which in turn excludes the possibility of having more than one solution. On the other hand, if the rank of the augmented matrix exceeds (necessarily by one, if at all) the rank of the coefficient matrix, then the equations will jointly contradict each other, which excludes the possibility of having any solution.
Finding the solution set of an indeterminate linear system
Let the system of equations be written in matrix form as
where is the coefficient matrix, is the vector of unknowns, and is an vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all vectors generated by
where is the Moore–Penrose pseudoinverse of and is any vector.
See also
Indeterminate equation
Indeterminate form
Indeterminate (variable)
Linear algebra
Simultaneous equations
Independent equation
Identifiability
References
Further reading
Linear algebra
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https://en.wikipedia.org/wiki/Independent%20equation
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An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. The concept typically arises in the context of linear equations. If it is possible to duplicate one of the equations in a system by multiplying each of the other equations by some number (potentially a different number for each equation) and summing the resulting equations, then that equation is dependent on the others. But if this is not possible, then that equation is independent of the others.
If an equation is independent of the other equations in its system, then it provides information beyond that which is provided by the other equations. In contrast, if an equation is dependent on the others, then it provides no information not contained in the others collectively, and the equation can be dropped from the system without any information loss.
The number of independent equations in a system equals the rank of the augmented matrix of the system—the system's coefficient matrix with one additional column appended, that column being the column vector of constants.
The number of independent equations in a system of consistent equations (a system that has at least one solution) can never be greater than the number of unknowns. Equivalently, if a system has more independent equations than unknowns, it is inconsistent and has no solutions.
See also
Linear algebra
Indeterminate system
Independent variable
References
Linear algebra
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https://en.wikipedia.org/wiki/Vector%20notation
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In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is lower case, upright boldface type, as in . The International Organization for Standardization (ISO) recommends either bold italic serif, as in , or non-bold italic serif accented by a right arrow, as in .
In advanced mathematics, vectors are often represented in a simple italic type, like any variable.
History
In 1835 Giusto Bellavitis introduced the idea of equipollent directed line segments which resulted in the concept of a vector as an equivalence class of such segments.
The term vector was coined by W. R. Hamilton around 1843, as he revealed quaternions, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion q = a + bi + cj + dk, Hamilton used two projections: S q = a, for the scalar part of q, and V q = bi + cj + dk, the vector part. Using the modern terms cross product (×) and dot product (.), the quaternion product of two vectors p and q can be written pq = –p.q + p×q. In 1878, W. K. Clifford severed the two products to make the quaternion operation useful for students in his textbook Elements of Dynamic. Lecturing at Yale University, Josiah Willard Gibbs supplied notation for the scalar product and vector products, which was introduced in Vector Analysis.
In 1891, Oliver Heaviside argued for Clarendon to distinguish vectors from scalars. He criticized the use of Greek letters by Tait and Gothic letters by Maxwell.
In 1912, J.B. Shaw contributed his "Comparative Notation for Vector Expressions" to the Bulletin of the Quaternion Society. Subsequently, Alexander Macfarlane described 15 criteria for clear expression with vectors in the same publication.
Vector ideas were advanced by Hermann Grassmann in 1841, and again in 1862 in the German language. But German mathematicians were not taken with quaternions as much as were English-speaking mathematicians. When Felix Klein was organizing the German mathematical encyclopedia, he assigned Arnold Sommerfeld to standardize vector notation. In 1950, when Academic Press published G. Kuerti’s translation of the second edition of volume 2 of Lectures on Theoretical Physics by Sommerfeld, vector notation was the subject of a footnote: "In the original German text, vectors and their components are printed in the same Gothic types. The more usual way of making a typographical distinction between the two has been adopted for this translation."
Rectangular coordinates
Given a Cartesian coordinate system, a vector may be specified by its Cartesian coordinates, which are a tuple of numbers.
Ordered set notation
A vector in can be specified using an ordered set of components, enclosed in either parentheses or angle brackets.
In a general sense, an n-dimensional vector v can be specified in either of the following forms:
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https://en.wikipedia.org/wiki/Surgery%20theory
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In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions.
More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.
The classification of exotic spheres by led to the emergence of surgery theory as a major tool in high-dimensional topology.
Surgery on a manifold
A basic observation
If X, Y are manifolds with boundary, then the boundary of the product manifold is
The basic observation which justifies surgery is that the space can be understood either as the boundary of or as the boundary of . In symbols,
,
where is the q-dimensional disk, i.e., the set of points in that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, is homeomorphic to the unit interval, while is a circle together with the points in its interior.
Surgery
Now, given a manifold M of dimension and an embedding , define another n-dimensional manifold to be
Since and from the equation from our basic observation before, the gluing is justified then
One says that the manifold M′ is produced by a surgery cutting out and gluing in , or by a p-surgery if one wants to specify the number p. Strictly speaking, M′ is a manifold with corners, but there is a canonical way to smooth them out. Notice that the submanifold that was replaced in M was of the same dimension as M (it was of codimension 0).
Attaching handles and cobordisms
Surgery is closely related to (but not the same as) handle attaching. Given an (n + 1)-manifold with boundary (L, ∂L) and an embedding : Sp × Dq → ∂L, where n = p + q, define another (n + 1)-manifold with boundary L′ by
The manifold L′ is obtained by "attaching a (p + 1)-handle", with ∂L′ obtained from ∂L by a p-surgery
A surgery on M not only produces a new manifold M′, but also a cobordism W between M and M′. The trace of the surgery is the cobordism (W; M, M′), with
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https://en.wikipedia.org/wiki/Amir%20Aczel
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Amir Dan Aczel (; November 6, 1950 – November 26, 2015) was an Israeli-born American lecturer in mathematics and the history of mathematics and science, and an author of popular books on mathematics and science.
Biography
Amir D. Aczel was born in Haifa, Israel. Aczel's father was the captain of a passenger ship that sailed primarily in the Mediterranean Sea. When he was ten, Aczel's father taught his son how to steer a ship and navigate. This inspired Aczel's book The Riddle of the Compass. Amir graduated from the Hebrew Reali School in Haifa, in 1969.
When Aczel was 21, he studied at the University of California, Berkeley. He graduated with a BA in mathematics in 1975, and received a Master of Science in 1976. Several years later Aczel earned a Ph.D. in statistics from the University of Oregon.
Aczel taught mathematics at universities in California, Alaska, Massachusetts, Italy, and Greece. He married his wife Debra in 1984 and had one daughter, Miriam, and one stepdaughter. He accepted a professorship at Bentley College in Massachusetts, where he taught classes on statistics and the history of science and history of mathematics. He authored two textbooks on statistics. While teaching at Bentley, Aczel wrote several non-technical books on mathematics and science, as well as two textbooks. His book, Fermat's Last Theorem (), was a United States bestseller and was nominated for a Los Angeles Times Book Prize. Aczel appeared on CNN, CNBC, The History Channel, and Nightline. Aczel was a 2004 Fellow of the John Simon Guggenheim Memorial Foundation, a visiting scholar in the History of Science at Harvard University (2007), and was awarded a Sloan Foundation grant to research his 2015 book Finding Zero (). In 2003, he became a research fellow at the Boston University Center for Philosophy and History of Science, and in Fall 2011 was teaching mathematics courses at University of Massachusetts Boston. He was a speaker at La Ciudad de las Ideas (The City of Ideas), Puebla, Mexico, in 2008 , 2010 , and 2011. He died in Nîmes, France in 2015 from cancer.
Works
Complete Business Statistics, 8th Edition, 2012.
Statistics: Concepts and Applications, 1995.
How to Beat the I.R.S. at Its Own Game: Strategies to Avoid and Fight an Audit, 1996.
Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, 1997.
God's Equation: Einstein, Relativity, and the Expanding Universe, 1999.
The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, 2000.
Probability 1: The Book That Proves There Is Life In Outer Space, Harvest Books, January 2000. .
The Riddle of the Compass: The Invention that Changed the World, 2001.
Entanglement: The Greatest Mystery in Physics, 2002. and
Pendulum: Léon Foucault and the Triumph of Science, 2003.
Chance: A Guide to Gambling, Love, and the Stock Market, 2004.
Descartes' Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe, 2005.
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https://en.wikipedia.org/wiki/Power%20iteration
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In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix , the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, .
The algorithm is also known as the Von Mises iteration.
Power iteration is a very simple algorithm, but it may converge slowly. The most time-consuming operation of the algorithm is the multiplication of matrix by a vector, so it is effective for a very large sparse matrix with appropriate implementation.
The method
The power iteration algorithm starts with a vector , which may be an approximation to the dominant eigenvector or a random vector. The method is described by the recurrence relation
So, at every iteration, the vector is multiplied by the matrix and normalized.
If we assume has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue, then a subsequence converges to an eigenvector associated with the dominant eigenvalue.
Without the two assumptions above, the sequence does not necessarily converge. In this sequence,
,
where is an eigenvector associated with the dominant eigenvalue, and . The presence of the term implies that does not converge unless . Under the two assumptions listed above, the sequence defined by
converges to the dominant eigenvalue (with Rayleigh quotient).
One may compute this with the following algorithm (shown in Python with NumPy):
#!/usr/bin/env python3
import numpy as np
def power_iteration(A, num_iterations: int):
# Ideally choose a random vector
# To decrease the chance that our vector
# Is orthogonal to the eigenvector
b_k = np.random.rand(A.shape[1])
for _ in range(num_iterations):
# calculate the matrix-by-vector product Ab
b_k1 = np.dot(A, b_k)
# calculate the norm
b_k1_norm = np.linalg.norm(b_k1)
# re normalize the vector
b_k = b_k1 / b_k1_norm
return b_k
power_iteration(np.array([[0.5, 0.5], [0.2, 0.8]]), 10)
The vector converges to an associated eigenvector. Ideally, one should use the Rayleigh quotient in order to get the associated eigenvalue.
This algorithm is used to calculate the Google PageRank.
The method can also be used to calculate the spectral radius (the eigenvalue with the largest magnitude, for a square matrix) by computing the Rayleigh quotient
Analysis
Let be decomposed into its Jordan canonical form: , where the first column of is an eigenvector of corresponding to the dominant eigenvalue . Since the dominant eigenvalue of is unique, the first Jordan block of is the matrix where is the largest eigenvalue of A in magnitude. The starting vector can be written as a linear combination of the columns of V:
By assumption, has a nonzero compone
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https://en.wikipedia.org/wiki/Tetrahedral%20prism
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In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Images
Alternative names
Tetrahedral dyadic prism (Norman W. Johnson)
Tepe (Jonathan Bowers: for tetrahedral prism)
Tetrahedral hyperprism
Digonal antiprismatic prism
Digonal antiprismatic hyperprism
Structure
The tetrahedral prism is bounded by two tetrahedra and four triangular prisms. The triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces.
Projections
The tetrahedron-first orthographic projection of the tetrahedral prism into 3D space has a tetrahedral projection envelope. Both tetrahedral cells project onto this tetrahedron, while the triangular prisms project to its faces.
The triangular-prism-first orthographic projection of the tetrahedral prism into 3D space has a projection envelope in the shape of a triangular prism. The two tetrahedral cells are projected onto the triangular ends of the prism, each with a vertex that projects to the center of the respective triangular face. An edge connects these two vertices through the center of the projection. The prism may be divided into three non-uniform triangular prisms that meet at this edge; these 3 volumes correspond with the images of three of the four triangular prismic cells. The last triangular prismic cell projects onto the entire projection envelope.
The edge-first orthographic projection of the tetrahedral prism into 3D space is identical to its triangular-prism-first parallel projection.
The square-face-first orthographic projection of the tetrahedral prism into 3D space has a cuboidal envelope (see diagram). Each triangular prismic cell projects onto half of the cuboidal volume, forming two pairs of overlapping images. The tetrahedral cells project onto the top and bottom square faces of the cuboid.
Related polytopes
It is the first in an infinite series of uniform antiprismatic prisms.
The tetrahedral prism, -131, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The tetrahedral prism is the vertex figure for the second, the rectified 5-simplex. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each uniform polytope in the sequence is the vertex figure of the next.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26)
Norman Johnson Uniform Polytopes, Manuscript (1991)
External links
4-polytopes
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https://en.wikipedia.org/wiki/Kernel%20principal%20component%20analysis
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In the field of multivariate statistics, kernel principal component analysis (kernel PCA)
is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space.
Background: Linear PCA
Recall that conventional PCA operates on zero-centered data; that is,
,
where is one of the multivariate observations.
It operates by diagonalizing the covariance matrix,
in other words, it gives an eigendecomposition of the covariance matrix:
which can be rewritten as
.
(See also: Covariance matrix as a linear operator)
Introduction of the Kernel to PCA
To understand the utility of kernel PCA, particularly for clustering, observe that, while N points cannot, in general, be linearly separated in dimensions, they can almost always be linearly separated in dimensions. That is, given N points, , if we map them to an N-dimensional space with
where ,
it is easy to construct a hyperplane that divides the points into arbitrary clusters. Of course, this creates linearly independent vectors, so there is no covariance on which to perform eigendecomposition explicitly as we would in linear PCA.
Instead, in kernel PCA, a non-trivial, arbitrary function is 'chosen' that is never calculated explicitly, allowing the possibility to use very-high-dimensional 's if we never have to actually evaluate the data in that space. Since we generally try to avoid working in the -space, which we will call the 'feature space', we can create the N-by-N kernel
which represents the inner product space (see Gramian matrix) of the otherwise intractable feature space. The dual form that arises in the creation of a kernel allows us to mathematically formulate a version of PCA in which we never actually solve the eigenvectors and eigenvalues of the covariance matrix in the -space (see Kernel trick). The N-elements in each column of K represent the dot product of one point of the transformed data with respect to all the transformed points (N points). Some well-known kernels are shown in the example below.
Because we are never working directly in the feature space, the kernel-formulation of PCA is restricted in that it computes not the principal components themselves, but the projections of our data onto those components. To evaluate the projection from a point in the feature space onto the kth principal component (where superscript k means the component k, not powers of k)
We note that denotes dot product, which is simply the elements of the kernel . It seems all that's left is to calculate and normalize the , which can be done by solving the eigenvector equation
where is the number of data points in the set, and and are the eigenvalues and eigenvectors of . Then to normalize the eigenvectors , we require that
Care must be taken regarding the fact that, whether or not has zero-mean in its original space, it is not guaranteed to be centered in the feat
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https://en.wikipedia.org/wiki/Elliptic%20unit
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In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.
Definition
A system of elliptic units may be constructed for an elliptic curve E with complex multiplication by the ring of integers R of an imaginary quadratic field F. For simplicity we assume that F has class number one. Let a be an ideal of R with generator α. For a Weierstrass model of E, define
where P is a point on E, Δ is the discriminant, and x is the X-coordinate on the Weierstrass model. The function Θ is independent of the choice of model, and is defined over the field of definition of E.
Properties
Let b be an ideal of R coprime to a and Q an R-generator of the b-torsion. Then Θa(Q) is defined over the ray class field K(b), and if b is not a prime power then Θa(Q) is a global unit: if b is a power of a prime p then Θa(Q) is a unit away from p.
The function Θa satisfies a distribution relation for b = (β) coprime to a:
See also
Modular unit
References
Robert, Gilles Unités elliptiques. (Elliptic units) Bull. Soc. Math. France, Supp. Mém. No. 36. Bull. Soc. Math. France, Tome 101. Société Mathématique de France, Paris, 1973. 77 pp.
Algebraic number theory
Modular forms
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https://en.wikipedia.org/wiki/Claire%20Voisin
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Claire Voisin (born 4 March 1962) is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of algebraic geometry at the Collège de France.
Work
She is noted for her work in algebraic geometry particularly as it pertains to variations of Hodge structures and mirror symmetry, and has written several books on Hodge theory. In 2002, Voisin proved that the generalization of the Hodge conjecture for compact Kähler varieties is false. The Hodge conjecture is one of the seven Clay Mathematics Institute Millennium Prize Problems which were selected in 2000, each having a prize of one million US dollars.
Voisin won the European Mathematical Society Prize in 1992 and the Servant Prize awarded by the Academy of Sciences in 1996. She received the Sophie Germain Prize in 2003 and the Clay Research Award in 2008 for her disproof of the Kodaira conjecture on deformations of compact Kähler manifolds. In 2007, she was awarded the Ruth Lyttle Satter Prize in Mathematics for, in addition to her work on the Kodaira conjecture, solving the generic case of Green's conjecture on the syzygies of the canonical embedding of an algebraic curve. This case of Green's conjecture had
received considerable attention from algebraic geometers for over two decades prior to its resolution by Voisin (the full conjecture for arbitrary curves is still partially open).
She was an invited speaker at the 1994 International Congress of Mathematicians in Zürich in the section 'Algebraic Geometry', and she was also invited as a plenary speaker at the 2010 International Congress of Mathematicians in Hyderabad.
In 2014, she was elected to the Academia Europaea.
She served on the Mathematical Sciences jury of the Infosys Prize from 2017 to 2019.
In 2009 she became a member of the German Academy of Sciences Leopoldina. In May 2016, she was elected as a foreign associate of the National Academy of Sciences. Also in 2016, she became the first female mathematician member of the Collège de France and is the first holder of the Chair of Algebraic Geometry. She received the Gold medal of the French National Centre for Scientific Research (CNRS) in September 2016. The latter is the highest scientific research award in France. In 2017, she received the Shaw Prize in Mathematical Sciences together with János Kollár. She was named MSRI Clay Senior Scholar for 2008-2009 and Spring 2019. She was elected Foreign Member of the Royal Society in 2021. She was elected International Honorary Member of the American Academy of Arts and Sciences in 2022.
Personal life
She is married to the applied mathematician Jean-Michel Coron. They have five children.
Selected publications
Hodge Theory and complex algebraic geometry. 2 vols., Cambridge University Press (Cambridge Studies in Advanced Mathematics), 2002, 2003, vol. 1, .
Mirror Symmetry. AMS 1999, .
Variations of Hodge Structure on Calabi Yau Threefolds. Edizioni Scuola Norm
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https://en.wikipedia.org/wiki/De%20Bruijn%20torus
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In combinatorial mathematics, a De Bruijn torus, named after Dutch mathematician Nicolaas Govert de Bruijn, is an array of symbols from an alphabet (often just 0 and 1) that contains every possible matrix of given dimensions exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where (one dimension).
One of the main open questions regarding De Bruijn tori is whether a De Bruijn torus for a particular alphabet size can be constructed for a given and . It is known that these always exist when , since then we simply get the De Bruijn sequences, which always exist. It is also known that "square" tori exist whenever and even (for the odd case the resulting tori cannot be square).
The smallest possible binary "square" de Bruijn torus, depicted above right, denoted as de Bruijn torus (or simply as ), contains all binary matrices.
B2
Apart from "translation", "inversion" (exchanging 0s and 1s) and "rotation" (by 90 degrees), no other de Bruijn tori are possible – this can be shown by complete inspection of all 216 binary matrices (or subset fulfilling constrains such as equal numbers of 0s and 1s).
The torus can be unrolled by repeating n−1 rows and columns. All n×n submatrices without wraparound, such as the one shaded yellow, then form the complete set:
{| class="wikitable"
| 1 || style="background:#ccc;"|0 || 1 || 1 || rowspan="5" style="border-left:solid; padding:0;"| || 1
|-
| 1 || style="background:#ccc;"|0 || style="background:#ccc;"|0 || style="background:#ccc;"|0 || 1
|-
| style="background:#ccc;"|0 || style="background:#ccc;"|0 || style="background:#ccc;"|0 || 1 || style="background:#ccc;"|0
|-
| 1 || 1 || style="background:#ccc;"|0 || style="background:#ff0;"|1 || style="background:#ff0;"|1
|- style="border-top:solid;"
| 1 || style="background:#ccc;"|0 || 1 || style="background:#ff0;"|1 || style="background:#ff0;"|1
|}
Larger example: B4
An example of the next possible binary "square" de Bruijn torus, (abbreviated as B4), has been explicitly constructed.
The image on the right shows an example of a de Bruijn torus / array, where the zeros have been encoded as white and the ones as red pixels respectively.
Binary de Bruijn tori of greater size
The paper in which an example of the de Bruijn torus was constructed contained over 10 pages of binary, despite its reduced font size, requiring three lines per row of array.
The subsequent possible binary de Bruijn torus, containing all binary matrices, would have entries, yielding a square array of dimension , denoted a de Bruijn torus or simply B6. This could easily be stored on a computer—if printed with pixels of side 0.1 mm, such a matrix would require an area of approximately 26×26 square metres.
The object B8, containing all binary matrices and denoted , has a total of 264 ≈ 18.447×1018 entries: storing such a matrix would require 18.
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https://en.wikipedia.org/wiki/Ravi%20Vakil
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Ravi D. Vakil (born February 22, 1970) is a Canadian-American mathematician working in algebraic geometry.
Education and career
Vakil attended high school at Martingrove Collegiate Institute in Etobicoke, Ontario, where he won several mathematical contests and olympiads. After earning a BSc and MSc from the University of Toronto in 1992, he completed a PhD in mathematics at Harvard University in 1997 under Joe Harris. He has since been an instructor at both Princeton University and MIT. Since the fall of 2001, he has taught at Stanford University, becoming a full professor in 2007.
Contributions
Vakil is an algebraic geometer and his research work spans over enumerative geometry, topology, Gromov–Witten theory, and classical algebraic geometry. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades.
Awards and honors
Vakil has received many awards, including an NSF CAREER Fellowship, a Sloan Research Fellowship, an American Mathematical Society Centennial Fellowship, a G. de B. Robinson prize for the best paper published (2000) in the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin, and the André-Aisenstadt Prize from the Centre de Recherches Mathématiques at the Université de Montréal (2005), and the Chauvenet Prize (2014)..
In 2012 he became a fellow of the American Mathematical Society.
Mathematics contests
He was a member of the Canadian team in three International Mathematical Olympiads, winning silver, gold (perfect score), and gold in 1986, 1987, and 1988 respectively. He was also the fourth person to be a four-time Putnam Fellow in the history of the contest. Also, he has been the coordinator of weekly Putnam preparation seminars at Stanford.
References
External links
Ravi Vakil's Home Page
The Rising Sea | Ravi's notes on algebraic geometry
1970 births
University of Toronto alumni
Harvard Graduate School of Arts and Sciences alumni
Stanford University faculty
Algebraic geometers
20th-century American mathematicians
21st-century American mathematicians
Canadian mathematicians
People from Etobicoke
Putnam Fellows
Living people
Canadian people of Indian descent
Fellows of the American Mathematical Society
International Mathematical Olympiad participants
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https://en.wikipedia.org/wiki/Dodecahedral%20prism
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In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces: 30 squares and 24 pentagons. It has 80 edges and 40 vertices.
It can be constructed by creating two coinciding dodecahedra in 3-space, and translating each copy in opposite perpendicular directions in 4-space until their separation equals their edge length.
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids or Archimedean solids.
Alternative names
Dodecahedral dyadic prism Norman W. Johnson
Dodecahedral hyperprism
Images
Structure
The dodecahedral prism consists of two dodecahedra connected to each other via 12 pentagonal prisms. The pentagonal prisms are joined to each other via their square faces.
Projections
The pentagonal-prism-first orthographic projection of the dodecahedral prism into 3D space has a decagonal envelope (see diagram). Two of the pentagonal prisms project to the center of this volume, each surrounded by 5 other pentagonal prisms. They form two sets (each consisting of a central pentagonal prism surrounded by 5 other non-uniform pentagonal prisms) that cover the volume of the decagonal prism twice. The two dodecahedra project onto the decagonal faces of the envelope.
The dodecahedron-first orthographic projection of the dodecahedral prism into 3D space has a dodecahedral envelope. The two dodecahedral cells project onto the entire volume of this envelope, while the 12 decagonal prismic cells project onto its 12 pentagonal faces.
External links
4-polytopes
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https://en.wikipedia.org/wiki/Star%20domain
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In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the line segment from to lies in This definition is immediately generalizable to any real, or complex, vector space.
Intuitively, if one thinks of as a region surrounded by a wall, is a star domain if one can find a vantage point in from which any point in is within line-of-sight. A similar, but distinct, concept is that of a radial set.
Definition
Given two points and in a vector space (such as Euclidean space ), the convex hull of is called the and it is denoted by
where for every vector
A subset of a vector space is said to be if for every the closed interval
A set is and is called a if there exists some point such that is star-shaped at
A set that is star-shaped at the origin is sometimes called a . Such sets are closed related to Minkowski functionals.
Examples
Any line or plane in is a star domain.
A line or a plane with a single point removed is not a star domain.
If is a set in the set obtained by connecting all points in to the origin is a star domain.
Any non-empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to any point in that set.
A cross-shaped figure is a star domain but is not convex.
A star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.
Properties
The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio the star domain can be dilated by a ratio such that the dilated star domain is contained in the original star domain.
The union and intersection of two star domains is not necessarily a star domain.
A non-empty open star domain in is diffeomorphic to
Given the set (where ranges over all unit length scalars) is a balanced set whenever is a star shaped at the origin (meaning that and for all and ).
See also
References
Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ,
C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, ,
External links
Convex analysis
Euclidean geometry
Functional analysis
Linear algebra
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https://en.wikipedia.org/wiki/Karna%20%28Chaulukya%20dynasty%29
|
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}
Karna (r. c. 1064–1092 CE) was an Indian king from the Chaulukya (Solanki) dynasty of Gujarat. He ruled the present-day Gujarat and surrounding areas, from his capital Anahilapataka (modern Patan).
Karna succeeded his father Bhima I, who had invaded the Paramara kingdom of Malwa at the time of Bhoja's death. Karna was forced to retreat from Malwa by Bhoja's brother Udayaditya. He annexed Lata to the Chaulukya territory by defeating a Kalachuri general, but lost it within a few years. He also suffered a defeat against the Chahamanas of Naddula, who raided the Chaulukya capital during his reign.
Karna is credited with defeating a Bhil chief of Ashapalli, and laying the foundation of the Karnavati city, identified with the modern Ahmedabad in western India. Karna married Mayanalladevi, who was the mother of his son and successor Jayasimha Siddharaja.
Early life
Karna was born to the Chaulukya monarch Bhima I and Queen Udayamati. According to the 12th century Jain chronicler Hemachandra, Bhima had three sons: Mularaja, Kshemaraja, and Karna. Mularaja died during Bhima's lifetime. Kshemaraja, the elder surviving son, renounced his rights to the throne, and retired to Dadhisthali as an ascetic. Bhima then placed Karna on the throne and retired. After becoming the king, Karna sent Kshemaraja's son Devaprasada to Dadisthali to take care of his father.
The veracity of Hemachandra is doubtful, and is not corroborated by any historical evidence. The 14th century chronicler Merutunga states that Bhima's three sons were Mularaja, Karna and Haripala. Of these, Haripala was born of a concubine named Bakuladevi. According to historian A. K. Majumdar, Merutunga's account appears to be more satisfactory, since voluntary rejections of thrones were very rare. Karna may have banished his half-brother and nephew to eliminate any rival claimants to the throne. Hemachandra was a royal courtier of Karna's son Jayasimha Siddharaja as well Kumarapala (a descendant of Kshemaraja/Haripala). Therefore, he probably invented a fictional narrative to avoid mentioning Bhima's illegitimate son as an ancestor of his patron. This theory is corroborated by the fact that Jayasimha Siddharaja hated Kumarapala.
Karna bore the title Trailokyamalla.
Military career
Paramaras of Malwa
Karna was a contemporary of his Kalachuri namesake Karna (also known as Lakshmi-Karna). Karna's father Bhima I had formed an al
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https://en.wikipedia.org/wiki/1904%E2%80%9305%20Belgian%20First%20Division
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Statistics of Belgian First Division in the 1904–05 season.
Overview
This season saw the two Groups merged back into one National Division: this was also the last season before promotion and relegation was introduced with the creation of the "Promotion" Division.
It was contested by 11 teams, and Union Saint-Gilloise won the championship.
League standings
Results
See also
1904–05 in Belgian football
References
Belgian Pro League seasons
Belgian First Division, 1913-14
1904–05 in Belgian football
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https://en.wikipedia.org/wiki/Overconvergent%20modular%20form
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In mathematics, overconvergent modular forms are special p-adic modular forms that are elements of certain p-adic Banach spaces (usually infinite dimensional)
containing classical spaces of modular forms as subspaces. They were introduced by Nicholas M. Katz in 1972.
References
Robert F. Coleman, Classical and overconvergent modular forms. Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993). J. Théor. Nombres Bordeaux 7 (1995), no. 1, 333–365.
Robert F. Coleman Classical and Overconvergent Modular Forms of Higher Level, J. Theor. Nombres Bordeaux 9 (1997), no. 2, 395–403.
Katz, Nicholas M. p-adic properties of modular schemes and modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69–190. Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973.
Modular forms
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https://en.wikipedia.org/wiki/Ignorability
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In statistics, ignorability is a feature of an experiment design whereby the method of data collection (and the nature of missing data) does not depend on the missing data. A missing data mechanism such as a treatment assignment or survey sampling strategy is "ignorable" if the missing data matrix, which indicates which variables are observed or missing, is independent of the missing data conditional on the observed data.
This idea is part of the Rubin Causal Inference Model, developed by Donald Rubin in collaboration with Paul Rosenbaum in the early 1970s. The exact definition differs between their articles in that period. In one of Rubins articles from 1978 Rubin discuss ignorable assignment mechanisms, which can be understood as the way individuals are assigned to treatment groups is irrelevant for the data analysis, given everything that is recorded about that individual. Later, in 1983 Rubin and Rosenbaum rather define strongly ignorable treatment assignment which is a stronger condition, mathematically formulated as , where is a potential outcome given treatment , is some covariates and is the actual treatment.
Pearl devised a simple graphical criterion, called back-door, that entails ignorability and identifies sets of covariates that achieve this condition.
Ignorability means we can ignore how one ended up in one vs. the other group (‘treated’ , or ‘control’ ) when it comes to the potential outcome (say ). It has also been called unconfoundedness, selection on the observables, or no omitted variable bias.
Formally it has been written as , or in words the potential outcome of person had they been treated or not does not depend on whether they have really been (observable) treated or not. We can ignore in other words how people ended up in one vs. the other condition, and treat their potential outcomes as exchangeable. While this seems thick, it becomes clear if we add subscripts for the ‘realized’ and superscripts for the ‘ideal’ (potential) worlds (notation suggested by David Freedman.
So: Y11/*Y01 are potential Y outcomes had the person been treated (superscript 1), when in reality they have actually been (Y11, subscript 1), or not (*Y01: the signals this quantity can never be realized or observed, or is fully contrary-to-fact or counterfactual, CF).
Similarly, are potential outcomes had the person not been treated (superscript ), when in reality they have been , subscript or not actually (.
Only one of each potential outcome (PO) can be realized, the other cannot, for the same assignment to condition, so when we try to estimate treatment effects, we need something to replace the fully contrary-to-fact ones with observables (or estimate them). When ignorability/exogeneity holds, like when people are randomized to be treated or not, we can ‘replace’ *Y01 with its observable counterpart Y11, and *Y10 with its observable counterpart Y00, not at the individual level Yi’s, but when it comes to averages like E[Yi1 – Yi0], whic
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https://en.wikipedia.org/wiki/Cyclotomic%20unit
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In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζ − 1) for ζ an nth root of unity and 0 < a < n.
Properties
The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field. The index of this subgroup of real cyclotomic units (those cyclotomic units in the maximal real subfield) within the full real unit group is equal to the class number of the maximal real subfield of the cyclotomic field.
If is the power of a prime, then is not a unit; however the numbers for , and ±ζ generate the group of cyclotomic units.
If is a composite number having two or more distinct prime factors, then is a unit. The subgroup of cyclotomic units generated by with is not of finite index in general.
The cyclotomic units satisfy distribution relations. Let be a rational number prime to and let denote . Then for we have
Using these distribution relations and the symmetry relation a basis Bn of the cyclotomic units can be constructed with the property that for .
See also
Elliptic unit
Modular unit
Notes
References
Algebraic number theory
Cyclotomic fields
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https://en.wikipedia.org/wiki/Bondi%20k-calculus
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Bondi k-calculus is a method of teaching special relativity popularised by Sir Hermann Bondi, that has been used in university-level physics classes (e.g. at the University of Oxford), and in some relativity textbooks.
The usefulness of the k-calculus is its simplicity. Many introductions to relativity begin with the concept of velocity and a derivation of the Lorentz transformation. Other concepts such as time dilation, length contraction, the relativity of simultaneity, the resolution of the twins paradox and the relativistic Doppler effect are then derived from the Lorentz transformation, all as functions of velocity.
Bondi, in his book Relativity and Common Sense, first published in 1964 and based on articles published in The Illustrated London News in 1962, reverses the order of presentation. He begins with what he calls "a fundamental ratio" denoted by the letter (which turns out to be the radial Doppler factor). From this he explains the twins paradox, and the relativity of simultaneity, time dilation, and length contraction, all in terms of . It is not until later in the exposition that he provides a link between velocity and the fundamental ratio . The Lorentz transformation appears towards the end of the book.
History
The k-calculus method had previously been used by E. A. Milne in 1935. Milne used the letter to denote a constant Doppler factor, but also considered a more general case involving non-inertial motion (and therefore a varying Doppler factor). Bondi used the letter instead of and simplified the presentation (for constant only), and introduced the name "k-calculus".
Bondi's k-factor
Consider two inertial observers, Alice and Bob, moving directly away from each other at constant relative velocity. Alice sends a flash of blue light towards Bob once every seconds, as measured by her own clock. Because Alice and Bob are separated by a distance, there is a delay between Alice sending a flash and Bob receiving a flash. Furthermore, the separation distance is steadily increasing at a constant rate, so the delay keeps on increasing. This means that the time interval between Bob receiving the flashes, as measured by his clock, is greater than seconds, say seconds for some constant . (If Alice and Bob were, instead, moving directly towards each other, a similar argument would apply, but in that case .)
Bondi describes as “a fundamental ratio”, and other authors have since called it "the Bondi k-factor" or "Bondi's k-factor".
Alice's flashes are transmitted at a frequency of Hz, by her clock, and received by Bob at a frequency of Hz, by his clock. This implies a Doppler factor of . So Bondi's k-factor is another name for the Doppler factor (when source Alice and observer Bob are moving directly away from or towards each other).
If Alice and Bob were to swap roles, and Bob sent flashes of light to Alice, the Principle of Relativity (Einstein's first postulate) implies that the k-factor from Bob to Alice would be the sa
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https://en.wikipedia.org/wiki/Jali
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A jali or jaali (jālī, meaning "net") is the term for a perforated stone or latticed screen, usually with an ornamental pattern constructed through the use of calligraphy, geometry or natural patterns. This form of architectural decoration is common in Indo-Islamic architecture and more generally in Indian architecture. It is closely related to mashrabiya in Islamic architecture.
According to Yatin Pandya, the jali allows light and air while minimizing the sun and the rain, as well as providing cooling through passive ventilation.> The holes are often nearly of the same width or smaller than the thickness of the stone, thus providing structural strength. It has been observed that humid areas like Kerala and Konkan have larger holes with overall lower opacity than compared with the dry climate regions of Gujarat and Rajasthan.
With the widespread use of glass in the late 19th century, and compactness of the residential areas in the modern India, jalis became less frequent for privacy and security matters. In the 21st century, it has gained popularity again as a low-energy building solution for the environmental footprint of energy use by buildings.
History
The earliest sanctuaries in India, dedicated to Buddhism, Jainism, and Hinduism, were often dimly lit and confined, resembling natural caves. Worshippers gathered in front of the sanctuaries' doorways for prayers and offerings. To control the harsh daylight entering the temples, screens known as "jalis" were used to filter and soften the light, encouraging devotion and directing attention to the sacred images. The tradition of using jalis persisted in later Indian architecture, including Hindu and Jain temples. Over time, the designs of jalis evolved, incorporating geometric and naturalistic patterns. With the advent of Islamic architecture in Gujarat, the use of jalis expanded and became a prominent feature in mosques and tombs, following the same symbolic importance of light in Islam. The adoption of jalis in Islamic buildings shows a fusion of architectural styles and motifs from Hindu, Jain, and Islamic traditions, largely influenced by the guilds of masons working for patrons across different cultural backgrounds.
Early jali work with multiple geometric shapes was built by carving into stone, in geometric patterns, first appearing in the Alai Darwaza of 1305 at Delhi besides the Qutub Minar, while later the Mughals used very finely carved plant-based designs, as at the Taj Mahal. They also often added pietra dura inlay to the surrounds, using marble and semi-precious stones.
In the Gwalior fort, near the Urwahi gate, there is a 17 line inscription dated Samvat 1553, mentioning names of some craftsmen and their creations. One of them is Khedu, who was an expert in "Gwaliyai jhilmili" i.e. jali screen crafted in the Gwalior style. The Mughal period tomb of Muhammad Ghaus built in 1565 AD at Gwalior is remarkable for its stone jalis. Many of the Gwalior's 19th century houses used stone
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https://en.wikipedia.org/wiki/Symplectic%20sum
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In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum.
The symplectic sum is the inverse of the symplectic cut, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone in algebraic geometry.
The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the Gromov–Witten invariants of symplectic manifolds.
Definition
Let and be two symplectic -manifolds and a symplectic -manifold, embedded as a submanifold into both and via
such that the Euler classes of the normal bundles are opposite:
In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation-reversing isomorphism
there is a canonical isotopy class of symplectic structures on the connected sum
meeting several conditions of compatibility with the summands . In other words, the theorem defines a symplectic sum operation whose result is a symplectic manifold, unique up to isotopy.
To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism is composed with an orientation-reversing symplectic involution of the normal bundles of (or rather their corresponding punctured unit disk bundles); then this composition is used to glue to along the two copies of .
Generalizations
In greater generality, the symplectic sum can be performed on a single symplectic manifold containing two disjoint copies of , gluing the manifold to itself along the two copies. The preceding description of the sum of two manifolds then corresponds to the special case where consists of two connected components, each containing a copy of .
Additionally, the sum can be performed simultaneously on submanifolds of equal dimension and meeting transversally.
Other generalizations also exist. However, it is not possible to remove the requirement that be of codimension two in the , as the following argument shows.
A symplectic sum along a submanifold of codimension requires a symplectic involution of a -dimensional annulus. If this involution exists, it can be used to patch two -dimensional balls together to form a symplectic -dimensional sphere. Because the sphere is a compact manifold, a symplectic form on it induces a nonzero cohomology class
But this second cohomology group is zero unless . So the symplectic sum is possible only along a submanifold of codimension two.
Identity element
Given with codimension-two symplectic submanifold , one may projectively complete the normal bundle of in to the -bundle
This
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https://en.wikipedia.org/wiki/Division%20No.%205%2C%20Newfoundland%20and%20Labrador
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Census Division No. 5 is a Statistics Canada statistical division composed of the areas of the province of Newfoundland and Labrador called Humber Valley, Bay of Islands, and White Bay. It covers a land area of 10,365.63 km² (4,002.19 sq mi), and had a population of 42,014 according to the 2016 census.
Cities
Corner Brook
Towns
Cormack
Cox's Cove
Deer Lake
Gillams
Hampden
Howley
Hughes Brook
Humber Arm South
Irishtown-Summerside
Jackson's Arm
Lark Harbour
Massey Drive
McIvers
Meadows
Mount Moriah
Pasadena
Reidville
Steady Brook
York Harbour
Unorganized subdivisions
Subdivision A (Includes: St. Jude's, and Hinds Lake)
Subdivision C (Includes: Spruce Brook, George's Lake, Pinchgut Lake)
Subdivision D (Includes: North Arm, Middle Arm, Goose Arm, Serpentine Lake)
Subdivision E (Includes: Galeville, The Beaches)
Subdivision F (Includes: Pynn's Brook, Little Rapids, Humber Valley Resort)
Subdivision G (Includes: Pollards Point, Sops Arm)
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 5 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Sources
005
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https://en.wikipedia.org/wiki/Walther%20Mayer
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Walther Mayer (11 March 1887 – 10 September 1948) was an Austrian mathematician, born in Graz, Austria-Hungary. With Leopold Vietoris he is the namesake of the Mayer–Vietoris sequence in topology. He served as an assistant to Albert Einstein, and was nicknamed "Einstein's calculator".
Biography
Mayer studied at the Federal Institute of Technology in Zürich and the University of Paris before receiving his doctorate in 1912 from the University of Vienna; his thesis concerned the Fredholm integral equation. He served in the military between 1914 and 1919, during which he found time to complete a habilitation on differential geometry. Because he was Jewish, he had little opportunity for an academic career in Austria, and left the country; however, in 1926, with help from Einstein, he returned to a position at the University of Vienna as Privatdozent (lecturer). He made a name for himself in topology with the Mayer–Vietoris sequence, and with an axiomatic treatment of homology predating the Eilenberg–Steenrod axioms. He also published a book on Riemannian geometry in 1930, the second volume of a textbook on differential geometry that had been started by Adalbert Duschek with a volume on curves and surfaces.
In 1929, on the recommendation of Richard von Mises, he became Albert Einstein's assistant with the explicit understanding that he work with him on distant parallelism, and from 1931 to 1936, he collaborated with Albert Einstein on the theory of relativity. In 1933, after Hitler's assumption of power, he followed Einstein to the United States and became an associate in mathematics at the Institute for Advanced Study in Princeton, New Jersey. He continued working on mathematics at the Institute, and died in Princeton in 1948.
References
External links
Portrait of Walther Mayer (1940), United States Holocaust Memorial Museum
1887 births
1948 deaths
Austrian mathematicians
Austrian Jews
Mathematicians from Austria-Hungary
University of Vienna alumni
Topologists
Institute for Advanced Study people
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https://en.wikipedia.org/wiki/Kummer%20sum
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In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy.
Definition
A Kummer sum is therefore a finite sum
taken over r modulo p, where χ is a Dirichlet character taking values in the cube roots of unity, and where e(x) is the exponential function exp(2πix). Given p of the required form, there are two such characters, together with the trivial character.
The cubic exponential sum K(n,p) defined by
is easily seen to be a linear combination of the Kummer sums. In fact it is 3P where P is one of the Gaussian periods for the subgroup of index 3 in the residues mod p, under multiplication, while the Gauss sums are linear combinations of the P with cube roots of unity as coefficients. However it is the Gauss sum for which the algebraic properties hold. Such cubic exponential sums are also now called Kummer sums.
Statistical questions
It is known from the general theory of Gauss sums that
In fact the prime decomposition of G(χ) in the cyclotomic field it naturally lies in is known, giving a stronger form. What Kummer was concerned with was the argument
of G(χ). Unlike the quadratic case, where the square of the Gauss sum is known and the precise square root was determined by Gauss, here the cube of G(χ) lies in the Eisenstein integers, but its argument is determined by that of the Eisenstein prime dividing p, which splits in that field.
Kummer made a statistical conjecture about θp and its distribution modulo 2π (in other words, on the argument of the Kummer sum on the unit circle). For that to make sense, one has to choose between the two possible χ: there is a distinguished choice, in fact, based on the cubic residue symbol. Kummer used available numerical data for p up to 500 (this is described in the 1892 book Theory of Numbers by George B. Mathews). There was, however, a 'law of small numbers' operating, meaning that Kummer's original conjecture, of a lack of uniform distribution, suffered from a small-number bias. In 1952 John von Neumann and Herman Goldstine extended Kummer's computations, on ENIAC. The calculations were programmed and coded by Hedvig Selberg but her work was only acknowledged at the end of the paper, similarly as with Mary Tsingou on the Fermi–Pasta–Ulam–Tsingou problem (formerly the Fermi–Pasta–Ulam problem).
In the twentieth century, progress was finally made on this question, which had been left untouched for over 100 years. Building on work of Tomio Kubota, S. J. Patterson and Roger Heath-Brown in 1978 disproved Kummer conjecture and proved a modified form of Kummer conjecture. In fact they showed that there was equidistribution of the θp. This work involved automorphic forms for the metaplectic group, and Vaughan's lemma in analytic numb
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https://en.wikipedia.org/wiki/Time%20Squared%20Academy
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Times2 STEM Academy is a charter school in Providence, Rhode Island that specializes in teaching science, technology, engineering, and mathematics.
Current
The elementary, middle school and high school (grades K–12) was established in 1998. It uses standards-based instruction and computer technology to teach science and mathematics. Algebra I and general science are taught to eighth grade students.
High school students are introduced to project-based learning – for example at the laboratory sciences at Providence College, a lead higher education partner. Seniors undertake internships in engineering, math, science, technology, and enrichment courses at local colleges and universities.
The current elementary principal is Glenn Piros who joined in August 2013. He replaced Tom Lombardi who had replaced Antonio DiManna, who was principal until June 2011. The current executive director is Dr. Rudy Moseley, who joined in October 2015.
Educational program
Students are taught using practices such as student-generated projects, enrichment initiatives, multiple assessments, and university and business partnerships.
Times2 STEM Academy uses the New Standards for curriculum development and implementation in its core subject areas. By using literacy and writing across all curricular areas, teachers build a foundation of learning that supports the school's program.
To assist the middle and upper school in identifying those competencies and qualities that are crucial for students to realize the school's mission, the administration and Board of Directors approved a K–5 Children's Academy. It is the feeder school to the middle and upper school.
The curriculum used in grades K–3 (and eventually K–5) and the middle and upper school emphasizes standards-based skills. Math Investigations teach students to integrate literacy skills with mathematics competencies.
Middle and high school teachers use the University of Chicago Math Project to develop modules that assist students in grades 6–12 to move through a mathematics sequence that culminates with AP Calculus in the 12th grade.
The science curriculum follows the standards established by the National Science Foundation. Using research to direct its decision, the Academy's Science Committee inverted its curriculum offerings where conceptual physics is offered to 9th graders, chemistry to sophomores, biology to juniors, and AP Physics to seniors.
Enrichment program
Grades 9–12 are given formally scheduled after-school enrichment courses and activities, and Saturday Academics. These include Web Club, Robotics, Tech Club, ACE Program, Aviation Club, Chess Club, Mock Trial, Yearbook, Studio Productions, Science Olympiad, and computer technology mini-courses such as Flowcharting, HTML, and Sketch Pad. Also, through work with scientists and experts in their field, students get a better understanding of what skills are needed and invested in certain careers.
Most after-school activities take place on Tuesday to T
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https://en.wikipedia.org/wiki/Pro-simplicial%20set
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In mathematics, a pro-simplicial set is an inverse system of simplicial sets.
A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups.
Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory).
References
.
.
Simplicial sets
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https://en.wikipedia.org/wiki/Honeycomb%20%28geometry%29
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In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Classification
There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered.
The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.
Uniform 3-honeycombs
A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs.
A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Two are quasiregular (made from two types of regular cells):
The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers.
Space-filling polyhedra
A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of the Platonic solids other than the cube.
Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra:
Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped)
Hexagonal prismatic honeycomb
Rhombic dodecahedral honeycomb
Elongated dodecahedral honeycomb
Bitruncated cubic honeycomb
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https://en.wikipedia.org/wiki/Period%20mapping
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In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
Ehresmann's theorem
Let be a holomorphic submersive morphism. For a point b of B, we denote the fiber of f over b by Xb. Fix a point 0 in B. Ehresmann's theorem guarantees that there is a small open neighborhood U around 0 in which f becomes a fiber bundle. That is, is diffeomorphic to . In particular, the composite map
is a diffeomorphism. This diffeomorphism is not unique because it depends on the choice of trivialization. The trivialization is constructed from smooth paths in U, and it can be shown that the homotopy class of the diffeomorphism depends only on the choice of a homotopy class of paths from b to 0. In particular, if U is contractible, there is a well-defined diffeomorphism up to homotopy.
The diffeomorphism from Xb to X0 induces an isomorphism of cohomology groups
and since homotopic maps induce identical maps on cohomology, this isomorphism depends only on the homotopy class of the path from b to 0.
Local unpolarized period mappings
Assume that f is proper and that X0 is a Kähler variety. The Kähler condition is open, so after possibly shrinking U, Xb is compact and Kähler for all b in U. After shrinking U further we may assume that it is contractible. Then there is a well-defined isomorphism between the cohomology groups of X0 and Xb. These isomorphisms of cohomology groups will not in general preserve the Hodge structures of X0 and Xb because they are induced by diffeomorphisms, not biholomorphisms. Let denote the pth step of the Hodge filtration. The Hodge numbers of Xb are the same as those of X0, so the number is independent of b. The period map is the map
where F is the flag variety of chains of subspaces of dimensions bp,k for all p, that sends
Because Xb is a Kähler manifold, the Hodge filtration satisfies the Hodge–Riemann bilinear relations. These imply that
Not all flags of subspaces satisfy this condition. The subset of the flag variety satisfying this condition is called the unpolarized local period domain and is denoted . is an open subset of the flag variety F.
Local polarized period mappings
Assume now not just that each Xb is Kähler, but that there is a Kähler class that varies holomorphically in b. In other words, assume there is a class ω in such that for every b, the restriction ωb of ω to Xb is a Kähler class. ωb determines a bilinear form Q on Hk(Xb, C) by the rule
This form varies holomorphically in b, and consequently the image of the period mapping satisfies additional constraints which again come from the Hodge–Riemann bilinear relations. These are:
Orthogonality: is orthogonal to with respect to Q.
Positive definiteness: For all , the restriction of to the primitive classes of type is positive definite.
The polarized local period domain is the subset of the unpolarized local period domain whose flags satisfy these additi
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https://en.wikipedia.org/wiki/Tightness
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Tightness may refer to:
In mathematics,
Tightness of (a collection of) measures is a concept in measure (and probability), theory in mathematics
Tightness (topology) is also a cardinal function used in general topology
In economics,
Tightness refers to the degree to which the number of unemployed workers exceed the number of posted job vacancies (or vice versa).
Market tightness, a point in time where it is very difficult to invest, but it is far easier to sell or to remove investments in return of monetary rewards
In other fields,
Tightness of a rope indicates the rope is under tension
Tightness of a sealing means it is impermeable, that it seals well
Tightness is the art of being 'tight' (i.e., stingy or miserly)
Tightness can refer to something being cool
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https://en.wikipedia.org/wiki/Gobelin
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Gobelin was the name of a family of dyers, who in all probability came originally from Reims, France, and who in the middle of the 15th century established themselves in the Faubourg Saint Marcel, Paris, on the banks of the Bièvre.
The first head of the firm was named Jehan Gobelin (d. 1476). He discovered a peculiar kind of scarlet dyestuff, and he expended so much money on his establishment that it was named by the common people la folie Gobelin. To the dye-works there was added in the 16th century a manufactory of tapestry.
The family's wealth increased so rapidly that in the third or fourth generation some of them forsook their trade and purchased titles of nobility. More than one of their number held offices of state, among others Balthasar, who became successively treasurer general of artillery, treasurer extraordinary of war, councillor secretary of the king, chancellor of the exchequer, councillor of state and president of the chamber of accounts, and who in 1601 received from Henry IV the lands and lordship of Brie-Comte-Robert. He died in 1603. The name of the Gobelins as dyers cannot be found later than the end of the 17th century.
In 1662, the works in the Faubourg Saint Marcel, with the adjoining grounds, were purchased by Jean-Baptiste Colbert on behalf of Louis XIV and transformed into a general upholstery manufactory, the Gobelins Manufactory.
In various languages 'gobelin' is synonymous for 'tapestry'.
See also
Gobelins Manufactory
References
External links
Gobelins Database Marketplace
The Gobelin and Handicraft Art Association
French families
French tapestry artists
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https://en.wikipedia.org/wiki/Regular%20homotopy
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In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions are homotopic if they represent points in the same path-components of the mapping space , given the compact-open topology. The space of immersions is the subspace of consisting of immersions, denoted by . Two immersions are regularly homotopic if they represent points in the same path-component of .
Examples
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy.
The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if their Gauss maps have the same degree/winding number.
Stephen Smale classified the regular homotopy classes of a k-sphere immersed in – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing. More precisely, the set of regular homotopy classes of embeddings of sphere in is in one-to-one correspondence with elements of group . In case we have . Since is path connected, and and due to Bott periodicity theorem we have and since then we have . Therefore all immersions of spheres and in euclidean spaces of one more dimension are regular homotopic. In particular, spheres embedded in admit eversion if . A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in . In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".
Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the homotopy principle (or h-principle) approach.
Non-degenerate homotopy
For locally convex, closed space curves, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish). There are 2 distinct non-degenerate homotopy classes. Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.
References
Differential topology
Algebraic topology
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https://en.wikipedia.org/wiki/Jos%C3%A9%20Adem
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José Adem (27 October 1921 – 14 February 1991) was a Mexican mathematician who worked in algebraic topology, and proved the Adem relations between Steenrod squares.
Life and education
Born José Adem Chahín in Tuxpan, Veracruz, (published his works as José Adem), Adem showed an interest in mathematics from an early age, and moved to Mexico City in 1941 to pursue a degree in engineering and mathematics. He obtained his B.S. in mathematics from the National Autonomous University of Mexico (UNAM) in 1949. During this time met Solomon Lefschetz, a famous algebraic topologist who was spending prolonged periods of time in Mexico. Lefschetz recognized Adem's mathematical talent, and sent him as a doctoral student to Princeton University where he graduated in 1952. His dissertation, Iterations of the squaring operations in algebraic topology, was written under the supervision of Norman Steenrod and introduced what are now called the Adem relations.
His brother is geophysicist Julián Adem, who obtained a Ph.D. in applied mathematics from Brown University in 1953. Julián's son is topologist Alejandro Adem.
Career
Adem became a researcher at the Mathematics Institute of UNAM (1954–1961), and then head of the Mathematics Department at the Instituto Politécnico Nacional (1961–1973).
He was elected to El Colegio Nacional on 4 April 1960.
In 1951 he was awarded a Guggenheim Fellowship. In 1956, Adem started the second series of the Boletín de la Sociedad Matemática Mexicana.
Publications
References
External links
1921 births
1991 deaths
People from Tuxpan, Veracruz
20th-century Mexican mathematicians
Topologists
National Autonomous University of Mexico alumni
Princeton University alumni
Academic staff of the National Autonomous University of Mexico
Academic staff of the Instituto Politécnico Nacional
Members of El Colegio Nacional (Mexico)
Mexican expatriates in the United States
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https://en.wikipedia.org/wiki/Class%20formation
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In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.
Definitions
A formation is a topological group G together with a topological G-module A on which G acts continuously.
A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if
F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic.
If E is a subgroup of G, then AE is defined to be the elements of A fixed by E.
We write
Hn(E/F)
for the Tate cohomology group
Hn(E/F, AF) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.)
In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure.
A class formation is a formation
such that for every normal layer E/F
H1(E/F) is trivial, and
H2(E/F) is cyclic of order |E/F|.
In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2(E/F),
called fundamental classes, that are compatible with each other in the sense that
the restriction (of cohomology classes) of a fundamental class is another fundamental class.
Often the fundamental classes are considered to be part of the structure of a class formation.
A formation that satisfies just the condition H1(E/F)=1 is sometimes called a field formation.
For example, if G is any finite group acting on a field L and A=L×, then this is a field formation by Hilbert's theorem 90.
Examples
The most important examples of class formations (arranged roughly in order of difficulty) are as follows:
Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.
Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers.
Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group.
Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group.
Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the G
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https://en.wikipedia.org/wiki/Istv%C3%A1n%20Hatvani
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István Hatvani (1718–1786) was a Hungarian mathematician. He worked on developing some of the earliest elements of probability theory.
External links
Biography at University of St Andrews, Scotland
1718 births
1786 deaths
18th-century Hungarian mathematicians
Probability theorists
Istvan
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https://en.wikipedia.org/wiki/Jen%C5%91%20Hunyady
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Jenő Hunyady (28 April 1838 in Pest – 26 December 1889 in Budapest) was a Hungarian mathematician noted for his work on conic sections and linear algebra, specifically on determinants. He received his Ph.D. in Göttingen (1864). He worked at the University of Technology of Budapest. He was elected a corresponding member (1867), member (1883) of the Hungarian Academy of Sciences. From 1885 he actively participated in the informal meetings of what became later the Mathematical and Physical Society of Hungary.
References
Hunyady Jenõ, Magyar Életrajzi Lexikon
Márton Sain: Matematikatörténeti ABC, Typotex, Budapest, 1993
19th-century Hungarian mathematicians
1838 births
1889 deaths
Members of the Hungarian Academy of Sciences
Mathematicians from Austria-Hungary
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https://en.wikipedia.org/wiki/Apex%20%28geometry%29
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In geometry, an apex (: apices) is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some "base". The word is derived from the Latin for 'summit, peak, tip, top, extreme end'.
Isosceles triangles
In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side.
Pyramids and cones
In a pyramid or cone, the apex is the vertex at the "top" (opposite the base). In a pyramid, the vertex is the point that is part of all the lateral faces, or where all the lateral edges meet.
References
Parts of a triangle
Polyhedra
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https://en.wikipedia.org/wiki/Emil%20Weyr
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Emil Weyr (31 August / 1 September 1848 – 25 January 1894) was an Austrian-Czech mathematician, known for his numerous publications on geometry.
Born in Prague, Weyr attended the Prague Polytechnic, where he was taught by Heinrich Durège and Otto Wilhelm Fiedler.
Biography
Early life
The birthdate of Weyr is disputed. Parish records indicate that Emil was born in Prague, on 1 September 1848, whereas his family believed he was born on 31 August 1848 and celebrated his birthday on this date. His parents were František Weyr and Marie Rumplova. František was a mathematics professor at a secondary school Emil would later attend. Emil's parents had ten children, five sons and five daughters. One of his brothers Eduard Weyr, was also a mathematician.
Early Education
Emil attended Our Lady of the Snows primary school in 1854. Following this he would study at the German secondary school on Mikulandská street in Prague from 1859. By this age Emil was already fascinated by maths through his father. In 1865 he would start attending the Prague Polytechnic, where he would study for three years. He was taught by O.W.Fiedler.
Weyr was assistant to the mathematics chair Professor H.Durège in 1868. The following year Emil earned his Doctorate in Philosophy at Leipzig University. Ernst Mach suggested to Emil that he should make a request for Habilitation at the Prague University. On 3 May 1870 he was given the senior role in the geometry faculty of the Prague Polytechnic. Here he would attain his Ph.D. and publish over 30 papers over the period from 1869 to 1870.
Trip to Italy
Emil had planned to travel to Paris to undertake a study break during the autumn of 1870, but due to the German-French War the destination for this venture changed to Milan. On 7 November he travelled by train to Trieste, then Venice where he stayed for ten days before arriving in Milan on 17 November. In February he would attend lectures given by Luigi Cremona and Felice Casorati. Later this year he got the chance to put his studies on hold to travel around Italy and visit the different universities and meet fellow colleagues, some of whom he would stay in touch with.
His relationship with Luigi Cremona emphasises how significant this trip was for Weyr, highlighted by their constant letters to each other in the twenty-one-year period following their first meeting. Emil originally wrote letters to Cremona as one would talk to a mentor, in a position of less prestige. But as the years went on the communication between the two became academic in tone. In the following years he would publish numerous papers in Italian (with help from his Italian colleagues) which would further highlights the impact of this trip to Italy.
Life in Education
On 6 February 1870 Weyr was inducted into the Union of Czech Mathematicians, which he would be given the chairman role of on 7 July 1872. In this role he created a library of scientific resources and would participate in events centred on publis
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https://en.wikipedia.org/wiki/Leopold%20Gegenbauer
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Leopold Bernhard Gegenbauer (2 February 1849, Asperhofen – 3 June 1903, Gießhübl) was an Austrian mathematician remembered best as an algebraist. Gegenbauer polynomials are named after him.
Leopold Gegenbauer was the son of a doctor. He studied at the University of Vienna from 1869 until 1873. He then went to Berlin where he studied from 1873 to 1875 working under Weierstrass and Kronecker.
After graduating from Berlin, Gegenbauer was appointed to the position of extraordinary professor at the University of Czernowitz in 1875. Czernowitz, on the upper Prut River in the Carpathian foothills, was at that time in the Austrian Empire but after World War I it was in Romania, then after 1944 it became Chernivtsi, Ukraine. Czernowitz University was founded in 1875 and Gegenbauer was the first professor of mathematics there. He remained in Czernowitz for three years before moving to the University of Innsbruck where he worked with Otto Stolz. Again he held the position of extraordinary professor in Innsbruck.
After three years teaching in Innsbruck Gegenbauer was appointed full professor in 1881, then he was appointed full professor at the University of Vienna in 1893. During the session 1897–98 he was Dean of the university. He remained at Vienna until his death. Among the students who studied with him at Vienna were the Slovenian Josip Plemelj, the American James Pierpont, Ernst Fischer, and Lothar von Rechtenstamm.
Gegenbauer had many mathematical interests such as number theory, complex analysis, and the theory of integration, but he was chiefly an algebraist. He is remembered for the Gegenbauer polynomials, a class of orthogonal polynomials. They are obtained from the hypergeometric series in certain cases where the series is in fact finite. The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
Gegenbauer also gave his name to arithmetic functions studied in analytic number theory. The Gegenbauer functions Ρ and ρ (upper case and lower case rho) are defined as follows.
In 1973 in Vienna in the district of Floridsdorf (21. Bezirk) a street was named in his honor the Gegenbauerweg.
Selected works
Einige Sätze über Determinanten hohen Ranges, 1890
Über den größten gemeinsamen Theiler, 1892
References
1849 births
1903 deaths
Mathematicians from Austria-Hungary
Academic staff of Chernivtsi University
Mathematicians from Austria-Hungary
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https://en.wikipedia.org/wiki/Pieter%20Hendrik%20Schoute
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Pieter Hendrik Schoute (21 January 1846, Wormerveer – 18 April 1913, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry.
He started his career as a civil engineer, but became a professor of mathematics at Groningen and published some thirty papers on polytopes between 1878 and his death in 1913. He collaborated with Alicia Boole Stott on describing the sections of the regular 4-polytopes.
In 1886, he became member of the Royal Netherlands Academy of Arts and Sciences.
Citations
References
Pieter Hendrik Schoute, Analytical treatment of the polytopes regularly derived from the regular polytopes., 1911, published by J. Muller in Amsterdam, Written in English. - 82 pages
External links
19th-century Dutch mathematicians
20th-century Dutch mathematicians
Geometers
1846 births
1923 deaths
Members of the Royal Netherlands Academy of Arts and Sciences
People from Zaanstad
Delft University of Technology alumni
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https://en.wikipedia.org/wiki/Hendrik%20Kloosterman
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Hendrik Douwe Kloosterman (9 April 1900 – 6 May 1968) was a Dutch mathematician, known for his work in number theory (in particular, for introducing Kloosterman sums) and in representation theory.
After completing his master's degree at Leiden University from 1918–1922 he studied at the University of Copenhagen with Harald Bohr and the University of Oxford with G. H. Hardy. In 1924 he received his Ph.D. in Leiden under the supervision of J. C. Kluyver. From 1926 to 1928 he studied at the Universities of Göttingen and Hamburg, and he was an assistant at the University of Münster from 1928-1930. Kloosterman was appointed lector (associate professorship) at Leiden University in 1930 and full professor in 1947. In 1950 he was elected a member of the Royal Netherlands Academy of Arts and Sciences.
References
External links
1900 births
1968 deaths
20th-century Dutch mathematicians
Leiden University alumni
Academic staff of Leiden University
Members of the Royal Netherlands Academy of Arts and Sciences
Number theorists
People from Smallingerland
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https://en.wikipedia.org/wiki/Homogeneous%20%28large%20cardinal%20property%29
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In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function f if f is constant in finite subsets of S. More precisely, given a set D, let be the set of all finite subsets of D (see Powerset#Subsets of limited cardinality) and let be a function defined in this set. On these conditions, S is homogeneous for f if, for every natural number n, f is constant in the set . That is, f is constant on the unordered n-tuples of elements of S.
See also
Ramsey's theorem
Ramsey_cardinal
Large cardinals
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https://en.wikipedia.org/wiki/Monty%20Hall%20problem
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The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:
Savant's response was that the contestant should switch to the other door. Under the standard assumptions, the switching strategy has a probability of winning the car, while the strategy of sticking with the initial choice has only a probability.
When the player first makes their choice, there is a chance that the car is behind one of the doors not chosen. This probability does not change after the host reveals a goat behind one of the unchosen doors. When the host provides information about the two unchosen doors (revealing that one of them does not have the car behind it), the chance of the car being behind one of the unchosen doors rests on the unchosen and unrevealed door, as opposed to the chance of the car being behind the door the contestant chose initially.
The given probabilities depend on specific assumptions about how the host and contestant choose their doors. A key insight is that, under these standard conditions, there is more information about doors 2 and 3 than was available at the beginning of the game when door 1 was chosen by the player: the host's action adds value to the door not eliminated, but not to the one chosen by the contestant originally. Another insight is that switching doors is a different action from choosing between the two remaining doors at random, as the former action uses the previous information and the latter does not. Other possible behaviors of the host than the one described can reveal different additional information, or none at all, and yield different probabilities.
Many readers of Savant's column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling Savant wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating Savant's predicted result.
The problem is a paradox of the veridical type, because the solution is so counterintuitive it can seem absurd but is nevertheless demonstrably true. The Monty Hall problem is mathematically closely related to the earlier Three Prisoners problem and to the much older Bertrand's box paradox.
Paradox
Steve Selvin wrote a letter to the American Statistician in 1975, describing a problem based on the game show Let's Ma
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https://en.wikipedia.org/wiki/Lexicographic%20product
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In mathematics, a lexicographical or lexicographic product may be formed of
graphs – see lexicographic product of graphs
orders – see lexicographical order
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https://en.wikipedia.org/wiki/David%20J.%20Asher
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David J. Asher (born 1966, Edinburgh) is a British astronomer, who works at the Armagh Observatory (IAU code 981) in Northern Ireland.
He studied mathematics at Cambridge and received his doctorate from Oxford.
He is known for the meteor research that he conducts with Robert McNaught.
In 1999 and 2000, they accurately gauged when the Leonids meteor shower would peak, while underestimating the peak intensities.
The Mars-crosser asteroid 6564 Asher, discovered by Robert McNaught in 1992, was named in his honor.
References
External links
David Asher at star.arm.ac.uk
1966 births
21st-century British astronomers
20th-century British astronomers
Discoverers of asteroids
Living people
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https://en.wikipedia.org/wiki/Arnold%20Zellner
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Arnold Zellner (January 2, 1927 – August 11, 2010) was an American economist and statistician specializing in the fields of Bayesian probability and econometrics. Zellner contributed pioneering work in the field of Bayesian analysis and econometric modeling.
In Bayesian analysis, Zellner not only provided many applications of it but also a new information-theoretic derivation of rules that are 100% efficient information processing rules — this class includes Bayes's theorem. In econometric modeling, he, in association with Franz Palm, developed the structural time-series approach for constructing new models and for checking the adequacy of old models. In addition, he was involved in many important applied econometric and statistical studies.
Born in Brooklyn, New York, to Ukrainian immigrant parents, Zellner earned his A.B. in physics from Harvard University in 1949 and his Ph.D. in economics from the University of California, Berkeley, under supervision of George Kuznets, in 1957. He held honorary degrees from the Autonomous University of Madrid in Spain, the Universidade Técnica de Lisboa in Portugal, the University of Kiel in Germany, and the Erasmus School of Economics at Erasmus University Rotterdam in the Netherlands.
He was H.G.B. Alexander Distinguished Service Professor Emeritus of Economics and Statistics at the Graduate School of Business of the University of Chicago. He was the founder of the International Society for Bayesian Analysis and also served as President of the American Statistical Association in 1991.
He died on August 11, 2010, in his home in Hyde Park, Chicago.
Bibliography
References
External links
Arnold Zellner's webpage
ET Interviews: Professor Arnold Zellner on the Econometric Theory page.
An Interview with Arnold Zellner (2004), Kathy Morrison
1927 births
2010 deaths
American people of Ukrainian descent
Econometricians
American economists
American statisticians
Harvard College alumni
Presidents of the American Statistical Association
Fellows of the American Statistical Association
Bayesian statisticians
Bayesian econometricians
Fellows of the Econometric Society
Distinguished Fellows of the American Economic Association
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https://en.wikipedia.org/wiki/Antiderivative%20%28complex%20analysis%29
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In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .
As such, this concept is the complex-variable version of the antiderivative of a real-valued function.
Uniqueness
The derivative of a constant function is the zero function. Therefore, any constant function is an antiderivative of the zero function. If is a connected set, then the constant functions are the only antiderivatives of the zero function. Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of (those constants need not be equal).
This observation implies that if a function has an antiderivative, then that antiderivative is unique up to addition of a function which is constant on each connected component of .
Existence
One can characterize the existence of antiderivatives via path integrals in the complex plane, much like the case of functions of a real variable. Perhaps not surprisingly, g has an antiderivative f if and only if, for every γ path from a to b, the path integral
Equivalently,
for any closed path γ.
However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions of a complex variable. For example, consider the reciprocal function, g(z) = 1/z which is holomorphic on the punctured plane C\{0}. A direct calculation shows that the integral of g along any circle enclosing the origin is non-zero. So g fails the condition cited above. This is similar to the existence of potential functions for conservative vector fields, in that Green's theorem is only able to guarantee path independence when the function in question is defined on a simply connected region, as in the case of the Cauchy integral theorem.
In fact, holomorphy is characterized by having an antiderivative locally, that is, g is holomorphic if for every z in its domain, there is some neighborhood U of z such that g has an antiderivative on U. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.
Various versions of Cauchy integral theorem, an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic g,
vanishes for any closed path γ (which may be, for instance, that the domain of g be simply connected or star-convex).
Necessity
First we show that if f is an antiderivative of g on U, then g has the path integral property given above. Given any piecewise C1 path γ : [a, b] → U, one can express
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https://en.wikipedia.org/wiki/Newman%27s%20lemma
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In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent.
Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph.
Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980. Newman's original proof was considerably more complicated.
Diamond lemma
In general, Newman's lemma can be seen as a combinatorial result about binary relations → on a set A (written backwards, so that a → b means that b is below a) with the following two properties:
→ is a well-founded relation: every non-empty subset X of A has a minimal element (an element a of X such that a → b for no b in X). Equivalently, there is no infinite chain . In the terminology of rewriting systems, → is terminating.
Every covering is bounded below. That is, if an element a in A covers elements b and c in A in the sense that and , then there is an element d in A such that and , where denotes the reflexive transitive closure of →. In the terminology of rewriting systems, → is locally confluent.
The lemma states that if the above two conditions hold, then → is confluent: whenever and , there is an element d such that and . In view of the termination of →, this implies that every connected component of → as a graph contains a unique minimal element a, moreover for every element b of the component.
Notes
References
M. H. A. Newman. On theories with a combinatorial definition of "equivalence". Annals of Mathematics, 43, Number 2, pages 223–243, 1942.
Textbooks
Term Rewriting Systems, Terese, Cambridge Tracts in Theoretical Computer Science, 2003. (book weblink)
Term Rewriting and All That, Franz Baader and Tobias Nipkow, Cambridge University Press, 1998 (book weblink)
John Harrison, Handbook of Practical Logic and Automated Reasoning, Cambridge University Press, 2009, , chapter 4 "Equality".
External links
"Newman's Proof of Newman's Lemma", a PDF on the original proof,
Wellfoundedness
Lemmas
Rewriting systems
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https://en.wikipedia.org/wiki/Division%20No.%202%2C%20Saskatchewan
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Division No. 2 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the south-southeastern part of the province, on the United States border. The most populous community in this division is Weyburn.
Demographics
In the 2021 Canadian census conducted by Statistics Canada, Division No. 2 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 2.
Cities
Weyburn
Towns
Bengough
Midale
Milestone
Ogema
Radville
Yellow Grass
Villages
Avonlea
Ceylon
Creelman
Fillmore
Goodwater
Halbrite
Lang
Macoun
McTaggart
Minton
Osage
Pangman
Torquay
Rural municipalities
RM No. 6 Cambria
RM No. 7 Souris Valley
RM No. 8 Lake Alma
RM No. 9 Surprise Valley
RM No. 10 Happy Valley
RM No. 36 Cymri
RM No. 37 Lomond
RM No. 38 Laurier
RM No. 39 The Gap
RM No. 40 Bengough
RM No. 66 Griffin
RM No. 67 Weyburn
RM No. 68 Brokenshell
RM No. 69 Norton
RM No. 70 Key West
RM No. 96 Fillmore
RM No. 97 Wellington
RM No. 98 Scott
RM No. 99 Caledonia
RM No. 100 Elmsthorpe
Indian reserves
Piapot Cree Nation
Piapot Cree First Nation 75H
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
SARM Division No. 2
Notes
References
Division. No. 2, Saskatchewan Statistics Canada
02
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https://en.wikipedia.org/wiki/Herbrand%20quotient
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In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
Definition
If G is a finite cyclic group acting on a G-module A, then the cohomology groups Hn(G,A) have period 2 for n≥1; in other words
Hn(G,A) = Hn+2(G,A),
an isomorphism induced by cup product with a generator of H2(G,Z). (If instead we use the Tate cohomology groups then the periodicity extends down to n=0.)
A Herbrand module is an A for which the cohomology groups are finite. In this case, the Herbrand quotient h(G,A) is defined to be the quotient
h(G,A) = |H2(G,A)|/|H1(G,A)|
of the order of the even and odd cohomology groups.
Alternative definition
The quotient may be defined for a pair of endomorphisms of an Abelian group, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q(f,g) is defined as
if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A, then we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 + ... .
Properties
The Herbrand quotient is multiplicative on short exact sequences. In other words, if
0 → A → B → C → 0
is exact, and any two of the quotients are defined, then so is the third and
h(G,B) = h(G,A)h(G,C)
If A is finite then h(G,A) = 1.
For A is a submodule of the G-module B of finite index, if either quotient is defined then so is the other and they are equal: more generally, if there is a G-morphism A → B with finite kernel and cokernel then the same holds.
If Z is the integers with G acting trivially, then h(G,Z) = |G|
If A is a finitely generated G-module, then the Herbrand quotient h(A) depends only on the complex G-module C⊗A (and so can be read off from the character of this complex representation of G).
These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.
See also
Class formation
Notes
References
See section 8.
Algebraic number theory
Abelian group theory
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https://en.wikipedia.org/wiki/Euler%20operator
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In mathematics Euler operators may refer to:
Euler–Lagrange differential operators d/dx: see Lagrangian system
Cauchy–Euler operators e.g. x·d/dx
quantum white noise conservation or QWN-Euler operator
Euler operator (digital geometry), a local operation on a mesh which preserves topology
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https://en.wikipedia.org/wiki/Meteorite%20fall%20statistics
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Meteorite fall statistics are frequently used by planetary scientists to approximate the true flux of meteorites on Earth. Meteorite falls are those meteorites that are collected soon after being witnessed to fall, whereas meteorite finds are discovered at a later time. Although there are 30 times as much finds than falls, their raw distribution of types does not accurately reflect what falls to Earth. The reasons for this include the following:
Some meteorite types are easier to find than others.
Some meteorite types are degraded by weathering more quickly than others.
Some meteorites, especially iron meteorites, may have been collected by people in the past who recognized them as being unusual and/or useful, thereby removing them from the scientific record.
Many meteorites fall as showers of many stones, but when they are collected long after the event it may be difficult to tell which ones were part of the same fall.
Many meteorites are found by people who sell meteorites... valuable, rare types become known to science quickly, while those of low value may never be described.
There have been many attempts to correct statistical analyses of meteorite finds for some of these effects, especially to estimate the frequency with which rare meteorite types fall. For example, there are over 100 known lunar meteorite finds, but none has ever been observed to fall. However, for abundant types, meteorite fall statistics are generally preferred.
These statistics are current through June 9, 2012.
Statistics by material
For most meteorite falls, even those that occurred long ago or for which material has never received complete scientific characterization, it is known whether the object was a stone, stony iron, or iron meteorite. Here are the numbers and percentages of each type, based on literature data.
Statistics by major category
The traditional way of subdividing meteorites (see Meteorites classification) is into irons, stony-irons, and two major groups of stony meteorites, chondrites and achondrites. For some of the less-studied stony meteorite falls, it is not known whether the object is chondritic; thus the number of meteorites that can be so grouped is 4% lower than shown above. These numbers are shown in the next table. One could make a slight correction for the undercounting of stony meteorites (e.g., the percentage of irons would decrease by a 0.2%), but this was not done.
Statistics by meteorite group
Probably the most useful statistical breakdown of meteorite falls is by group, which is the fundamental way that meteorites are classified. About 5% of the meteorites in the table just above have not been sufficiently classified to allow them to be put into such groups. Again, a small adjustment could be made to the percentages to correct for this effect, but it does not greatly change the results. Note that a number of meteorite groups are only represented by a small number of falls; the percentages of falls belonging to these gro
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https://en.wikipedia.org/wiki/Cuspidal%20representation
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In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups.
When the group is the general linear group , the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.
Formulation
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. The group G(K) embeds diagonally in the group G(A) by sending g in G(K) to the tuple (gp)p in G(A) with g = gp for all (finite and infinite) primes p. Let Z denote the center of G and let ω be a continuous unitary character from Z(K) \ Z(A)× to C×.
Fix a Haar measure on G(A) and let L20(G(K) \ G(A), ω) denote the Hilbert space of complex-valued measurable functions, f, on G(A) satisfying
f(γg) = f(g) for all γ ∈ G(K)
f(gz) = f(g)ω(z) for all z ∈ Z(A)
for all unipotent radicals, U, of all proper parabolic subgroups of G(A) and g ∈ G(A).
The vector space L20(G(K) \ G(A), ω) is called the space of cusp forms with central character ω on G(A). A function appearing in such a space is called a cuspidal function.
A cuspidal function generates a unitary representation of the group G(A) on the complex Hilbert space generated by the right translates of f. Here the action of g ∈ G(A) on is given by
.
The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations of L20(G(K) \ G(A), ω) and the m are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (, V) for some ω.
The groups for which the multiplicities m all equal one are said to have the multiplicity-one property.
See also
Jacquet module
References
James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Section 5 of Lecture 2.
Representation theory of algebraic groups
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https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac%20theorem
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In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of
is the standard normal distribution. ( is sequence A001221 in the OEIS.) This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size .
Precise statement
For any fixed a < b,
where is the normal (or "Gaussian") distribution, defined as
More generally, if f(n) is a strongly additive function () with for all prime p, then
with
Kac's original heuristic
Intuitively, Kac's heuristic for the result says that if n is a randomly chosen large integer, then the number of distinct prime factors of n is approximately normally distributed with mean and variance log log n. This comes from the fact that given a random natural number n, the events "the number n is divisible by some prime p" for each p are mutually independent.
Now, denoting the event "the number n is divisible by p" by , consider the following sum of indicator random variables:
This sum counts how many distinct prime factors our random natural number n has. It can be shown that this sum satisfies the Lindeberg condition, and therefore the Lindeberg central limit theorem guarantees that after appropriate rescaling, the above expression will be Gaussian.
The actual proof of the theorem, due to Erdős, uses sieve theory to make rigorous the above intuition.
Numerical examples
The Erdős–Kac theorem means that the construction of a number around one billion requires on average three primes.
For example, 1,000,000,003 = 23 × 307 × 141623. The following table provides a numerical summary of the growth of the average number of distinct prime factors of a natural number with increasing .
Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% are constructed from between 7 and 13 primes.
A hollow sphere the size of the planet Earth filled with fine sand would have around 1033 grains. A volume the size of the observable universe would have around 1093 grains of sand. There might be room for 10185 quantum strings in such a universe.
Numbers of this magnitude—with 186 digits—would require on average only 6 primes for construction.
It is very difficult if not impossible to discover the Erdös-Kac theorem empirically, as the Gaussian only shows up when starts getting to be around . More precisely, Rényi and Turán showed that the best possible uniform asymptotic bound on the error in the approximation to a Gaussian is
References
External links
Timothy Gowers: The Importance of Mathematics (part 6, 4 mins in) and (part 7)
Kac theorem
Normal distribution
Theorems about prime numbers
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https://en.wikipedia.org/wiki/Vidar%20Nisja
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Vidar Nisja (born 21 August 1986) was a Norwegian midfielder who retired 18 December 2018.
Career statistics
Personal
Nisja has actively participated in KRIK work and has studied at Stavanger Mission College.
References
External links
1986 births
Living people
People from Hå
Norwegian men's footballers
Norway men's youth international footballers
Norway men's under-21 international footballers
Bryne FK players
Viking FK players
Sandnes Ulf players
Norwegian First Division players
Eliteserien players
Men's association football midfielders
Footballers from Rogaland
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https://en.wikipedia.org/wiki/Johan%20Wikmanson
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Johan Wikmanson (28 December 1753 – 10 January 1800) was a Swedish organist and composer.
Biography
Wikmanson was born in Stockholm and, except for 18 months spent in Copenhagen studying mathematics and instrument making, lived his entire life in the Swedish capital. He was reputed to be a superb organist and for many years held the post of organist at the Storkyrkan, Stockholm's principal church. He was also an accomplished cellist. His teachers included Abbé Georg Joseph Vogler and Joseph Martin Kraus.
Nonetheless, like most Swedish musicians of this era, he was unable to earn his living solely as a practicing musician and was forced to find employment as a government accountant, working for the Royal Swedish National Lottery. He did, however, obtain some recognition during his lifetime. In 1788, he was made a member of the Swedish Royal Academy and later was put in charge of its music program.
As a composer, Wikmanson is remembered for his five string quartets, all published after his early death from tuberculosis in 1800. His close friend, Gustav Silverstolpe, published what he felt were the three best at his own expense, titling them Opus 1. Later, Silverstolpe gave the rights to the well-known German publisher Breitkopf and Härtel, hoping they would publish the quartets and hence give them wider circulation. However, this appears not to have happened. No new edition appeared for more than 170 years. In the 1970s the Swedish firm of Edition Reimers published all three quartets of the Op.1 and recently (2006) Edition Silvertrust brought out a new edition of String Quartet No.1 in d minor, Op.1 No.1
It is not known exactly when Wikmanson composed the Opus 1 string quartets, as Silverstolpe called them. They were not, however, his first work, and probably were among his last works. Of the five quartets, most scholars believe the so-called First Quartet was probably his fifth and last. Evidence supports this, as Silverstolpe placed what he considered the strongest work first in the set of three that he published. This was common practice, because it was generally felt that the first work of a published set had to be strong to interest players in the others in the set. The weakest was usually placed in the middle and another strong work at the end. The Op.1 Quartets were dedicated to Haydn, albeit posthumously. Though Wikmanson did not know Haydn personally, it is clear that he was familiar with Haydn's quartets, including the Op.76, published in 1799, the year before his death. Haydn for his part, was impressed by these works and tried, unsuccessfully, to stimulate interest in them.
String Quartet No.1 is in four movements—Allegro—Adagio—Minuetto and Allegro. Critics consider it the equal of any of Haydn's Op.64 quartets and, in some ways, in advance of them, particularly in its excellent use of the viola and cello. The most striking movement is the Adagio, a powerful funeral march—which was performed at Wikmanson's own funeral. It is remi
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https://en.wikipedia.org/wiki/Average%20and%20over
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Average and over, often abbreviated A&O, refers to two baseball statistics used in the 1850s and 1860s by the National Association of Base Ball Players. They referred to a player's average performance over a number of games, and were among the first baseball statistics ever reported and tracked. The term and the reporting method were borrowed from cricket.
The basic statistic was presented as a whole number (the "average") and a remainder (the "over").
Average and over (runs) was the average number of runs a player scored per game, expressed as a whole number and a remainder. If a player scored 29 runs in nine games, his average runs per game would be three, with two left over. This would be written as 3,2.
Average and over (hands lost) was the number of times a player was called out, divided by the number of games he played, once again expressed as a whole number and a remainder. If a player was called out 17 times in eight games, his average and over for hands lost would be 2,1.
When statistics for hits and total bases were introduced in 1868, their totals were expressed in the same way.
In 1870, most teams began presenting these statistics in decimal form. Continuing the examples above, the player with 29 runs in nine games would have this reported as an average of 3.22 Runs per game. The man called out 17 times in eight games would have an average Hands Lost of 2.12.
References
Wright, Marshall. D. "The National Association of Base Ball Players, 1857-1870." Jefferson, North Carolina: McFarland & Company, Inc.: 2000.
Batting statistics
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https://en.wikipedia.org/wiki/Morris%20Berman
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Morris Berman (born August 3, 1944) is an American historian and social critic. He earned a BA in mathematics at Cornell University in 1966 and a PhD in the history of science at Johns Hopkins University in 1971. Berman is an academic humanist cultural critic who specializes in Western cultural and intellectual history.
Life and work
Berman has served on the faculties of a number of universities in the U.S., Canada, and Europe. Berman emigrated from the U.S. to Mexico in 2006, where he was a visiting professor at the Tecnologico de Monterrey in Mexico City from 2008 to 2009. During this period he continued writing for various publications including Parteaguas, a quarterly magazine.
Although an academic, Berman has written several books for a general audience. They deal with the state of Western civilization and with an ethical, historically responsible, or enlightened approach to living within it. His work emphasizes the legacies of the European Enlightenment and the historical place of present-day American culture.
He wrote a trilogy on consciousness and spirituality, published between 1981 and 2000, and another trilogy on the American decline, published between 2000 and 2011. Book reviewer George Scialabba commented:
Participating consciousness
The term participating consciousness was introduced by Berman in The Re-enchantment of the World (1981) expanding on Owen Barfield's concept of "original participation," to describe an ancient mode of human thinking that does not separate the perceiver from the world he or she perceives. Berman says that this original world view has been replaced during the past 400 years with the modern paradigm called Cartesian, Newtonian, or scientific, which depends on an isolated observer, proposing that we can understand the world only by distancing ourselves from it.
Max Weber, early 20th-century German sociologist, was concerned with the "disenchantment" he associated with the rise of modernity, capitalism, and scientific consciousness. Berman traces the history of this disenchantment. He argues that the modern consciousness is destructive to both the human psyche and the planetary environment. Berman challenges the supremacy of the modern world view and argues for some new form of the older holistic tradition, which he describes as follows:
The concept of participating consciousness has been used and further developed by philosophers and analytical psychologists, among others, and the idea of re-enchantment is a recurring theme among scholars. Some compare participating consciousness to the thinking of non-Western indigenous peoples. Others link it to esoteric traditions or religious thought.
Recognition
In 1990, Berman received the Governor's Writers Award (Washington State) for his book Coming to Our Senses. In 1992, he was the recipient of the first annual Rollo May Center Grant for Humanistic Studies. In 2000, Berman's book The Twilight of American Culture was named one of the ten most recommended b
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https://en.wikipedia.org/wiki/Myron%20E.%20Witham
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Myron Ellis Witham (October 29, 1880 – March 7, 1973) was an American football player, coach of football and baseball, and mathematics professor. He served as the head football coach at Purdue University in 1906 and at the University of Colorado at Boulder from 1920 to 1931, compiling a career college football record of 63–31–7. He was also the head baseball coach Colorado from 1920 to 1925, tallying a mark of 29–25. Witham was born in Pigeon Cove, Massachusetts, on October 29, 1880. He attended Dartmouth College and was captain of the football team there in 1903. Witham taught mathematics at Purdue, Colorado, the University of Vermont, and Saint Michael's College. He died on March 7, 1973, in Burlington, Vermont.
Head coaching record
Football
References
1880 births
1973 deaths
Colorado Buffaloes baseball coaches
Colorado Buffaloes football coaches
Dartmouth Big Green football players
Purdue Boilermakers football coaches
All-American college football players
Purdue University faculty
University of Colorado faculty
University of Vermont faculty
People from Rockport, Massachusetts
Coaches of American football from Massachusetts
Players of American football from Essex County, Massachusetts
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https://en.wikipedia.org/wiki/Distributive%20category
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In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects , the canonical map
is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms . It follows that and aforementioned canonical maps are equal for each choice of objects.
In particular, if the functor has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.
Example
The category of sets is distributive. Let , , and be sets. Then
where denotes the coproduct in Set, namely the disjoint union, and denotes a bijection. In the case where , , and are finite sets, this result reflects the distributive property: the above sets each have cardinality .
The categories Grp and Ab are not distributive, even though they have both products and coproducts.
An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.
References
Further reading
Category theory
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https://en.wikipedia.org/wiki/Scale%20analysis
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Scale analysis may refer to:
Scale analysis (mathematics)
Scale analysis (statistics)
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https://en.wikipedia.org/wiki/Scale%20analysis%20%28statistics%29
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In statistics, scale analysis is a set of methods to analyze survey data, in which responses to questions are combined to measure a latent variable. These items can be dichotomous (e.g. yes/no, agree/disagree, correct/incorrect) or polytomous (e.g. disagree strongly/disagree/neutral/agree/agree strongly). Any measurement for such data is required to be reliable, valid, and homogeneous with comparable results over different studies.
Constructing scales
The item-total correlation approach is a way of identifying a group of questions whose responses can be combined into a single measure or scale. This is a simple approach that works by ensuring that, when considered across a whole population, responses to the questions in the group tend to vary together and, in particular, that responses to no individual question are poorly related to an average calculated from the others.
Measurement models
Measurement is the assignment of numbers to subjects in such a way that the relations between the objects are represented by the relations between the numbers (Michell, 1990).
Traditional models
Likert scale
Semantic differential (Osgood) scale
Reliability analysis, see also Classical test theory and Cronbach's alpha
Factor analysis
Modern models based on Item response theory
Guttman scale
Mokken scale
Rasch model
(Circular) Unfolding analysis
Circumplex model
Other models
Latent class analysis
Multidimensional scaling
NOMINATE (scaling method)
References
Michell, J (1990). An Introduction to the logic of Psychological Measurement. Hillsdales, NJ: Lawrences Erlbaum Associates Publ.
Survey methodology
Sampling (statistics)
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https://en.wikipedia.org/wiki/Berlekamp%27s%20algorithm
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In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems.
Overview
Berlekamp's algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree with coefficients in a finite field and gives as output a polynomial with coefficients in the same field such that divides . The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of into powers of irreducible polynomials (recalling that the ring of polynomials over a finite field is a unique factorization domain).
All possible factors of are contained within the factor ring
The algorithm focuses on polynomials which satisfy the congruence:
These polynomials form a subalgebra of R (which can be considered as an -dimensional vector space over ), called the Berlekamp subalgebra. The Berlekamp subalgebra is of interest because the polynomials it contains satisfy
In general, not every GCD in the above product will be a non-trivial factor of , but some are, providing the factors we seek.
Berlekamp's algorithm finds polynomials suitable for use with the above result by computing a basis for the Berlekamp subalgebra. This is achieved via the observation that Berlekamp subalgebra is in fact the kernel of a certain matrix over , which is derived from the so-called Berlekamp matrix of the polynomial, denoted . If then is the coefficient of the -th power term in the reduction of modulo , i.e.:
With a certain polynomial , say:
we may associate the row vector:
It is relatively straightforward to see that the row vector corresponds, in the same way, to the reduction of modulo . Consequently, a polynomial is in the Berlekamp subalgebra if and only if (where is the identity matrix), i.e. if and only if it is in the null space of .
By computing the matrix and reducing it to reduced row echelon form and then easily reading off a basis for the null space, we may find a basis for the Berlekamp subalgebra and hence construct polynomials in it. We then need to successively compute GCDs of the form above until we find a non-trivial factor. Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm.
Conceptual algebraic explanation
With some abstract algebra, the idea behind Berlekamp's algorithm becomes conceptually clear. We represent a finite field , where for some prime , as . We can assume that is square free, by taking all possible pth roots and then computing the gcd with its derivative.
Now, suppose that is the fact
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https://en.wikipedia.org/wiki/Tetradecagon
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In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
Regular tetradecagon
A regular tetradecagon has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges.
The area of a regular tetradecagon of side length a is given by
Construction
As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis with use of the angle trisector, or with a marked ruler, as shown in the following two examples.
Symmetry
The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih7, Dih2, and Dih1, and 4 cyclic group symmetries: Z14, Z7, Z2, and Z1.
These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order. Full symmetry of the regular form is r28 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g14 subgroup has no degrees of freedom but can seen as directed edges.
The highest symmetry irregular tetradecagons are d14, an isogonal tetradecagon constructed by seven mirrors which can alternate long and short edges, and p14, an isotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular tetradecagon.
Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetradecagon, m=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a Petrie polygon projection of a 7-cube, with 21 of 672 faces. The list defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.
Numismatic use
The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, the number of sides representing the 14 states of the Malaysian Federation.
Related figures
A tetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regular star polygons: {14/3} and {14/5}, using the same vertices, but connecting every third or fifth points. There are also three compounds: {14/2} is reduced to 2{7} as two heptagons, while {14/4} and {14/6} are reduced to 2{7/2} a
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https://en.wikipedia.org/wiki/Cantor%E2%80%93Zassenhaus%20algorithm
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In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields).
The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor and Hans Zassenhaus in 1981.
It is arguably the dominant algorithm for solving the problem, having replaced the earlier Berlekamp's algorithm of 1967. It is currently implemented in many computer algebra systems.
Overview
Background
The Cantor–Zassenhaus algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree n with coefficients in a finite field whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, is a squarefree polynomial with the same factors as , so that the Cantor–Zassenhaus algorithm can be used to factor arbitrary polynomials). It gives as output a polynomial with coefficients in the same field such that divides . The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of into powers of irreducible polynomials (recalling that the ring of polynomials over any field is a unique factorisation domain).
All possible factors of are contained within the factor ring
. If we suppose that has irreducible factors , all of degree d, then this factor ring is isomorphic to the direct product of factor rings . The isomorphism from R to S, say , maps a polynomial to the s-tuple of its reductions modulo each of the , i.e. if:
then . It is important to note the following at this point, as it shall be of critical importance later in the algorithm: Since the are each irreducible, each of the factor rings in this direct sum is in fact a field. These fields each have degree .
Core result
The core result underlying the Cantor–Zassenhaus algorithm is the following: If is a polynomial satisfying:
where is the reduction of modulo as before, and if any two of the following three sets is non-empty:
then there exist the following non-trivial factors of :
Algorithm
The Cantor–Zassenhaus algorithm computes polynomials of the same type as above using the isomorphism discussed in the Background section. It proceeds as follows, in the case where the field is of odd-characteristic (the process can be generalised to characteristic 2 fields in a fairly straightforward way ). Select a random polynomial such that . Set and compute . Since is an isomorphism, we have (using our now-established notation):
Now, each is an element of a field of order , as noted earlier. The multiplicative subgroup of this field has order and so, unless , we have for each i and hence for each i. If , then of course . Hence is a polynomial of the same type as above. Further, since , at least two of the sets and C are non-empty and by com
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https://en.wikipedia.org/wiki/Universal%20coding
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Universal coding may refer to one of two concepts in data compression:
Universal code (data compression), a fixed prefix code that, for any probability mass function, has a data compression ratio within a constant of the optimal prefix code
Universal source coding, a data compression method that asymptotically approaches the data compression ratio of the optimal data compression method, e.g., LZ77 and LZ78
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https://en.wikipedia.org/wiki/Almost%20%28disambiguation%29
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Almost is a term in mathematics (especially in set theory) used to mean all the elements except for finitely many.
Almost may also refer to:
Songs
"Almost" (Bowling for Soup song), 2005
"Almost", by DNCE from DNCE, 2016
"Almost" (George Morgan song), 1952
"Almost (Sweet Music)", by Hozier, 2019
"Almost", by Jewel from Freewheelin' Woman, 2022
"Almost", by Orchestral Manoeuvres in the Dark from Orchestral Manoeuvres in the Dark, 1980
"Almost", by Sarah Close, 2019
"Almost", by Sarah Harmer from All of Our Names, 2004
"Almost", by Soraya, a B-side of the single "Casi", 2003
"Almost", by the Staple Singers from The Staple Swingers, 1971
"Almost" (Tamia song), 2007
"Almost", by Thomas Rhett from Center Point Road, 2019
"Almost", by Tracy Chapman from Let It Rain, 2002
Other uses
The Almost, an American Christian rock band
Almost Skateboards, an American skateboard company
See also
Approximation
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https://en.wikipedia.org/wiki/Gheorghe%20Vr%C4%83nceanu
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Gheorghe Vrănceanu (June 30, 1900 – April 27, 1979) was a Romanian mathematician, best known for his work in differential geometry and topology. He was titular member of the Romanian Academy and vice-president of the International Mathematical Union.
Biography
He was born in 1900 in Valea Hogei, then a village in Vaslui County, now a component of Lipova commune, in Bacău County. He was the eldest of five children in his family. After attending primary school in his village and high school in Vaslui, he went to study mathematics at the University of Iași in 1919. There, he took courses with , Vera Myller, , Victor Vâlcovici, and Simion Stoilow. After graduating in 1922, he went in 1923 to the University of Göttingen, where he studied under David Hilbert. Thereafter, he went to the University of Rome, where he studied under Tullio Levi-Civita, obtaining his doctorate on November 5, 1924, with thesis Sopra una teorema di Weierstrass e le sue applicazioni alla stabilita. The thesis defense committee was composed of 11 faculty, and was headed by Vito Volterra.
Vrănceanu returned to Iași, where he was appointed a lecturer at the university. In 1927–1928, he was awarded a Rockefeller Foundation scholarship to study in France and the United States, where he was in a contact with Élie Cartan and Oswald Veblen. In 1929, he returned to Romania, and was appointed professor at the University of Cernăuți. In 1939, he moved to the University of Bucharest, where he was appointed Head of the Geometry and Topology department in 1948, a position he held until his retirement in 1970. His doctoral students include Henri Moscovici and .
Vrănceanu was elected to the Romanian Academy as a corresponding member in 1946, then as a full member in 1955. From 1964 he was president of the Mathematics Section of the Romanian Academy. Also from 1964, he was an editor of the journal Revue Roumaine de mathématiques pures et appliquées, founded that year. At the International Congress of Mathematicians held in Vancouver, Canada in 1974, he was elected vice-president of the International Mathematical Union, a position he held from 1975 to 1978. He died in Bucharest in 1979 of an intestinal obstruction and was buried at the city's Bellu Cemetery.
A high school in Bacău (Colegiul Național "Gheorghe Vrânceanu") is named after him, and so is a school in Lipova.
Research
During his career, Vrănceanu published over 300 articles in journals throughout the world. His work covers a whole range of modern geometry, from the classical theory of surfaces, to the notion of non-holonomic spaces, which he discovered.
In 1928 he gave an invited talk at the International Congress of Mathematicians in Bologna, titled Parallelisme et courbure dans une variété non holonome. In it, he introduced the notion of "non-holonomic manifolds," which are smooth manifolds provided with a smooth distribution that is generally not integrable.
Publications
Notes
References
1900 births
1979 deaths
Peo
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https://en.wikipedia.org/wiki/Spatial%20descriptive%20statistics
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Spatial descriptive statistics is the intersection of spatial statistics and descriptive statistics; these methods are used for a variety of purposes in geography, particularly in quantitative data analyses involving Geographic Information Systems (GIS).
Types of spatial data
The simplest forms of spatial data are gridded data, in which a scalar quantity is measured for each point in a regular grid of points, and point sets, in which a set of coordinates (e.g. of points in the plane) is observed. An example of gridded data would be a satellite image of forest density that has been digitized on a grid. An example of a point set would be the latitude/longitude coordinates of all elm trees in a particular plot of land. More complicated forms of data include marked point sets and spatial time series.
Measures of spatial central tendency
The coordinate-wise mean of a point set is the centroid, which solves the same variational problem in the plane (or higher-dimensional Euclidean space) that the familiar average solves on the real line — that is, the centroid has the smallest possible average squared distance to all points in the set.
Measures of spatial dispersion
Dispersion captures the degree to which points in a point set are separated from each other. For most applications, spatial dispersion should be quantified in a way that is invariant to rotations and reflections. Several simple measures of spatial dispersion for a point set can be defined using the covariance matrix of the coordinates of the points. The trace, the determinant, and the largest eigenvalue of the covariance matrix can be used as measures of spatial dispersion.
A measure of spatial dispersion that is not based on the covariance matrix is the average distance between nearest neighbors.
Measures of spatial homogeneity
A homogeneous set of points in the plane is a set that is distributed such that approximately the same number of points occurs in any circular region of a given area. A set of points that lacks homogeneity may be spatially clustered at a certain spatial scale. A simple probability model for spatially homogeneous points is the Poisson process in the plane with constant intensity function.
Ripley's K and L functions
Ripley's K and L functions introduced by Brian D. Ripley are closely related descriptive statistics for detecting deviations from spatial homogeneity. The K function (technically its sample-based estimate) is defined as
where dij is the Euclidean distance between the ith and jth points in a data set of n points, t is the search radius, λ is the average density of points (generally estimated as n/A, where A is the area of the region containing all points) and I is the indicator function (1 if its operand is true, 0 otherwise). In 2 dimensions, if the points are approximately homogeneous, should be approximately equal to πt2.
For data analysis, the variance stabilized Ripley K function called the L function is generally used. The sa
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https://en.wikipedia.org/wiki/Reversible-jump%20Markov%20chain%20Monte%20Carlo
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In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard Markov chain Monte Carlo (MCMC) methodology, introduced by Peter Green, which allows simulation of the posterior distribution on spaces of varying dimensions.
Thus, the simulation is possible even if the number of parameters in the model is not known.
Let
be a model indicator and the parameter space whose number of dimensions depends on the model . The model indication need not be finite. The stationary distribution is the joint posterior distribution of that takes the values .
The proposal can be constructed with a mapping of and , where is drawn from a random component
with density on . The move to state can thus be formulated as
The function
must be one to one and differentiable, and have a non-zero support:
so that there exists an inverse function
that is differentiable. Therefore, the and must be of equal dimension, which is the case if the dimension criterion
is met where is the dimension of . This is known as dimension matching.
If then the dimensional matching
condition can be reduced to
with
The acceptance probability will be given by
where denotes the absolute value and is the joint posterior probability
where is the normalising constant.
Software packages
There is an experimental RJ-MCMC tool available for the open source BUGs package.
The Gen probabilistic programming system automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of its Involution MCMC feature.
References
Computational statistics
Markov chain Monte Carlo
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https://en.wikipedia.org/wiki/Hans%20Fitting
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Hans Fitting (13 November 1906 in München-Gladbach (now Mönchengladbach) – 15 June 1938 in Königsberg (now Kaliningrad))
was a mathematician who worked in group theory. He proved Fitting's theorem and Fitting's lemma, and defined the Fitting subgroup
in finite group theory and the Fitting decomposition for Lie algebras and Fitting ideals in ring theory.
After finishing his undergraduate work in 1931, he wrote his dissertation with the help of Emmy Noether, who helped him secure a grant from the Notgemeinschaft der Deutschen Wissenschaften (Emergency Society for German Sciences). He died at the age of 31 from a sudden bone disease.
References
External links
Biography (in German)
1906 births
1938 deaths
20th-century German mathematicians
Group theorists
People from the Rhine Province
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https://en.wikipedia.org/wiki/Tate%20cohomology%20group
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In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by , and are used in class field theory.
Definition
If G is a finite group and A a G-module, then there is a natural map N from to
taking a representative a to (the sum over all G-conjugates of a). The Tate cohomology groups are defined by
for ,
quotient of by norms of elements of A,
quotient of norm 0 elements of A by principal elements of A,
for .
Properties
If
is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
If A is an induced G module then all Tate cohomology groups of A vanish.
The zeroth Tate cohomology group of A is
(Fixed points of G on A)/(Obvious fixed points of G acting on A)
where by the "obvious" fixed point we mean those of the form . In other words, the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A.
The Tate cohomology groups are characterized by the three properties above.
Tate's theorem
Tate's theorem gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. There are several slightly different versions of it; a version that is particularly convenient for class field theory is as follows:
Suppose that A is a module over a finite group G and a is an element of , such that for every subgroup E of G
is trivial, and
is generated by , which has order E.
Then cup product with a is an isomorphism:
for all n; in other words the graded Tate cohomology of A is isomorphic to
the Tate cohomology with integral coefficients, with the degree shifted by 2.
Tate-Farrell cohomology
F. Thomas Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension. In Farrell's theory, the groups
are isomorphic to the usual cohomology groups whenever n is greater than the virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell's cohomology groups are the same as those of Tate.
See also
Herbrand quotient
Class formation
References
M. F. Atiyah and C. T. C. Wall, "Cohomology of Groups", in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich , Chapter IV. See section 6.
Class field theory
Homological algebra
Finite groups
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https://en.wikipedia.org/wiki/Rob%20Eastaway
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Rob Eastaway is an English author. He is active in the popularisation of mathematics and was awarded the Zeeman medal in 2017 for excellence in the promotion of maths. He is best known for his books, including the bestselling Why Do Buses Come in Threes? and Maths for Mums and Dads. His first book was What is a Googly?, an explanation of cricket for Americans and other newcomers to the game.
Eastaway is a keen cricketer and was one of the originators of the International Rankings of Cricketers. He is also a puzzle setter and adviser for New Scientist magazine and he has appeared frequently on BBC Radio 4 and 5 Live.
He is the director of Maths Inspiration, a national programme of maths lectures for teenagers which involves some of the UK’s leading maths speakers. He was president of the UK Mathematical Association for 2007/2008. He is a former pupil of The King's School, Chester, and has a degree in engineering and management science from the University of Cambridge.
Books
1992: What is a Googly?
1995: The Guinness Book of Mindbenders, co-author David Wells
1998: Why do Buses Come in Threes?, co-author Jeremy Wyndham, foreword by Tim Rice
1999: The Memory Kit
2002: How Long is a Piece of String?, co-author Jeremy Wyndham
2004: How to Remember
2005: How to Take a Penalty, co-author John Haigh
2007: How to Remember (Almost) Everything Ever
2007: Out of the Box
2008: How Many Socks Make a Pair?
2009: Improve Your Memory Today, with Dr Hilary Jones
2010: Maths for Mums and Dads, co-author Mike Askew
2011: The Hidden Mathematics of Sport (new edition of Beating the Odds)
2013: More Maths for Mums and Dads, co-author Mike Askew
2016: Maths on the Go, co-author Mike Askew
2017: Any ideas? Tips and Techniques to Help You Think Creatively
2018: 100 Maddening Mindbending Puzzles
2019: Maths On The Back of an Envelope
References
External links
Rob Eastaway's Official Website
Maths Inspiration Website
Alumni of Christ's College, Cambridge
Living people
Mathematics popularizers
People educated at The King's School, Chester
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Jonathan%20Mestel
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Andrew Jonathan Mestel (born 13 March 1957 in Cambridge, England) is British mathematician and chess player. He holds the position of Professor of Applied Mathematics at Imperial College London. He worked on magnetohydrodynamics and biological fluid dynamics. He obtained his PhD with the thesis "Magnetic Levitation of Liquid Metals" at University of Cambridge.
A distinguished chess player, he was the first person to be awarded chess Grandmaster titles by FIDE in both over-the-board play and problem solving. He has also represented England at contract bridge.
He announced his arrival on the international chess scene by winning the World Cadet Championship in 1974 at Pont-Sainte-Maxence. In the same year he nearly won the British Chess Championship, figuring in a seven-way play-off at Clacton, but failing to clinch the title at the last hurdle. Playing in Tjentiste in 1975, he took the bronze medal at the World Junior Championship, finishing behind Valery Chekhov and Larry Christiansen. There followed a string of British Championship successes, where he took the title in 1976, 1983 and 1988. His victory at Portsmouth in 1976 was remarkable for a start of nine consecutive wins, a record for the competition.
Along the way, Mestel was awarded the Grandmaster (GM) title in 1982 and became a Chess Solving Grandmaster and the World Chess Solving Champion in 1997. With fellow GM John Nunn, he is a medal-winning member of the British Chess Solving Team.
Between 1976 and 1988 he was a frequent member of the English Chess Olympiad squad, winning three team medals (two silver and one bronze). In 1984, he earned an individual gold medal for an outstanding (7/9, 78%) performance on his board. Other notable results for English teams occurred in 1978 at the World Student Olympiad in Mexico and at the 1983 European Team Chess Championship in Plovdiv. Both of these events yielded gold-medal-winning performances, the latter being exceptional for the highest percentage score (6/7, 85%) on any board. As a player of league chess, he has been a patron of the 4NCL since its earliest days and represented The Gambit ADs in the 2008/9 season.
In international tournaments, his best early result was equal second at London 1977, tying with Miguel Quinteros and Michael Stean, behind winner Vlastimil Hort. There were also good results at Esbjerg, where he won the North Sea Cup in 1979 (with Laszlo Vadasz) and finished with a share of second in 1984 (after Nigel Short, with Lars Karlsson). At Marbella in 1982, he was the co-winner of a zonal tournament, with Nunn, Stean and John van der Wiel. At Hastings, there were good results in 1977/78 (a share of fifth in a strong field) and in 1983/84, when he shared third place (after joint winners Karlsson and Jon Speelman).
Aside from his academic and chess activities, he wrote the mainframe computer game Brand X with Peter Killworth, which was later rewritten for Microsoft Windows and released commercially as Philosopher's Quest.
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https://en.wikipedia.org/wiki/Transcendental%20equation
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In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function.
Examples include:
A transcendental equation need not be an equation between elementary functions, although most published examples are.
In some cases, a transcendental equation can be solved by transforming it into an equivalent algebraic equation.
Some such transformations are sketched below; computer algebra systems may provide more elaborated transformations.
In general, however, only approximate solutions can be found.
Transformation into an algebraic equation
Ad hoc methods exist for some classes of transcendental equations in one variable to transform them into algebraic equations which then might be solved.
Exponential equations
If the unknown, say x, occurs only in exponents:
applying the natural logarithm to both sides may yield an algebraic equation, e.g.
transforms to , which simplifies to , which has the solutions
This will not work if addition occurs "at the base line", as in
if all "base constants" can be written as integer or rational powers of some number q, then substituting y=qx may succeed, e.g.
transforms, using y=2x, to which has the solutions , hence is the only real solution.
This will not work if squares or higher power of x occurs in an exponent, or if the "base constants" do not "share" a common q.
sometimes, substituting y=xex may obtain an algebraic equation; after the solutions for y are known, those for x can be obtained by applying the Lambert W function, e.g.:
transforms to which has the solutions hence , where and the denote the real-valued branches of the multivalued function.
Logarithmic equations
If the unknown x occurs only in arguments of a logarithm function:
applying exponentiation to both sides may yield an algebraic equation, e.g.
transforms, using exponentiation to base to which has the solutions If only real numbers are considered, is not a solution, as it leads to a non-real subexpression in the given equation.
This requires the original equation to consist of integer-coefficient linear combinations of logarithms w.r.t. a unique base, and the logarithm arguments to be polynomials in x.
if all "logarithm calls" have a unique base and a unique argument expression then substituting may lead to a simpler equation, e.g.
transforms, using to which is algebraic and has the single solution . After that, applying inverse operations to the substitution equation yields
Trigonometric equations
If the unknown x occurs only as argument of trigonometric functions:
applying Pythagorean identities and trigonometric sum and multiple formulas, arguments of the forms with integer might all be transformed to arguments of the form, say, . After that, substituting yields an algebraic equation, e.g.
transforms to , and, after substitution, to which is algebraic and can be solved. Af
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https://en.wikipedia.org/wiki/Biholomorphism
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In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definition
Formally, a biholomorphic function is a function defined on an open subset U of the -dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set in Cn and the inverse is also holomorphic. More generally, U and V can be complex manifolds. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11 or Corollary E.10 pg. 57).
If there exists a biholomorphism , we say that U and V are biholomorphically equivalent or that they are biholomorphic.
Riemann mapping theorem and generalizations
If every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (this is the Riemann mapping theorem). The situation is very different in higher dimensions. For example, open unit balls and open unit polydiscs are not biholomorphically equivalent for In fact, there does not exist even a proper holomorphic function from one to the other.
Alternative definitions
In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is biholomorphic. Notice that per definition of biholomorphisms, nothing is assumed about their derivatives, so, this equivalence contains the claim that a homeomorphism that is complex differentiable must actually have nonzero derivative everywhere. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, but without requiring that the map be injective. According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. For example, if f: U → U is defined by f(z) = z2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1.
References
Several complex variables
Algebraic geometry
Complex manifolds
Functions and mappings
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https://en.wikipedia.org/wiki/Kevin%20Lano
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Kevin C. Lano (born 1963) is a British computer scientist.
Life and work
Kevin Lano studied at the University of Reading, attaining a first class degree in Mathematics and Computer Science, and the University of Bristol where he completed his doctorate. He was an originator of formal object-oriented techniques (Z++), and developed a combination of UML and formal methods in a number of papers and books. He was one of the founders of the Precise UML group, who influenced the definition of UML 2.0.
Lano published the book Advanced Systems Design with Java, UML and MDA (Butterworth-Heinemann, ) in 2005. He is also the editor of UML 2 Semantics and Applications, published by Wiley in October 2009, among a number of computer science books.
Lano was formerly a Research Officer at the Oxford University Computing Laboratory (now the Oxford University Department of Computer Science). He is a reader at the Department of Informatics at King's College London.
In 2008, Lano and his co-authors Andy Evans, Robert France, and Bernard Rumpe, were awarded the Ten Year Most Influential Paper Award at the MODELS 2008 Conference on Model Driven Engineering Languages and Systems for the 1998 paper "The UML as a Formal Modeling Notation".
Selected publications
Books
Reverse Engineering and Software Maintenance (McGraw-Hill, 1993)
Object-oriented Specification Case Studies (Prentice Hall, 1993)
Formal Object-oriented Development (Springer, 1995)
The B Language and Method: A Guide to Practical Formal Development (Springer, 1996)
Software Design in Java 2 (Palgrave, 2002)
UML 2 Semantics and Applications (Wiley, 2009), editor
Model-Driven Development using UML and Java (Cengage, 2009)
Agile MBD using UML-RSDS (Taylor & Francis, 2016)
Financial Software Engineering (Springer, 2019), with Howard Haughton
References
External links
Kevin Lano home page
1963 births
Living people
Alumni of the University of Reading
Alumni of the University of Bristol
English computer scientists
Formal methods people
Computer science writers
Members of the Department of Computer Science, University of Oxford
Academics of King's College London
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https://en.wikipedia.org/wiki/Pivotal%20quantity
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In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.
More formally, let be a random sample from a distribution that depends on a parameter (or vector of parameters) . Let be a random variable whose distribution is the same for all . Then is called a pivotal quantity (or simply a pivot).
Pivotal quantities are commonly used for normalization to allow data from different data sets to be compared. It is relatively easy to construct pivots for location and scale parameters: for the former we form differences so that location cancels, for the latter ratios so that scale cancels.
Pivotal quantities are fundamental to the construction of test statistics, as they allow the statistic to not depend on parameters – for example, Student's t-statistic is for a normal distribution with unknown variance (and mean). They also provide one method of constructing confidence intervals, and the use of pivotal quantities improves performance of the bootstrap. In the form of ancillary statistics, they can be used to construct frequentist prediction intervals (predictive confidence intervals).
Examples
Normal distribution
One of the simplest pivotal quantities is the z-score; given a normal distribution with mean and variance , and an observation x, the z-score:
has distribution – a normal distribution with mean 0 and variance 1. Similarly, since the n-sample sample mean has sampling distribution the z-score of the mean
also has distribution Note that while these functions depend on the parameters – and thus one can only compute them if the parameters are known (they are not statistics) – the distribution is independent of the parameters.
Given independent, identically distributed (i.i.d.) observations from the normal distribution with unknown mean and variance , a pivotal quantity can be obtained from the function:
where
and
are unbiased estimates of and , respectively. The function is the Student's t-statistic for a new value , to be drawn from the same population as the already observed set of values .
Using the function becomes a pivotal quantity, which is also distributed by the Student's t-distribution with degrees of freedom. As required, even though appears as an argument to the function , the distribution of does not depend on the parameters or of the normal probability distribution that governs the observations .
This can be used to compute a prediction interval for the next observation see Prediction interval: Normal distribution.
Bivariate normal distribution
In more complicated cases, it is impossible to construct exact pivots. However, having
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https://en.wikipedia.org/wiki/Psephos
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Psephos: Adam Carr's Electoral Archive is an online archive of election statistics, and claims to be the world's largest online resource of such information. Psephos is maintained by Dr Adam Carr, of Melbourne, Australia, a historian and former aide to Australian MP Michael Danby and Senator David Feeney. It includes detailed statistics for presidential and legislative elections from 182 countries, with at least some statistics for every country that has what Carr considers to be genuine national elections.
"Psephos" is a Greek word meaning "pebble", a reference to the Ancient Greek method of voting by dropping pebbles into urns, and is the root of the word psephology, the study of elections.
Carr began accumulating Australian election statistics in the mid-1980s, with the intention of publishing a complete print edition of Australian national elections statistics dating back to 1901. With the advent of the World Wide Web, Carr abandoned this idea and began to place election statistics at his personal website. In 2001 he founded Psephos, and began to include statistics from other countries. He has also included historical statistics, such as figures for all U.S. presidential elections and for British House of Commons elections since 1900.
In 2015 Carr added a complete archive of election statistics for the Australian state of Victoria, dating back to 1843.
Psephos is more noted, however, for locating less easily accessible statistics and placing them online. These include constituency-level figures for Japanese legislative elections, which are not available in English anywhere else on the internet, and figures from countries as obscure as Andorra, Equatorial Guinea, São Tomé and Príncipe and Tuvalu. These figures are obtained from a variety of sources, including government election websites, media websites and political party websites. Increasingly, statistics are sent directly to Carr by psephologists, academics and political activists in the countries concerned.
The archive also includes Carr's original electoral maps at constituency level for a number or countries, including Australia, Canada, France, Germany, India, Japan, Mexico and the U.S.
Matthew M. Singer, a specialist in election studies at Duke University, wrote:
Psephos: Adam Carr's Electoral Archive is maintained by Australian journalist Adam Carr and contains election results for 163 countries as well as many sub-national entities. While not all the results are at the district level, in many cases the Psephos archive is the only online source for electoral data (especially for small countries). Another strength of the Psephos archive is that while many national-sources remove the links to previous election results as the new elections approach, the Psephos archive does not delete old-election results.
This archive, however, has two small drawbacks. First, small parties are often compiled into an "Other" category that for some studies, especially those interested in representa
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https://en.wikipedia.org/wiki/Completely%20positive%20map
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In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let and be C*-algebras. A linear map is called positive map if maps positive elements to positive elements: .
Any linear map induces another map
in a natural way. If is identified with the C*-algebra of -matrices with entries in , then acts as
is called k-positive if is a positive map and completely positive if is k-positive for all k.
Properties
Positive maps are monotone, i.e. for all self-adjoint elements .
Since for all self-adjoint elements , every positive map is automatically continuous with respect to the C*-norms and its operator norm equals . A similar statement with approximate units holds for non-unital algebras.
The set of positive functionals is the dual cone of the cone of positive elements of .
Examples
Every *-homomorphism is completely positive.
For every linear operator between Hilbert spaces, the map is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional (in particular every state) is automatically completely positive.
Given the algebras and of complex-valued continuous functions on compact Hausdorff spaces , every positive map is completely positive.
The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let denote this map on . The following is a positive matrix in : The image of this matrix under is which is clearly not positive, having determinant −1. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.) Incidentally, a map Φ is said to be co-positive if the composition Φ T is positive. The transposition map itself is a co-positive map.
See also
Choi's theorem on completely positive maps
References
C*-algebras
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https://en.wikipedia.org/wiki/Human%20settlement
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In geography, statistics and archaeology, a settlement, locality or populated place is a community of people living in a particular place. The complexity of a settlement can range from a minuscule number of dwellings grouped together to the largest of cities with surrounding urbanized areas. Settlements may include hamlets, villages, towns and cities. A settlement may have known historical properties such as the date or era in which it was first settled, or first settled by particular people. The process of settlement involves human migration.
In the field of geospatial predictive modeling, settlements are "a city, town, village or other agglomeration of buildings where people live and work".
A settlement conventionally includes its constructed facilities such as roads, enclosures, field systems, boundary banks and ditches, ponds, parks and woodlands, wind and water mills, manor houses, moats and churches.
An unincorporated area is a related designation used in the United States.
History
The earliest geographical evidence of a human settlement was Jebel Irhoud, where early modern human remains of eight individuals date back to the Middle Paleolithic around 300,000 years ago.
The oldest remains that have been found of constructed dwellings are remains of huts that were made of mud and branches around 17,000 BC at the Ohalo site (now underwater) near the edge of the Sea of Galilee. The Natufians built houses, also in the Levant, around 10,000 BC. Remains of settlements such as villages become much more common after the invention of agriculture.
In landscape history
Landscape history studies the form (morphology) of settlements – for example whether they are dispersed or nucleated. Urban morphology can thus be considered a special type of cultural-historical landscape studies. Settlements can be ordered by size, centrality or other factors to define a settlement hierarchy. A settlement hierarchy can be used for classifying settlement all over the world, although a settlement called a 'town' in one country might be a 'village' in other countries; or a 'large town' in some countries might be a 'city' in others.
Statistics
Australia
Geoscience Australia defines a populated place as "a named settlement with a population of 200 or more persons".
The Committee for Geographical Names in Australasia used the term localities for rural areas, while the Australian Bureau of Statistics uses the term "urban centres/localities" for urban areas.
Bosnia and Herzegovina
The Agency for Statistics in Bosnia and Herzegovina uses the term "populated place""settled place" for rural (or urban as an administrative center of some Municipality/City), and "Municipality" and "City" for urban areas.
Bulgaria
The Bulgarian Government publishes a National Register of Populated places (NRPP).
Canada
The Canadian government uses the term "populated place" in the Atlas of Canada, but does not define it. Statistics Canada uses the term localities for historically na
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https://en.wikipedia.org/wiki/Yum-Tong%20Siu
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Yum-Tong Siu (; born May 6, 1943, in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University.
Siu is a prominent figure in the study of functions of several complex variables. His research interests involve the intersection of complex variables, differential geometry, and algebraic geometry. He has resolved various conjectures by applying estimates of the complex Neumann problem and the theory of multiplier ideal sheaves to algebraic geometry.
Education and career
Siu obtained his B.A. in mathematics from the University of Hong Kong in 1963, his M.A. from the University of Minnesota, and his Ph.D. from Princeton University in 1966. Siu completed his doctoral dissertation, titled "Coherent Noether-Lasker decomposition of subsheaves and sheaf cohomology", under the supervision of Robert C. Gunning. Before joining Harvard, he taught at Purdue University, the University of Notre Dame, Yale, and Stanford. In 1982 he joined Harvard as a Professor, of Mathematics. He previously served as the Chairman of the Harvard Math Department.
In 2006, Siu published a proof of the finite generation of the pluricanonical ring.
Awards, honors and professional memberships
In 1993, Siu received the Stefan Bergman Prize of the American Mathematical Society. He has holds honorary doctorates from the University of Hong Kong, University of Bochum, Germany, and University of Macau. He is a Corresponding Member of the Goettingen Academy of Sciences (elected 1993); a Foreign member of the Chinese Academy of Sciences (elected 2004); and a member of the American Academy of Arts & Sciences (elected 1998), the National Academy of Sciences (elected 2002), and Academia Sinica, Taiwan (elected 2004). He has been an invited speaker at the International Congress of Mathematicians in Helsinki (1978), Warsaw (1983) and Beijing (2002).
Currently, Siu is a member of the Scientific Advisory Board of the Clay Mathematics Institute (since 2003); the Advisory Committee for the Shaw Prize In Mathematical Sciences (since 2010); the Advisory Committee for the Millennium Prize Problems under the sponsorship of the Clay Mathematics Institute; the Scientific Advisory Board for the Institute for Mathematics Sciences, National University of Singapore (since 2009) and the Institute of Advanced Studies, Nanyang Technological University, Singapore (since 2006).
See also
Göttingen Academy of Sciences
Siu's semicontinuity theorem
List of graduates of University of Hong Kong
Math 55
Notes
External links
In 2003 and 2004, the Asian Journal of Mathematics dedicated several issues to Siu:
Vol. 7 #4
Vol. 8 #1 and 2
english.cas.cn
1943 births
Living people
Members of the United States National Academy of Sciences
Foreign members of the Chinese Academy of Sciences
20th-century American mathematicians
21st-century American mathematicians
Alumni of the University of Hong Kong
Alumni of St. John's College, University of Hong Kong
Differential geometers
Unive
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https://en.wikipedia.org/wiki/Silver%20Valley%20High%20School
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Silver Valley High School is a public high school in Yermo, California, in the High Desert of Southern California. The school is in the Silver Valley Unified School District.
Academic statistics
The school serves an area of approximately , equivalent in size to the combined states of Rhode Island and Delaware. It provides educational services to the communities of Calico, Daggett, Fort Irwin, Ludlow, Newberry Springs and Yermo. Many of the families who live in the valley work in agriculture, railroading, trucking, local businesses, or on military bases. Sixty-five percent of the families are military or military-related.
General information
Total students (as of 2009): 533
Full-time teachers: 27
Student/teacher ratio: 18:1
Race distribution
Asian: 5%
Black: 18%
Hispanic: 20%
Native American: 1%
White/Other: 53%
Hall of Fame
Johntavious "Jay" Jones # 7 Football C'15
Brian Ili #2 Football
Dauntarius "Will" Williams #8 Football, Track
Student graduation data (2005-2006)
Number of 12th grade graduates: 59
Number of 12th grade graduates who also completed UC/CSU courses: 18
Student dropouts (2005-2006)
Number of dropouts: 4
Staff (2006-2007)
Number of full-time paraprofessional staff: 5
Number of full-time office/clerical staff: 5
Number of full-time other staff: 4
Number of part-time other staff: 5
Select enrollment (2006-2007 data)
Enrollment in algebra or algebra II: 78
Enrollment in advanced math course: 121
Enrollment in first-year chemistry: 73
Enrollment in first-year physics: 20
Career-technical education enrollment: 216
Facilities
School Technology Infrastructure (2006-2007 data)
Number of computers used for instructionally-related purposes: 235
Number of classrooms or other instructional settings with internet access: 28
Special events
Clubs and activities
Academic Decathlon
Associated Student Body
Color Guard
Envirothon
Homeroom
Make a Difference Day
Step Team
Trojan Times, School Newspaper
Yearbook
Basketball
References
External links
Silver Valley Unified School District
Educational institutions established in 1979
High schools in San Bernardino County, California
Public high schools in California
1979 establishments in California
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https://en.wikipedia.org/wiki/Jacket%20matrix
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In mathematics, a jacket matrix is a square symmetric matrix of order n if its entries are non-zero and real, complex, or from a finite field, and
where In is the identity matrix, and
where T denotes the transpose of the matrix.
In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as:
The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.
Motivation
As shown in the table, i.e. in the series, for example with n=2, forward: , inverse : , then, . That is, there exists an element-wise inverse.
Example 1.
:
or more general
:
Example 2.
For m x m matrices,
denotes an mn x mn block diagonal Jacket matrix.
Example 3.
Euler's formula:
, and .
Therefore,
.
Also,
,.
Finally,
A·B = B·A = I
Example 4.
Consider be 2x2 block matrices of order
.
If and are pxp Jacket matrix, then is a block circulant matrix if and only if , where rt denotes the reciprocal transpose.
Example 5.
Let and , then the matrix is given by
,
⇒
where U, C, A, G denotes the amount of the DNA nucleobases and the matrix is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.
References
[1] Moon Ho Lee, "The Center Weighted Hadamard Transform", IEEE Transactions on Circuits Syst. Vol. 36, No. 9, PP. 1247–1249, Sept. 1989.
[2] Kathy Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007.
[3] Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing, LAP LAMBERT Publishing, Germany, Nov. 2012.
[4] Moon Ho Lee, et. al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix," US patent, no. US 009356671B1, May, 2016.
[5] S. K. Lee and M. H. Lee, “The COVID-19 DNA-RNA Genetic Code Analysis Using Information Theory of Double Stochastic Matrix,” IntechOpen, Book Chapter, April 17th, 2022. [Available in Online: https://www.intechopen.com/chapters/81329].
External links
Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices
Jacket Matrix and Its Fast Algorithms for Cooperative Wireless Signal Processing
Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing
Matrices
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https://en.wikipedia.org/wiki/E8%20manifold
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{{DISPLAYTITLE:E8 manifold}}
In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.
History
The manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the manifold is not even triangulable as a simplicial complex.
Construction
The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for . This results in , a 4-manifold with boundary equal to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the manifold.
See also
References
4-manifolds
Geometric topology
Manifold
|
https://en.wikipedia.org/wiki/Herta%20Freitag
|
Herta Freitag ( Taussig; December 6, 1908 – January 25, 2000) was an Austrian-American mathematician, a professor of mathematics at Hollins College, known for her work on the Fibonacci numbers.
Life
She was born as Herta Taussig in Vienna, earning a master's degree from the University of Vienna in 1934. She took a teaching position at the university. However, her father (the editor of Die Neue Freie Presse) had publicly opposed the Nazis.
Herta and her parents decided to move to a summer cottage in the mountains outside Vienna, to give themselves some time to make plans for the future. Herta's brother, Walter Taussig, a musician, was touring the United States and decided to remain in the U.S. (Walter later became an assistant conductor for the Metropolitan Opera Company.) Herta and her parents immediately started to work on finding a sponsor to bring them to the United States. However, even when they identified a possible sponsor, they had to wait until their quota number was called up.
In 1938, she and her parents emigrated to England. She took a job as a maid as British immigration laws prevented her from entering the country as a teacher. In 1944, she, her brother, and her mother moved to the United States. (Her father had died a year earlier in England). She began teaching mathematics again at the Greer School in Dutchess County, New York.
She earned a second master's degree in 1948 from Columbia University, and a doctorate from Columbia in 1953. Meanwhile, in 1948, she had joined the faculty at Hollins, where she eventually became a full professor and department chair. In 1962 she served as a section president for the Mathematical Association of America, the first woman in her section to do so. She retired in 1971, but returned to teaching again in 1979 after the death of her husband, Arthur Freitag, whom she had married in 1950.
Recognition
Freitag was named a Fellow of the American Association for the Advancement of Science in 1959.
After her retirement, she became a frequent contributor to the Fibonacci Quarterly, and the journal honored her in 1996 by dedicating an issue to her on the occasion of her 89th birthday (89 being a Fibonacci number).
References
Austrian mathematicians
20th-century American mathematicians
American women mathematicians
Scientists from Vienna
University of Vienna alumni
Columbia University alumni
1908 births
2000 deaths
20th-century American women scientists
Mathematicians from Austria-Hungary
20th-century women mathematicians
Fellows of the American Association for the Advancement of Science
Austrian emigrants to the United States
|
https://en.wikipedia.org/wiki/List%20of%20the%20busiest%20airports%20in%20the%20Nordic%20countries
|
This is a list of the 100 busiest airports in the Nordic countries by passengers per year, aircraft movements per year and freight and mail tonnes per year.
The list also includes yearly statistics for the busiest metropolitan airport systems and the busiest air-routes for 2012.
This transport-related list is intended to be regularly updated as new statistics become available from the relevant official authorities.
Nordic countries
The Nordic countries make up a region in Northern Europe and the North Atlantic which consists of Denmark, Finland, Iceland, Norway and Sweden and their associated territories which include the Faroe Islands, Greenland and Åland. "Scandinavia" is sometimes used as a synonym for the Nordic countries, although within the Nordic countries the terms are considered distinct, especially since Scandinavia is by definition made up of the countries Denmark, Norway and Sweden.
The region's five sovereign states and three autonomous regions share much common history as well as common traits in their respective societies, such as political systems and the Nordic model.
Politically, Nordic countries do not form a separate entity, but they co-operate in the Nordic Council. Linguistically, the area is heterogeneous, with three unrelated language groups, the North Germanic branch of Indo-European languages and the Finnic and Sami branches of Uralic languages as well as the Eskimo–Aleut language Greenlandic spoken in Greenland.
The Nordic countries have a combined population of approximately 27 million spread over a land area of 3.5 million km2 (Greenland accounts for 60% of the total area).
Passengers
At a glance
2022 statistics
Airports for which figures are not available but are likely to be among top 100 include:
Stord Airport
Kangerlussuaq Airport
Nuuk Airport
Ilulissat Airport
Ísafjörður Airport
The ranking does not include these airports.
2021 statistics
Airports for which figures are not available but are likely to be among top 100 include:
Kangerlussuaq Airport
Nuuk Airport
Ilulissat Airport
Ísafjörður Airport
The ranking does not include these airports.
2020 statistics
Airports for which figures are not available but are likely to be among top 100 include:
Kangerlussuaq Airport
Nuuk Airport
Ilulissat Airport
Narsarsuaq Airport
Ísafjörður Airport
The ranking does not include these airports.
Greenland airport administration publishes a figure of 87,672 for Kangerlussuaq and Narsarsuaq together, and 124,762 for all other airports together.
2020 COVID-19 statistics
For the purpose of documenting the large decline in air travel during the COVID-19 pandemic, figures for April 2020 are compared with April 2019 (for the 50 busiest airports).
2019 statistics
2018 statistics
International and domestic
2017 statistics
International and domestic
2016 statistics
2015 statistics
2014 statistics
International and domestic
2013 statistics
2012 statistics
Aircraft movements
Note: The statistics provided below do
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https://en.wikipedia.org/wiki/Fundamental%20sequence
|
The mathematical term fundamental sequence can refer to:
In analysis, Cauchy sequence.
In discrete mathematics and computer science, Unary coding.
In set theory, a fundamental sequence for an ordinal is a sequence of ordinals approaching the limit ordinal from below.
|
https://en.wikipedia.org/wiki/Nikolay%20Govorun
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Nikolay Nikolayevich Govorun (1930–1989) was a Soviet mathematician known best for his contributions to computational mathematics.
Bibliography
Николай Николаевич Говорун (1930—1989). Дубна, Объединенный институт ядерных исследований (Библиография научных работ Н. Н. Говоруна). 1990.
Николай Николаевич Говорун. Книга воспоминаний. Под общей редакцией В. П. Ширикова, Е. М. Молчанова. Сост. А. Г. Заикина, Т. А. Стриж. Дубна, Объединенный институт ядерных исследований, 1999.
References
1930 births
1989 deaths
20th-century Russian mathematicians
Communist Party of the Soviet Union members
Recipients of the Order of the Red Banner of Labour
Numerical analysts
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