source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Such%C3%A1%C5%88
|
Sucháň () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
It is also the best name ever!
Villages and municipalities in Veľký Krtíš District
|
https://en.wikipedia.org/wiki/I%C5%BE
|
Iž (; , ) is an island in the Zadar Archipelago within the Croatian reaches of the Adriatic Sea.
Geography
Geology and topology
The island is situated between Ugljan on the north-east and Dugi Otok on the south-west. From all islands of Zadar Archipelago, the closest one to Iž is the island of Rava, situated between Iž and Dugi Otok. Iž and Rava are separated by the channel Iški kanal (average width about 2.5 km; 1.5 miles). Iž has length of 12.2 km (7.5 miles) and average width of 2.5 km (1.5 miles). It has an area of 17.59 square kilometers (6.8 square miles) and a population of 615 (according to 2011 census), so it is considered one of the small islands of Zadar's island group. Length of the coast is 35.1 km (21.8 miles). Iž, like the other islands of Zadar Archipelago, lies in the direction northwest–southeast (NW-SE) meaning it is parallel with the mainland. Its mineralogy is composed mainly of limestone and dolomite. The highest peak of the island is Korinjak (height: 168 m; 551 ft). Iž is surrounded by more than 10 very small, uninhabited islands, largest of which is Knežak.
The vegetation of the island is Mediterranean, as on other islands of Zadar, which means that the forests are composed of coniferous trees. Due to the relatively high temperatures, Mediterranean plants are evergreen. The exploitation of forests created a macchia that is richer in flora in the southwestern part of the island (on limestone) than in the northeastern part (on the dolomites). About 60% of the island is covered with pine forest; the first afforestation of the island with aleppo pine begins in the 20th century, more precisely in 1931. The island's oldest and most important cultivated plants are olives, vines and figs.
The main soil types are terra rossa (Croatian: crvenica; crljenica) associated with limestone (cultivated and rich of hummus in the gardens of settlements) and sandy soils on the dolomites.
Settlements are located exclusively on island's eastern part, facing Ugljan. The main settlement, Veli Iž, is situated in the bay on the north-western shore, while Mali Iž is situated on the south-eastern shore and consists of three hamlets — Muće, Makovac and Porovac — located on three hills, below which are two bays — the bay of Knež below Porovac and the bay of Komoševa below Makovac.
Climate
Iž belongs to the area which has a borderline humid subtropical and Mediterranean climate. Summers are dry, warm or hot and winters are mild and rainy. Average annual air temperature on the island is 15 degrees Celsius (59 degrees Fahrenheit).
The island is relatively low and spatially small so that significant day and night winds can form there. It is relatively far from the mainland, surrounded on all sides by the sea and protected by neighboring higher islands. The most common winds are bora (Croatian: bura) during winter, sirocco (Croatian: jugo) during spring, autumn and winter and maestral - a constant humid breeze of moderate intensity - during summ
|
https://en.wikipedia.org/wiki/Se%C4%BEany
|
Seľany () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
|
https://en.wikipedia.org/wiki/Senn%C3%A9%2C%20Ve%C4%BEk%C3%BD%20Krt%C3%AD%C5%A1%20District
|
Senné () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
|
https://en.wikipedia.org/wiki/%C5%A0ir%C3%A1kov
|
Širákov () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
Statistics
Villages and municipalities in Veľký Krtíš District
|
https://en.wikipedia.org/wiki/%C5%A0u%C4%BEa
|
Šuľa () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
|
https://en.wikipedia.org/wiki/Introduction%20to%20the%20mathematics%20of%20general%20relativity
|
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
For an introduction based on the example of particles following circular orbits about a large mass, nonrelativistic and relativistic treatments are given in, respectively, Newtonian motivations for general relativity and Theoretical motivation for general relativity.
Vectors and tensors
Vectors
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or – as here – simply a vector) is a geometric object that has both a magnitude (or length) and direction. A vector is what is needed to "carry" the point to the point ; the Latin word vector means "one who carries". The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from to . Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.
Tensors
A tensor extends the concept of a vector to additional directions. A scalar, that is, a simple number without a direction, would be shown on a graph as a point, a zero-dimensional object. A vector, which has a magnitude and direction, would appear on a graph as a line, which is a one-dimensional object. A vector is a first-order tensor, since it holds one direction.
A second-order tensor has two magnitudes and two directions, and would appear on a graph as two lines similar to the hands of a clock. The "order" of a tensor is the number of directions contained within, which is separate from the dimensions of the individual directions. A second-order tensor in two dimensions might be represented mathematically by a 2-by-2 matrix, and in three dimensions by a 3-by-3 matrix, but in both cases the matrix is "square" for a second-order tensor. A third-order tensor has three magnitudes and directions, and would be represented by a cube of numbers, 3-by-3-by-3 for directions in three dimensions, and so on.
Applications
Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (in 2 dimensions with the positive
|
https://en.wikipedia.org/wiki/List%20of%20career%20achievements%20by%20Michael%20Jordan
|
This page details statistics, records, and other achievements pertaining to Michael Jordan.
College statistics
The three-point line did not exist during Michael Jordan's freshman and junior seasons in North Carolina in the NCAA. During his sophomore season, the three-point line was tested within ACC play. Many other conferences also tested with the line during this season, but again, only within their respective conference competition.
Averages
Totals
NBA career statistics
Averages
Totals
Source: basketball-reference.com and nba.com
Playoffs
Source: basketball-reference.com
Awards and accomplishments
NBA achievements
Naismith Memorial Basketball Hall of Fame Class of 2009
6× NBA champion: 1991, 1992, 1993, 1996, 1997, 1998
5× NBA Most Valuable Player: 1987–88, 1990–91, 1991–92, 1995–96, 1997–98
6× NBA Finals Most Valuable Player: 1991, 1992, 1993, 1996, 1997, 1998
10× Scoring leader: 1986–87, 1987–88, 1988–89, 1989–90, 1990–91, 1991–92, 1992–93, 1995–96, 1996–97, 1997–98
NBA Defensive Player of the Year: 1987–88
NBA Rookie of the Year: 1984–85
14× NBA All-Star: 1985, 1986 (selected but injured), 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1996, 1997, 1998, 2002, 2003
3× NBA All-Star Game Most Valuable Player: 1988, 1996, 1998
2× NBA Slam Dunk Contest champion: 1987, 1988
Runner-up in 1985
3× Steals leader: 1987–88, 1989–90, 1992–93
2× Minutes leader: 1987–88, 1988–89
2× IBM Award winner: 1985, 1989
11× All-NBA selection:
First Team: 1987–93, 1996–98
Second Team: 1985
9× All-Defensive selection:
First Team: 1988–93, 1996–98
NBA All-Rookie selection:
First Team: 1985
7× The Sporting News Most Valuable Player: 1987–88, 1988–89, 1990–91, 1991–92, 1995–96, 1996–97, 1997–98
The Sporting News Rookie of the Year: 1985
Sports Illustrated'' Sportsman of the Year: 1991
Ranked #1 by SLAM Magazine's Top 50 Players of All-time Ranked #1 by ESPN SportsCentury's Top North American Athletes of the 20th Century Selected in 1996 as one of the "50 Greatest Players in NBA History"
Selected in 1996 as member of two of the "Top 10 Teams in NBA History"
1991–92 Chicago Bulls (67–15; .817)
1995–96 Chicago Bulls (72–10; .878)
25 NBA Player of the Week
16 NBA Player of the Month
United States National Team
2× Olympic gold medals: 1984, 1992
Tournament of the Americas gold medal: 1992
Pan American Games gold medal: 1983
3× USA Basketball Male Athlete of the Year: 1983, 1984 (with Sam Perkins), 1992 (as a part of the 1992 Olympic Team)
FIBA Hall of Fame (Class of 2015)
College
NCAA National Championship – University of North Carolina: 1981–82
3× Atlantic Coast Conference regular season champions: 1981–82, 1982–83, 1983–84 (undefeated)
1982 ACC tournament champions
ACC Rookie of the Year: 1982
Naismith College Player of the Year: 1984
John R. Wooden Award: 1984
Adolph Rupp Trophy: 1984
USBWA College Player of the Year: 1984
AP College Basketball Player of the Year: 1984
ACC Athlete of the Year: 1984
ACC Men's Bask
|
https://en.wikipedia.org/wiki/Maurice%20Kraitchik
|
Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser. His main interests were the theory of numbers and recreational mathematics.
He was born to a Jewish family in Minsk. He wrote several books on number theory during 1922–1930 and after the war, and from 1931 to 1939 edited Sphinx, a periodical devoted to recreational mathematics. During World War II, he emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations."
Kraïtchik was agrégé of the Free University of Brussels, engineer at the Société Financière de Transports et d'Entreprises Industrielles (Sofina), and director of the Institut des Hautes Etudes de Belgique. He died in Brussels.
Kraïtchik is famous for having inspired the two envelopes problem in 1953, with the following puzzle in La mathématique des jeux:
Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "Suppose that I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man?
Among his publications were the following:
Théorie des Nombres, Paris: Gauthier-Villars, 1922
Recherches sur la théorie des nombres, Paris: Gauthier-Villars, 1924
La mathématique des jeux ou Récréations mathématiques, Paris: Vuibert, 1930, 566 pages
Mathematical Recreations, New York: W. W. Norton, 1942 and London: George Allen & Unwin Ltd, 1943, 328 pages (revised edition New York: Dover, 1953)
Alignment Charts, New York: Van Nostrand, 1944
References
1882 births
1957 deaths
20th-century American mathematicians
Recreational mathematicians
Mathematics popularizers
Belgian mathematicians
Belarusian Jews
Emigrants from the Russian Empire to Belgium
Belgian emigrants to the United States
|
https://en.wikipedia.org/wiki/Presentation%20of%20a%20monoid
|
In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set of generators and a set of relations on the free monoid (or the free semigroup ) generated by . The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory.
As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).
A presentation should not be confused with a representation.
Construction
The relations are given as a (finite) binary relation on . To form the quotient monoid, these relations are extended to monoid congruences as follows:
First, one takes the symmetric closure of . This is then extended to a symmetric relation by defining if and only if = and = for some strings with . Finally, one takes the reflexive and transitive closure of , which then is a monoid congruence.
In the typical situation, the relation is simply given as a set of equations, so that . Thus, for example,
is the equational presentation for the bicyclic monoid, and
is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as for integers i, j, k, as the relations show that ba commutes with both a and b.
Inverse monoids and semigroups
Presentations of inverse monoids and semigroups can be defined in a similar way using a pair
where
is the free monoid with involution on , and
is a binary relation between words. We denote by (respectively ) the equivalence relation (respectively, the congruence) generated by T.
We use this pair of objects to define an inverse monoid
Let be the Wagner congruence on , we define the inverse monoid
presented by as
In the previous discussion, if we replace everywhere with we obtain a presentation (for an inverse semigroup) and an inverse semigroup presented by .
A trivial but important example is the free inverse monoid (or free inverse semigroup) on , that is usually denoted by (respectively ) and is defined by
or
Notes
References
John M. Howie, Fundamentals of Semigroup Theory (1995), Clarendon Press, Oxford
M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, .
Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, , chapter 7, "Algebraic Properties"
Semigroup theory
|
https://en.wikipedia.org/wiki/Concordant%20pair
|
In statistics, a concordant pair is a pair of observations, each on two variables, (X1,Y1) and (X2,Y2), having the property that
where "sgn" refers to whether a number is positive, zero, or negative (its sign). Specifically, the signum function, often represented as sgn, is defined as:
That is, in a concordant pair, both elements of one pair are either greater than, equal to, or less than the corresponding elements of the other pair.
In contrast, a discordant pair is a pair of two-variable observations such that
That is, if one pair contains a higher value of X then the other pair contains a higher value of Y.
Uses
The Kendall tau distance between two series is the total number of discordant pairs. The Kendall tau rank correlation coefficient, which measures how closely related two series of numbers are, is proportional to the difference between the number of concordant pairs and the number of discordant pairs. An estimate of Goodman and Kruskal's gamma, another measure of rank correlation, is given by the ratio of the difference to the sum of the numbers of concordant and discordant pairs. Somers' D is another similar but asymmetric measure given by the ratio of the difference in the number of concordant and discordant pairs to the number of pairs with unequal values for one of the two variables.
See also
Spearman's rank correlation coefficient
References
Abdi, Hervé (2007). "The Kendall Rank Correlation Coefficient". In: Neil Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.
Kendall, M. (1948) Rank Correlation Methods, Charles Griffin & Company Limited
Kendall, M. (1938) "A New Measure of Rank Correlation", Biometrika, 30:81-89.
External links
MacTutor: David George Kendall
Janus: The Papers of Professor David Kendall
Nonparametric statistics
|
https://en.wikipedia.org/wiki/Laymen%27s%20Evangelical%20Fellowship%20International
|
Laymen's Evangelical Fellowship International is a Christian organization founded in 1935 in Madras, India by N. Daniel (1897-1963/12/18), a former mathematics teacher at McLaurin High School in Kakinada, Andhra Pradesh, was headed from 1963 to 2014 by his son Joshua Daniel (1928/02/06 - 2014/10/18), and now by grandson John Daniel (1962/09/30 - ).
Headquartered in Chennai, India, the Church has centres in many parts of India, with the majority in the states of Tamil Nadu and Andhra Pradesh, as well as churches in Arunachal Pradesh, Assam, Delhi, Goa, Gujarat, Haryana, Himachal Pradesh, Kashmir, Jharkhand, Kerala, Manipur, Meghalaya, Maharashtra and several other Indian states.
Additionally, it has "tentmaker" missionaries (supported through their own work rather than by the organisation) in various countries such as Cyprus, Guyana, Venezuela, Ireland and the Brixton area of London, and radio broadcasts named "The Lord's Challenge" from Guyana, United Kingdom, France, Germany, Australia, and parts of the United States: in Buffalo, New York, Atlanta, Georgia, Council Bluffs, Iowa, and Detroit, Michigan). Since January 2007, a weekly series of half-hour television broadcasts with same title as the radio broadcasts (The Lord's Challenge), have been going forth from Novi, Michigan. A RealVideo copy of each broadcast is available for play or download from the organisation's website.
Beautiful Books is the literature division of the Laymen's Evangelical Fellowship which publishes books in English and various Indian languages. Some of its popular publications are Another Daniel the biography of Mr. Daniel, Faith Is The Victory a daily-devotional guide which contains 366 messages by N. Daniel and Joshua Daniel, as well as Indian editions of their magazine Christ Is Victor. They are printed at Beulah Gardens, its retreat centre in Sirinium village, from Red Hills, Chennai. Other editions of the magazine are printed abroad.
Annual retreats are held in "Beulah Gardens" located near Chennai. Approximately 7000 to 9000 people attend these retreats every year. Live webcast of retreats (including annual retreat, generally in May of the year) has been available from May 2008.
References
My Conversion by Joshua Daniel
https://www.youtube.com/watch?v=m8fN0gX9FuI
Father of the great Revival in India
https://web.archive.org/web/20141210193712/http://www.formations.org.in/OrgForCofl/index.php?tmpl=component&view=article&id=180
External links
LEFI Website
Lord's Challenge MP3 broadcast
Beautiful Books
Christian organisations based in India
1935 establishments in India
|
https://en.wikipedia.org/wiki/An%20Intelligent%20Person%27s%20Guide%20to%20Atheism
|
An Intelligent Person's Guide to Atheism is the first book by Daniel Harbour, an Oxford maths and philosophy graduate, who at the time of writing was working for a PhD in linguistics at MIT.
Synopsis
Rather than a history of atheism, as the title may suggest, the book is a guide to why (according to the author) atheism is superior to theism and why the (a)theist discussion is important.
According to Harbour, atheism is "the plausible and probably correct belief that God does not exist", while theism is "the implausible and probably incorrect belief that God does exist", and anyone who cares about the truth should be an atheist. Harbour makes his case on the basis of two fundamental worldviews which he labels the Spartan Meritocracy and the Baroque Monarchy. Worldviews are the ways in which we look at and try to explain the world around us; as a result, the validity of our worldviews is extremely important because it determines the validity and reasonableness of our beliefs.
The Spartan Meritocracy makes minimal assumptions, that are subject to criticism and possible revision, when trying to explain the world - focusing more upon a proper method of inquiry than on reaching any particular or prejudicial conclusions. The Baroque Monarchy, however, relies upon elaborate dogmatic assumptions in the absence of any evidence — assumptions which are placed beyond question, critique or revision.
Harbour spends little time directly comparing atheism and theism; rather, he compares these two opposing worldviews and argues that the Spartan Meritocracy is more plausible, more reasonable, and helps make the world a better place to live. Thus, anyone who cares about the truth should be inclined to adopt it rather than blind obedience to dogmatism as in the Baroque Monarchy.
He does not, however, argue that there is a direct and necessary connection between these worldviews and either atheism or theism — he acknowledges that it is possible in theory for an atheist to adopt the Baroque Monarchy and for some types of theist to adopt the Spartan Meritocracy. Strictly speaking, then, the main thrust of his argument is that the Spartan Meritocracy is superior and anyone who cares about the truth should adopt this worldview. Nevertheless, he also argues that it is highly unlikely for theism ever to occur within the Spartan Meritocracy due to the evidence the world presents, and that, consequently, anyone who adopts the Spartan Meritocracy will almost inevitably be an atheist. Harbour constructs an argument throughout the book to demonstrate that the Spartan Meritocracy leads logically and naturally to atheism rather than theism.
Much of Daniel Harbour's book is focused on demonstrating the ways in which the Spartan Meritocracy does a better job of helping us to explain the world and make the world a better place to live in. The former involves analyzing the impact of science and technology, pursuits fundamentally based upon a spartan and meritocratic perspectiv
|
https://en.wikipedia.org/wiki/Faulhaber%27s%20formula
|
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers
as a polynomial in n. In modern notation, Faulhaber's formula is
Here, is the binomial coefficient "p + 1 choose k", and the Bj are the Bernoulli numbers with the convention that .
The result: Faulhaber's formula
Faulhaber's formula concerns expressing the sum of the p-th powers of the first n positive integers
as a (p + 1)th-degree polynomial function of n.
The first few examples are well known. For p = 0, we have
For p = 1, we have the triangular numbers
For p = 2, we have the square pyramidal numbers
The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbers Bj. The Bernoulli numbers begin
where here we use the convention that . The Bernoulli numbers have various definitions (see Bernoulli_number#Definitions), such as that they are the coefficients of the exponential generating function
Then Faulhaber's formula is that
Here, the Bj are the Bernoulli numbers as above, and
is the binomial coefficient "p + 1 choose k".
Examples
So, for example, one has for ,
The first seven examples of Faulhaber's formula are
History
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients later discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below.
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the powers of the first integers as a ()th-degree polynomial function of , with coefficients involving numbers , now called Bernoulli numbers:
Introducing also the first two Bernoulli numbers (which Bernoulli did not), the previous formula becomes
using the Bernoulli number of the second kind for which , or
using the Bernoulli number of the first kind for which
Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers.
A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until , two centuries later.
Proof with exponential generating function
Let
denote the sum under consideration for integer
Define the following exponential generating function with (initially) indeterminate
We find
This is an entire function in so that can be taken to be any complex number.
We next recall the exponential generating function for the Bernoulli polynomials
where denotes the Bernoulli number with the convention . This may be converted to a generating function with the convention by the addition of to the coefficient of in each ( does not need to be changed):
It follows immediately that
for all .
Faulhaber polynomials
The term Faulhaber polynomia
|
https://en.wikipedia.org/wiki/Karl%20Rubin
|
Karl Cooper Rubin (born January 27, 1956) is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio State University between 1987 and 1999. His research interest is in elliptic curves. He was the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate–Shafarevich groups. It is widely believed that these groups are always finite.
Education and career
Rubin graduated from Princeton University in 1976, and obtained his Ph.D. from Harvard in 1981. His thesis advisor was Andrew Wiles. He was a Putnam Fellow in 1974, and a Sloan Research Fellow in 1985.
In 1988, Rubin received a National Science Foundation Presidential Young Investigator award, and in 1992 won the American Mathematical Society Cole Prize in number theory. In 2012 he became a fellow of the American Mathematical Society. Rubin's parents were mathematician Robert Joshua Rubin and astronomer Vera Rubin. Rubin is brother to astronomer and physicist Judith Young.
See also
CEILIDH
Torus-based cryptography
Euler system
Stark conjectures
References
External links
Karl Cooper Rubin at the Mathematics Genealogy Project
Karl Rubin's Home page
Institute for Advanced Study visiting scholars
University of California, Irvine faculty
Ohio State University faculty
Columbia University faculty
1956 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Princeton University alumni
Harvard University alumni
Number theorists
Putnam Fellows
Fellows of the American Mathematical Society
|
https://en.wikipedia.org/wiki/%C3%89lisabeth%20Lutz
|
Élisabeth Lutz (May 14, 1914 – July 31, 2008) was a French mathematician. The Nagell–Lutz theorem in Diophantine geometry describes the torsion points of elliptic curves; it is named after Lutz and Trygve Nagell, who both published it in the 1930s.
Lutz was a student of André Weil at the University of Strasbourg, from 1934 to 1938. She earned a thesis for her research for him, on elliptic curves over -adic fields. She completed her doctorate (thèse d’état) on -adic Diophantine approximation at the University of Grenoble in 1951 under the supervision of Claude Chabauty; her dissertation was Sur les approximations diophantiennes linéaires -adiques.
She became a professor of mathematics at the University of Grenoble.
Selected publications
References
1914 births
2008 deaths
Number theorists
20th-century French mathematicians
21st-century French mathematicians
French women mathematicians
20th-century French women scientists
University of Strasbourg alumni
Grenoble Alpes University alumni
Academic staff of Grenoble Alpes University
20th-century women mathematicians
21st-century women mathematicians
21st-century French women
|
https://en.wikipedia.org/wiki/Helly%27s%20selection%20theorem
|
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence.
In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions.
It is named for the Austrian mathematician Eduard Helly.
A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point.
The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.
Statement of the theorem
Let (fn)n ∈ N be a sequence of increasing functions mapping the real line R into itself,
and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ fn ≤ b for every n ∈ N.
Then the sequence (fn)n ∈ N admits a pointwise convergent subsequence.
Generalisation to BVloc
Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
(fn) has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W ⊆ U with compact closure W̄ ⊆ U,
where the derivative is taken in the sense of tempered distributions;
and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set.
Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that
fnk converges to f pointwise;
and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U,
and, for W compactly embedded in U,
Further generalizations
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu:
Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that
for all t ∈ [0, T],
and, for all t ∈ [0, T],
and, for all 0 ≤ s < t ≤ T,
See also
Bounded variation
Fraňková-Helly selection theorem
Total variation
References
Compactness theorems
Theorems in analysis
|
https://en.wikipedia.org/wiki/James%20Malton
|
James Malton (1761–1803) was an Irish engraver and watercolourist, who once taught geometry and perspective. He worked briefly as a draughtsman in the office of the celebrated Irish architect James Gandon. He is best known for a series of prints, published in the 1790s as A Picturesque and Descriptive View of the City of Dublin, commonly known as Malton's Views of Dublin.
Early life
Born in 1761, James Malton was the son of the English architectural draughtsman Thomas Malton the elder and brother of Thomas Malton the younger. He moved to Ireland with his father and was living in Dublin by the 1780s. He was employed as a draughtsman in the office of the architect James Gandon for nearly three years during the building of the Custom House (built between 1781 and 1791), but was eventually dismissed.
Career
Malton is first recorded as an artist in 1790, when he sent two drawings to the Society of Artists in London from an address in Dublin. He is best known for A Picturesque and Descriptive View of the City of Dublin, a series of 25 prints originally published between 1792 and 1799. The plates were executed in etching and aquatint by Malton himself, after his own drawings. Each plate was accompanied by descriptive text with a dedication and a vignette in aquatint. Following the completion of the issue of the work in six parts, Malton republished the whole in a bound volume. The coloured prints from this work, which depict many of the new public buildings erected, capture the architectural changes which Dublin underwent in the 18th century.
Between 1792 and 1803, Malton showed 51 drawings of architectural subjects at the Royal Academy. They included 17 views of Dublin in Indian ink and watercolour, mostly depicting the same subjects as his published prints. They were not, however, the original drawings from which the plates were made, often being larger, and with the scenes populated with different figures.
In 1798, he published An Essay on British Cottage Architecture, described in its subtitle as "an attempt to perpetuate on principle, that peculiar mode of building, which was originally the effect of chance". His later publications include a practical treatise on perspective called The Young Painter's Mahlstick (1800), four aquatints after drawings by Francis Keenan, issued as A Select Collection of Views in the County of Devon (1800), and A Collection of Designs for Rural Retreats as Villas Principally in the Gothic and Castle Styles of Architecture (1802).
Malton died "of a brain-fever" in Norton Street, Marylebone, London, on 28 July 1803.
Gallery
Footnotes
While most sources give Malton's date of birth as 1761, some (such as the Dublin Historical Record) give 1763.
References
Notes
Sources
1761 births
1803 deaths
18th-century Irish painters
Irish male painters
Irish engravers
Irish emigrants to Kingdom of Great Britain
Artists from County Dublin
|
https://en.wikipedia.org/wiki/Mal%C3%A1%20Lehota
|
Malá Lehota () is a village and municipality in the Žarnovica District, Banská Bystrica Region in Slovakia.
External links
https://web.archive.org/web/20071217080336/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Žarnovica District
|
https://en.wikipedia.org/wiki/Ve%C4%BEk%C3%A1%20Lehota
|
Veľká Lehota () is a village and municipality in the Žarnovica District, Banská Bystrica Region in Slovakia.
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Žarnovica District
|
https://en.wikipedia.org/wiki/%C5%BDupkov
|
Župkov () is a village and municipality in the Žarnovica District, Banská Bystrica Region in Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Žarnovica District
|
https://en.wikipedia.org/wiki/Compact%20embedding
|
In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.
Definition (topological spaces)
Let (X, T) be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W, and write V ⊂⊂ W, if
V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closure of V, and Int(W) denotes the interior of W; and
Cl(V) is compact.
Definition (normed spaces)
Let X and Y be two normed vector spaces with norms ||•||X and ||•||Y respectively, and suppose that X ⊆ Y. We say that X is compactly embedded in Y, and write X ⊂⊂ Y, if
X is continuously embedded in Y; i.e., there is a constant C such that ||x||Y ≤ C||x||X for all x in X; and
The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm ||•||Y.
If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator.
When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of functions. Several of the Sobolev embedding theorems are compact embedding theorems. When an embedding is not compact, it may possess a related, but weaker, property of cocompactness.
References
.
.
.
Compactness (mathematics)
Functional analysis
General topology
|
https://en.wikipedia.org/wiki/Pict%20%28programming%20language%29
|
Pict is a statically typed programming language, one of the very few based on the π-calculus. Work on the language began at the University of Edinburgh in 1992, and development has been more or less dormant since 1998. The language is still at an experimental stage.
References
Sources
Benjamin C. Pierce and David N. Turner. Pict: A programming language based on the pi-calculus. Technical report, Computer Science Department, Indiana University, 1997
External links
, links to a compiler, manuals, tutorial
Experimental programming languages
Functional languages
|
https://en.wikipedia.org/wiki/Bochner%20space
|
In mathematics, Bochner spaces are a generalization of the concept of spaces to functions whose values lie in a Banach space which is not necessarily the space or of real or complex numbers.
The space consists of (equivalence classes of) all Bochner measurable functions with values in the Banach space whose norm lies in the standard space. Thus, if is the set of complex numbers, it is the standard Lebesgue space.
Almost all standard results on spaces do hold on Bochner spaces too; in particular, the Bochner spaces are Banach spaces for
Bochner spaces are named for the mathematician Salomon Bochner.
Definition
Given a measure space a Banach space and the Bochner space is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions such that the corresponding norm is finite:
In other words, as is usual in the study of spaces, is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a -measure zero subset of As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in rather than an equivalence class (which would be more technically correct).
Applications
Bochner spaces are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation: if the temperature is a scalar function of time and space, one can write to make a family (parametrized by time) of functions of space, possibly in some Bochner space.
Application to PDE theory
Very often, the space is an interval of time over which we wish to solve some partial differential equation, and will be one-dimensional Lebesgue measure. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region in and an interval of time one seeks solutions
with time derivative
Here denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); denotes the dual space of
(The "partial derivative" with respect to time above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)
See also
References
Functional analysis
Partial differential equations
Sobolev spaces
Lp spaces
|
https://en.wikipedia.org/wiki/Exchangeable%20random%20variables
|
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences
both have the same joint probability distribution.
It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling.
Definition
Formally, an exchangeable sequence of random variables is a finite or infinite sequence X1, X2, X3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the joint probability distribution of the permuted sequence
is the same as the joint probability distribution of the original sequence.
(A sequence E1, E2, E3, ... of events is said to be exchangeable precisely if the sequence of its indicator functions is exchangeable.) The distribution function FX1,...,Xn(x1, ..., xn) of a finite sequence of exchangeable random variables is symmetric in its arguments Olav Kallenberg provided an appropriate definition of exchangeability for continuous-time stochastic processes.
History
The concept was introduced by William Ernest Johnson in his 1924 book Logic, Part III: The Logical Foundations of Science. Exchangeability is equivalent to the concept of statistical control introduced by Walter Shewhart also in 1924.
Exchangeability and the i.i.d. statistical model
The property of exchangeability is closely related to the use of independent and identically distributed (i.i.d.) random variables in statistical models. A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form.
Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable. The converse can be established for infinite sequences, through an important representation theorem by Bruno de Finetti (later extended by other probability theorists such as Halmos and Savage). The extended versions of the theorem show that in any infinite sequence of exchangeable random variables, the random variables are conditionally independent and identically-distributed, given the underlying distributional form. This theorem is stated briefly below. (De Finetti's original theorem only showed this to be true for random indicator variables, but this was later extended to encompass all sequences of random variables.) Another way of putting this is that de Finetti's theorem characterizes exchangeable sequences as mixtures of i.i.d. sequences — while an exchangeable sequence need not itself be uncondition
|
https://en.wikipedia.org/wiki/Straight%20skeleton
|
In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves. However, both are homotopy-equivalent to the underlying polygon.
Straight skeletons were first defined for simple polygons by , and generalized to planar straight-line graphs (PSLG) by .
In their interpretation as projection of roof surfaces, they are already extensively discussed by .
Definition
The straight skeleton of a polygon is defined by a continuous shrinking process in which the edges of the polygon are moved inwards parallel to themselves at a constant speed. As the edges move in this way, the vertices where pairs of edges meet also move, at speeds that depend on the angle of the vertex. If one of these moving vertices collides with a nonadjacent edge, the polygon is split in two by the collision, and the process continues in each part. The straight skeleton is the set of curves traced out by the moving vertices in this process.
In the illustration the top figure shows the shrinking process and the middle figure depicts the straight skeleton in blue.
Algorithms
The straight skeleton may be computed by simulating the shrinking process by which it is defined; a number of variant algorithms for computing it have been proposed, differing in the assumptions they make on the input and in the data structures they use for detecting combinatorial changes in the input polygon as it shrinks.
The following algorithms consider an input that forms a polygon, a polygon with holes, or a PSLG. For a polygonal input we denote the number of vertices by n and the number of reflex (concave, i.e., angle greater than ) vertices by r. If the input is a PSLG then we consider the initial wavefront structure, which forms a set of polygons, and again denote by n the number of vertices and by r the number of reflex vertices w.r.t. the propagation direction. Most of the algorithms listed here are designed and analyzed in the real RAM model of computation.
Aichholzer et al. showed how to compute straight skeletons of PSLGs in time O(n3 log n), or more precisely time O((n2+f) log n), where n is the number of vertices of the input polygon and f is the number of flip events during the construction. The best known bound for f is O(n3).
An algorithm with a worst case running time in O(nr log n), or simply O(n2 log n), is given by , who argue that their approach is likely to run in near-linear time for many inputs.
Petr Felkel and Štěpán Obdržálek designed an algorithm for simple polygons that is said to have an efficiency of O(nr + n log r). However, it has been shown that their algorithm is incorrect.
By using data structures for the bichromatic closest pair problem, Eppstein and Erickson showed how to construct straight skeleton problems using a linear number of closest pair data structur
|
https://en.wikipedia.org/wiki/Cell%20Loss%20Priority
|
Cell Loss Priority (CLP) is a flag bit in the ATM cell header that determines the probability of a cell being discarded if the network becomes congested. Cells where the CLP = 0 are insured traffic and unlikely to be dropped. Cells with CLP = 1 are best-effort traffic, which may be discarded in congested conditions in order to free up resources to handle insured traffic.
CLP is used as a control for a network traffic "policing mechanism". Policing is a process that determines if the cells meet predefined restrictions as they enter an ATM network. These restrictions include traffic rates and "burst sizes" that are agreed upon by the customer and the network provider.
Link protocols
|
https://en.wikipedia.org/wiki/Saint%20Petersburg%20Lyceum%20239
|
Presidential Physics and Mathematics Lyceum No. 239 (), is a public high school in Saint Petersburg, Russia that specializes in mathematics and physics. The school opened in 1918 and it became a specialized city school in 1961. The school is noted for its strong academic programs. It is the alma mater of numerous winners of International Mathematical Olympiads and it has produced many notable alumni. The lyceum has been named the best school in Russia in 2015, 2016, and 2017.
History
The school was founded in 1918. Originally, it was located in the Lobanov-Rostovsky Palace, also known as "house with lions" at the corner of Saint Isaac's Square and Admiralteysky Prospect. It was one of only handful of schools to remain open during Siege of Leningrad. In 1961 the school was granted status of city's school with specialization in physics and mathematics. In 1964 the school moved to the building on Kazansky Street 48/1, which was previously occupied by school of working youth, and in 1966 it moved again to Moika River, 108. Finally, in 1975 the school relocated to its current location, into the historic Annenschule building.
In 1990, the Russian Ministry of Education granted school the status of physico-mathematical lyceum and experimental laboratory for standard of education in physics, mathematics and informatics in Saint Petersburg. In 1994, the school won the George Soros grant. The US Mathematical society voted the school as one of top ten schools of former Soviet Union. The first of January 2014 the school received a status of "Presidential Physics and Mathematics Lyceum №239".
Famous alumni
Yelena Bonner (1940) – human rights activist (widow of Andrei Sakharov)
Leonid Kharitonov – actor
Alisa Freindlich (c. 1942 – 1953) – major Russian movie and theater actress
Yuri Matiyasevich (1962–1963) – mathematician who solved Hilbert's tenth problem
Andrei Tolubeyev (?–1963) – theatrical and cinema actor, People's Artist of Russia
Natalia Kuchinskaya (1964–1966) – Olympic champion in gymnastics (Mexico City, 1968), the first of a number of young gymnastics champions
Boris Grebenshchikov (1968–1970) – rock musician, who is one of the "founding fathers" of Russian rock music
Mikhail Zurabov (?–1970) – minister of health of the Russian Federation, chair of the Russian pension fund administration
Sergey Fursenko – businessman, president of the football club Zenit Saint Petersburg
Grigori Perelman (1980–1982) – mathematician who was awarded Fields Medal for his proof of the Poincaré conjecture
Alexander Khalifman (1981–1983) – FIDE World Chess champion in 1999
Stanislav Smirnov (1985–1987) – mathematician and recipient of Fields Medal in 2010 for his work on the mathematical foundations of statistical physics, particularly finite lattice models
Yoel Matveyev (1990-?) – Yiddish poet, writer and journalist
Directors of the School
Matkovskaya Maria Vasilievna - from 1950 to 1976
Radionov Victor Evseevich - from 1976 to 1980
Golubeva Galin
|
https://en.wikipedia.org/wiki/Braunton%20Academy
|
Braunton Academy (formerly Braunton School and Community College) is a coeducational secondary school with academy status in Braunton, North Devon, England. The school specialises in mathematics and computing.
The school first opened in 1937 with 140 pupils, and now has around 740 pupils aged 11 to 16.
The school has various sports facilities, which include four tennis courts, a climbing wall, a dance studio, access to Tweedies Field and the North Devon Athletics Track. A new multi-usage games area was opened in 2014 supported by Fullabrook and is used by the school as well as being accessible to local primary schools and to other sports clubs.
Braunton Academy secured £750,000 from the government to build a new library, reception and supported study centre which opened in Spring 2015.
Principals
1992 – 1997: Mr Hunkin
1997 – 1999: Mr Roff (acting)
1999 – 2001: Mr Scutt
2002 – 2006: Mr V Game
2006 – 2014: David Sharratt
September 2014 – July 2021: Michael Cammack
September 2021 – Present: Fay Bowler
House and Tutor Systems
Houses
Braunton Academy has four houses, each with their own colour: Croyde (yellow), Putsborough (green), Saunton (red) and Woolacombe (blue). They are named after 4 of the beaches in the local North Devon area.
Tutors
Braunton Academy's tutor group system was originally vertical, with students from all 5 year groups making up a tutor. This was then changed to have distinct KS3 and KS4 groups, with students changing to their KS4 tutor at the end of Year 9.
In September 2018, the tutors were switched to a horizontal system, with students in the same year from all 4 houses making up a group, which they would stay in up until Year 11.
References
External links
Ofsted Profile
Secondary schools in Devon
Academies in Devon
Braunton
|
https://en.wikipedia.org/wiki/List%20of%20Asian%20American%20jurists
|
Research history
Studies led by California Supreme Court Justice Goodwin Liu (2017) and the Center for American Progress (2019) provided in-depth statistics into the issue.
Judicial officers
This is a dynamic list of Asian Americans who are or were judges, magistrate judges, court commissioners, or administrative law judges. If known, it will be listed if a judge has served on multiple courts.
Other topics of interest
List of first minority male lawyers and judges in the United States
List of first women lawyers and judges in the United States
List of African-American jurists
List of Hispanic and Latino American jurists
List of Jewish American jurists
List of LGBT jurists in the United States
List of Native American jurists
References
Sources
Asian American Bar Association of The Greater Bay Area
APAs In The Judiciary Resource Page
Asian Americans and Pacific Islanders on the Federal Bench
Current Asian Pacific American Federal Judges
Selected APA Judges in California
First Vietnamese American and Korean American Women Seated on State Judiciary, by Sam Chu Lin, AsianWeek, August 23, 2002
Vietnamese American Facts, Tieng Magazine
Two New APA Judges in Cook County, Illinois, AsianWeek, March 30, 2007
Asian-American issues
Jurists
Lists of American judges
|
https://en.wikipedia.org/wiki/Skorokhod%27s%20representation%20theorem
|
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Soviet mathematician A. V. Skorokhod.
Statement
Let be a sequence of probability measures on a metric space such that converges weakly to some probability measure on as . Suppose also that the support of is separable. Then there exist -valued random variables defined on a common probability space such that the law of is for all (including ) and such that converges to , -almost surely.
See also
Convergence in distribution
References
(see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)
Probability theorems
Theorems in statistics
|
https://en.wikipedia.org/wiki/Gabor%20atom
|
In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function.
Overview
In 1946, Dennis Gabor suggested the idea of using a granular system to produce sound. In his work, Gabor discussed the problems with Fourier analysis. Although he found the mathematics to be correct, it did not reflect the behaviour of sound in the world, because sounds, such as the sound of a siren, have variable frequencies over time. Another problem was the underlying supposition, as we use sine waves analysis, that the signal under concern has infinite duration even though sounds in real life have limited duration – see time–frequency analysis. Gabor applied ideas from quantum physics to sound, allowing an analogy between sound and quanta. He proposed a mathematical method to reduce Fourier analysis into cells. His research aimed at the information transmission through communication channels. Gabor saw in his atoms a possibility to transmit the same information but using less data. Instead of transmitting the signal itself it would be possible to transmit only the coefficients which represent the same signal using his atoms.
Mathematical definition
The Gabor function is defined by
where a and b are constants and g is a fixed function in L2(R), such that ||g|| = 1. Depending on , , and , a Gabor system may be a basis for L2(R), which is defined by translations and modulations. This is similar to a wavelet system, which may form a basis through dilating and translating a mother wavelet.
When one takes
one gets the kernel of the Gabor transform.
See also
Gabor filter
Gabor wavelet
Fourier analysis
Wavelet
Morlet wavelet
References
Further reading
Hans G. Feichtinger, Thomas Strohmer: "Gabor Analysis and Algorithms", Birkhäuser, 1998;
Hans G. Feichtinger, Thomas Strohmer: "Advances in Gabor Analysis", Birkhäuser, 2003;
Karlheinz Gröchenig: "Foundations of Time-Frequency Analysis", Birkhäuser, 2001;
External links
NuHAG homepage [Numerical Harmonic Analysis Group]
Wavelets
Fourier analysis
|
https://en.wikipedia.org/wiki/Generalized%20algebraic%20data%20type
|
In functional programming, a generalized algebraic data type (GADT, also first-class phantom type, guarded recursive datatype, or equality-qualified type) is a generalization of parametric algebraic data types.
Overview
In a GADT, the product constructors (called data constructors in Haskell) can provide an explicit instantiation of the ADT as the type instantiation of their return value. This allows defining functions with a more advanced type behaviour. For a data constructor of Haskell 2010, the return value has the type instantiation implied by the instantiation of the ADT parameters at the constructor's application.
-- A parametric ADT that is not a GADT
data List a = Nil | Cons a (List a)
integers :: List Int
integers = Cons 12 (Cons 107 Nil)
strings :: List String
strings = Cons "boat" (Cons "dock" Nil)
-- A GADT
data Expr a where
EBool :: Bool -> Expr Bool
EInt :: Int -> Expr Int
EEqual :: Expr Int -> Expr Int -> Expr Bool
eval :: Expr a -> a
eval e = case e of
EBool a -> a
EInt a -> a
EEqual a b -> (eval a) == (eval b)
expr1 :: Expr Bool
expr1 = EEqual (EInt 2) (EInt 3)
ret = eval expr1 -- False
They are currently implemented in the GHC compiler as a non-standard extension, used by, among others, Pugs and Darcs. OCaml supports GADT natively since version 4.00.
The GHC implementation provides support for existentially quantified type parameters and for local constraints.
History
An early version of generalized algebraic data types were described by and based on pattern matching in ALF.
Generalized algebraic data types were introduced independently by and prior by as extensions to ML's and Haskell's algebraic data types. Both are essentially equivalent to each other. They are similar to the inductive families of data types (or inductive datatypes) found in Coq's Calculus of Inductive Constructions and other dependently typed languages, modulo the dependent types and except that the latter have an additional positivity restriction which is not enforced in GADTs.
introduced extended algebraic data types which combine GADTs together with the existential data types and type class constraints.
Type inference in the absence of any programmer supplied type annotations is undecidable and functions defined over GADTs do not admit principal types in general. Type reconstruction requires several design trade-offs and is an area of active research (; .
In spring 2021, Scala 3.0 is released. This major update of Scala introduce the possibility to write GADTs with the same syntax as ADTs, which is not the case in other programming languages according to Martin Odersky.
Applications
Applications of GADTs include generic programming, modelling programming languages (higher-order abstract syntax), maintaining invariants in data structures, expressing constraints in embedded domain-specific languages, and modelling objects.
Higher-order abstract syntax
An important application of GADTs is
|
https://en.wikipedia.org/wiki/Polar%20curve
|
In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose tangent line passes through Q. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.
Definition
Let C be defined in homogeneous coordinates by f(x, y, z) = 0 where f is a homogeneous polynomial of degree n, and let the homogeneous coordinates of Q be (a, b, c). Define the operator
Then ΔQf is a homogeneous polynomial of degree n−1 and ΔQf(x, y, z) = 0 defines a curve of degree n−1 called the first polar of C with respect of Q.
If P=(p, q, r) is a non-singular point on the curve C then the equation of the tangent at P is
In particular, P is on the intersection of C and its first polar with respect to Q if and only if Q is on the tangent to C at P. For a double point of C, the partial derivatives of f are all 0 so the first polar contains these points as well.
Class of a curve
The class of C may be defined as the number of tangents that may be drawn to C from a point not on C (counting multiplicities and including imaginary tangents). Each of these tangents touches C at one of the points of intersection of C and the first polar, and by Bézout's theorem there are at most n(n−1) of these. This puts an upper bound of n(n−1) on the class of a curve of degree n. The class may be computed exactly by counting the number and type of singular points on C (see Plücker formula).
Higher polars
The p-th polar of a C for a natural number p is defined as ΔQpf(x, y, z) = 0. This is a curve of degree n−p. When p is n−1 the p-th polar is a line called the polar line of C with respect to Q. Similarly, when p is n−2 the curve is called the polar conic of C.
Using Taylor series in several variables and exploiting homogeneity, f(λa+μp, λb+μq, λc+μr) can be expanded in two ways as
and
Comparing coefficients of λpμn−p shows that
In particular, the p-th polar of C with respect to Q is the locus of points P so that the (n−p)-th polar of C with respect to P passes through Q.
Poles
If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1)2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1. Similarly, the locus of points whose polar lines pass through Q is the first polar of Q and this is also a curve of degree n−1. The polar line of a point is L if and only if it contains both P and Q, so the poles of L are exactly the points of intersection of the two first polars. By Bézout's theorem these curves have (n−1)2 points of intersection and these are the poles of L.
The Hessian
For a given point Q=(a, b, c), the polar conic is the locus of points P so that Q is on the seco
|
https://en.wikipedia.org/wiki/Linear%20%28disambiguation%29
|
Linear is used to describe linearity in mathematics.
Linear may also refer to:
Mathematics
Linear algebra
Linear code
Linear cryptanalysis
Linear equation
Linear function
Linear functional
Linear map
Linear programming, a type of optimization problem
Linear system
Linear system of equations
Linear transformation
Technology
Particularly in electronics, a device whose characteristic or transfer function is linear, in the mathematical sense, is called linear
Linear amplifier, a component of amateur radio equipment
Linear element, part of an electric circuit
Linear motor a type of electric motor
Linear phase, a property of an electronic filter
Linear Technology, an integrated circuit manufacturer
Linearity (computer and video games)
Other uses
A kind of leaf shape in botany
LINEAR, the Lincoln Near-Earth Asteroid Research project
Linear A, one of two scripts used in ancient Crete
Linear B, a script that was used for writing Mycenaean, an early form of Greek
Linear counterpoint in music
Linear narrative structure
Linear (group), a pop music group popular in the 1990s
Linear (album), their group's debut album
Linear (film), a film that was released with the U2 album No Line on the Horizon
Linear molecular geometry in chemistry
Linear motion, motion along a straight line
Linearity (writing), describing whether symbols in a writing system are composed of lines
A kind of typeface in the VOX-ATypI classification
See also
Curvilinear
Rectilinear (disambiguation)
Straight (disambiguation)
|
https://en.wikipedia.org/wiki/The%20Second%20Hand%20Stopped
|
The Second Hand Stopped is the debut studio album by American metalcore band Odd Project, released on July 13, 2004.
Track listing
Statistics Like Cigarettes – 3:41
The Phone Is Such A Blunt Object – 3:32
A Hero's Trial – 4:23
A Perfect Smile and Broken Wings – 3:33
Tear Stained Lies – 3:40
Love – 3:19
The Fashion Police Hate Robots – 3:31
Photographic Memories – 4:57
The Wanderer – 2:02
Silver Screen Lovers – 4:37
Personnel
Matt Lamb – lead vocals
Scott Zschomler – lead guitar, vocals
Eric Cline – bass guitar, backing vocals
Greg Pawloski – rhythm guitar
Christian Escobar – drums
2004 debut albums
Indianola Records albums
|
https://en.wikipedia.org/wiki/Riesz%20potential
|
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.
Definition
If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by
where the constant is given by
This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α. In fact, for any 1 ≤ p (p>1 is classical, due to Sobolev, while for p=1 see , the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
where is the vector-valued Riesz transform. More generally, the operators Iα are well-defined for complex α such that .
The Riesz potential can be defined more generally in a weak sense as the convolution
where Kα is the locally integrable function:
The Riesz potential can therefore be defined whenever f is a compactly supported distribution. In this connection, the Riesz potential of a positive Borel measure μ with compact support is chiefly of interest in potential theory because Iαμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rn.
Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier.
In fact, one has
and so, by the convolution theorem,
The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions
provided
Furthermore, if , then
One also has, for this class of functions,
See also
Bessel potential
Fractional integration
Sobolev space
Notes
References
.
Fractional calculus
Partial differential equations
Potential theory
Singular integrals
|
https://en.wikipedia.org/wiki/Kepler%E2%80%93Bouwkamp%20constant
|
In plane geometry, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following sequence. Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth.
The radius of the limiting circle is called the Kepler–Bouwkamp constant. It is named after Johannes Kepler and , and is the inverse of the polygon circumscribing constant.
Numerical value
The decimal expansion of the Kepler–Bouwkamp constant is
The natural logarithm of the Kepler-Bouwkamp constant is given by
where is the Riemann zeta function.
If the product is taken over the odd primes, the constant
is obtained .
References
Further reading
External links
Mathematical constants
Infinite products
|
https://en.wikipedia.org/wiki/David%20Norgrove
|
Sir David Ronald Norgrove (born 23 January 1948) is an English businessman and government official, who was chair of the UK Statistics Authority from 2017 to 2022. He was previously the first chairman of The Pensions Regulator, and then chair of the Low Pay Commission.
Early life
Norgrove was born on 23 January 1948 in Peckham, London. He was educated at Christ's Hospital School and read History at Exeter College, Oxford. He gained a diploma in Economics at Cambridge University and then a master's degree in Economics at the London School of Economics.
Career
Norgrove started his career as an economist at HM Treasury (1972–85), where his time included a secondment to the First National Bank of Chicago.
Norgrove was private secretary to Prime Minister Margaret Thatcher between 1985 and 1988.
In 1988 he joined Marks and Spencer, where he held several positions: from 1988–99 he was Director of Europe; Worldwide franchising; Menswear and Strategy. In September 1999 he became chairman of Marks & Spencer's Ventures Division and a year later he was appointed to the executive board as executive director for Strategy, International and Ventures.
Norgrove was heavily involved in the early recovery of Marks and Spencer but in January 2004 he was fired from his role as director of clothing following poor Christmas sales. He continued in his position as chair of the trustees of the Marks & Spencer pension fund until later in the year, playing a role in the attempt by Philip Green to acquire the company.
In March 2004 he was appointed to the board of the British Museum, later becoming deputy chairman. In November 2009 he became the Chairman of The British Museum Friends, where he retired as a trustee in 2012. He was a trustee of Amnesty International Charitable Trust from 2008 to 2014.
Norgrove was appointed as the first chair of The Pensions Regulator between 2005 and 2010 and subsequently named one of the hundred most influential people in the capital markets by Financial News. In 2011 he joined pension consultants PensionsFirst as chairman. From May 2009 until December 2016 he was the chairman of the Low Pay Commission. In April 2017 he became chair of the UK Statistics Authority.
In 2022, Sir Robert Chote was selected to replace Norgrove as chair of the UK Statistics Authority.
Family Justice Review
Norgrove chaired the Family Justice Review in 2011–12. This recommended substantial changes to speed up public law proceedings (cases concerning the protection of children) and to help family justice operate better as a system. In private law (divorce and separation) it recommended changes to help couples avoid the need to undertake court proceedings, for example increased use of mediation. Most media attention was paid to a recommendation that there should not be legislation to create a presumption around parental involvement in children's lives after separation. This was controversial, particularly with groups campaigning for the rights of fathers, an
|
https://en.wikipedia.org/wiki/Jaro%E2%80%93Winkler%20distance
|
In computer science and statistics, the Jaro–Winkler similarity is a string metric measuring an edit distance between two sequences. It is a variant of the Jaro distance metric metric (1989, Matthew A. Jaro) proposed in 1990 by William E. Winkler.
The Jaro–Winkler distance uses a prefix scale which gives more favourable ratings to strings that match from the beginning for a set prefix length .
The higher the Jaro–Winkler distance for two strings is, the less similar the strings are. The score is normalized such that 0 means an exact match and 1 means there is no similarity. The original paper actually defined the metric in terms of similarity, so the distance is defined as the inversion of that value (distance = 1 − similarity).
Although often referred to as a distance metric, the Jaro–Winkler distance is not a metric in the mathematical sense of that term because it does not obey the triangle inequality.
Definition
Jaro similarity
The Jaro similarity of two given strings and is
Where:
is the length of the string ;
is the number of matching characters (see below);
is the number of transpositions (see below).
Jaro similarity score is 0 if the strings do not match at all, and 1 if they are an exact match. In the first step, each character of is compared with all its matching characters in . Two characters from and respectively, are considered matching only if they are the same and not farther than characters apart. For example, the following two nine character long strings, FAREMVIEL and FARMVILLE, have 8 matching characters. 'F', 'A' and 'R' are in the same position in both strings. Also 'M', 'V', 'I', 'E' and 'L' are within three (result of ) characters away. If no matching characters are found then the strings are not similar and the algorithm terminates by returning Jaro similarity score 0.
If non-zero matching characters are found, the next step is to find the number of transpositions. Transposition is the number of matching characters that are not in the right order divided by two. In the above example between FAREMVIEL and FARMVILLE, 'E' and 'L' are the matching characters that are not in the right order. So the number of transposition is one.
Finally, plugging in the number of matching characters and number of transpositions the Jaro similarity of FAREMVIEL and FARMVILLE can be calculated,
Jaro–Winkler similarity
Jaro–Winkler similarity uses a prefix scale which gives more favorable ratings to strings that match from the beginning for a set prefix length . Given two strings and , their Jaro–Winkler similarity is:
where:
is the Jaro similarity for strings and
is the length of common prefix at the start of the string up to a maximum of 4 characters
is a constant scaling factor for how much the score is adjusted upwards for having common prefixes. should not exceed 0.25 (i.e. 1/4, with 4 being the maximum length of the prefix being considered), otherwise the similarity could become larger than 1
|
https://en.wikipedia.org/wiki/History%20of%20trigonometry
|
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics.
Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
Etymology
The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".
The modern words "sine" and "cosine" are derived from the Latin word via mistranslation from Arabic (see Sine and cosine#Etymology). Particularly Fibonacci's sinus rectus arcus proved influential in establishing the term.
The word tangent comes from Latin meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin "cutting" since the line cuts the circle.
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to "first small parts" and "second small parts".
Development
Ancient Near East
The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.
The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere. Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations won't apply. There is, however, much debate as to whether it is a table of Pythagorean tr
|
https://en.wikipedia.org/wiki/Quantum%20Magazine
|
Quantum: The Magazine of Math and Science was a United States-based bimonthly magazine of mathematics and science, primarily physics, designed for young readers. It was published by the National Science Teachers Association (NSTA) and Springer-Verlag and was headquartered in Washington DC.
Quantum was a sister publication of the Russian magazine Kvant. Quantum contained translations from Kvant and original material.
The magazine was founded in 1990. It ceased publication with its July/August 2001 issue.
Two books derived from Quantum materials have been published: Quantoons and Quantum Quandaries.
All articles from the magazine are indexed online by the NSTA.
References
External links
WorldCat info
Student magazines published in the United States
Bimonthly magazines published in the United States
Defunct magazines published in the United States
Education magazines
Magazines established in 1990
Magazines disestablished in 2001
Science education in the United States
Magazines published in Washington, D.C.
|
https://en.wikipedia.org/wiki/Yuri%20Prokhorov
|
Yuri Vasilyevich Prokhorov (; 15 December 1929 – 16 July 2013) was a Russian mathematician, active in the field of probability theory. He was a PhD student of Andrey Kolmogorov at the Moscow State University, where he obtained his PhD in 1956.
Prokhorov became a corresponding member of the Russian Academy of Sciences in 1966, a full member in 1972. He was a vice-president of the IMU. He received Lenin Prize in 1970, Order of the Red Banner of Labour in 1975 and 1979. He was also an editor of the Great Soviet Encyclopedia.
See also
Lévy–Prokhorov metric
Prokhorov's theorem
References
Larry Shepp, "A Conversation with Yuri Vasilyevich Prokhorov", Statistical Science, Vol. 7, No. 1 (February, 1992), pp. 123–130.
External links
Yuri Prokhorov — scientific works on the website Math-Net.Ru
Yuriĭ Vasilʹevich Prokhorov — scientific works in MathSciNet
Prokhorov's Biography (in Russian)
Yuri Vasilevich Prokhorov (in Russian)
Obituaries: Yuri Vasilyevich Prokhorov
20th-century Russian mathematicians
Soviet mathematicians
Probability theorists
Full Members of the USSR Academy of Sciences
Full Members of the Russian Academy of Sciences
Moscow State University alumni
1929 births
2013 deaths
Russian scientists
|
https://en.wikipedia.org/wiki/Langlands%20dual%20group
|
In representation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G (also called the L-group of G) is a group that controls the representation theory of G. If G is defined over a field k, then LG is an extension of the absolute Galois group of k by a complex Lie group. There is also a variation called the Weil form of the L-group, where the Galois group is replaced by a Weil group. Here, the letter L in the name also indicates the connection with the theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by in a letter to A. Weil.
The L-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group G, when k is a global field. It is not exactly G with respect to which automorphic forms and representations are functorial, but LG. This makes sense of numerous phenomena, such as 'lifting' of forms from one group to another larger one, and the general fact that certain groups that become isomorphic after field extensions have related automorphic representations.
Definition for separably closed fields
From a reductive algebraic group over a separably closed field K we can construct its root datum (X*, Δ,X*, Δv), where
X* is the lattice of characters of a maximal torus, X* the dual lattice (given by the 1-parameter subgroups), Δ the roots, and Δv the coroots. A connected reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.
For any root datum (X*, Δ,X*, Δv), we can define a dual root datum (X*, Δv,X*, Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group LG is the complex connected reductive group whose root datum is dual to that of G.
Examples:
The Langlands dual group LG has the same Dynkin diagram as G, except that components of type Bn are changed to components of type Cn and vice versa. If G has trivial center then LG is simply connected, and if G is simply connected then LG has trivial center. The Langlands dual of GLn(K) is GLn(C).
Definition for groups over more general fields
Now suppose that G is a reductive group over some field k with separable closure K. Over K, G has a root datum, and this comes with an action of the Galois group Gal(K/k). The identity component LGo of the L-group is the connected complex reductive group of the dual root datum; this has an induced action of the Galois group Gal(K/k). The full L-group LG is the semidirect product
LG = LGo×Gal(K/k)
of the connected component with the Galois group.
There are some variations of the definition of the L-group, as follows:
Instead of using the
|
https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov%20metric
|
In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.
Definition
Let be a metric space with its Borel sigma algebra . Let denote the collection of all probability measures on the measurable space .
For a subset , define the ε-neighborhood of by
where is the open ball of radius centered at .
The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be
For probability measures clearly .
Some authors omit one of the two inequalities or choose only open or closed ; either inequality implies the other, and , but restricting to open sets may change the metric so defined (if is not Polish).
Properties
If is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus, is a metrization of the topology of weak convergence on .
The metric space is separable if and only if is separable.
If is complete then is complete. If all the measures in have separable support, then the converse implication also holds: if is complete then is complete. In particular, this is the case if is separable.
If is separable and complete, a subset is relatively compact if and only if its -closure is -compact.
If is separable, then , where is the Ky Fan metric.
Relation to other distances
Let be separable. Then
, where is the total variation distance of probability measures
, where is the Wasserstein metric with and have finite th moment.
See also
Lévy metric
Prokhorov's theorem
Tightness of measures
Weak convergence of measures
Wasserstein metric
Radon distance
Total variation distance of probability measures
Notes
References
Measure theory
Metric geometry
Probability theory
Paul Lévy (mathematician)
|
https://en.wikipedia.org/wiki/L%C3%A9vy%20metric
|
In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.
Definition
Let be two cumulative distribution functions. Define the Lévy distance between them to be
Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to L(F, G).
A sequence of cumulative distribution functions weakly converges to another cumulative distribution function if and only if .
See also
Càdlàg
Lévy–Prokhorov metric
Wasserstein metric
References
Measure theory
Metric geometry
Theory of probability distributions
Paul Lévy (mathematician)
|
https://en.wikipedia.org/wiki/Don%20Weatherburn
|
Donald James Weatherburn PSM (born 14 May 1951) was Director of the NSW Bureau of Crime Statistics and Research in Sydney from 1988 until July 2019. He is a professor at the National Drug and Alcohol Research Centre at the University of New South Wales and a Fellow of the Academy of the Social Sciences in Australia.
Early life
Weatherburn attended Newington College (1964-1969) and the University of Sydney where in 1974 he received his BA with first class honours. He completed a Ph.D. at the University of Sydney in 1979 and lectured in the School of Justice Administration at Charles Sturt University.
Career
In 1983 Weatherburn was appointed Senior Research Officer at the NSW Bureau of Crime Statistics and Research (BOCSAR) and four years later he was appointed Foundation Director of Research at the NSW Judicial Commission. He was Director of BOCSAR from 1988 until 3 July 2019.
For his contribution to public debate about crime and justice, he was awarded the Public Service Medal in January 1998. In 2000 he received an Alumni Award for Community Service from the University of Sydney.
Following the introduction in February 2014 of the New South Wales lockout laws which limited the sale of alcohol in the Sydney CBD and Kings Cross, Weatherburn released a study arguing that the drop in the number of alcohol-related assaults since the new laws was "simply precipitous" and "one of the most dramatic effects I've seen in my time, of policy intervention to reduce crime".
In September 2018 Weatherburn admitted his Bureau was at fault for releasing misleading drug detection results. Figures in some cases were doubled, after BOCSAR had mistakenly added positive results to searches by NSW Police.
Publications
Delinquent-prone Communities (Cambridge University Press, 2001)
Law and Order in Australia: Rhetoric and Reality (Federation Press, 2004)
Arresting Incarceration: Pathways Out of Indigenous Imprisonment (Aboriginal Studies Press, 2014)
The Vanishing Criminal: Causes of Decline in Australia’s Crime Rate (Melbourne University Press, 2021) Joint author: Sara Rahman.
References
Further reading
Biography at the University of Sydney
1951 births
Living people
University of Sydney alumni
Academic staff of the University of New South Wales
People educated at Newington College
Recipients of the Public Service Medal (Australia)
|
https://en.wikipedia.org/wiki/Bureau%20of%20Crime%20Statistics%20and%20Research
|
The Bureau of Crime Statistics and Research (BOCSAR), also known as NSW Bureau of Crime Statistics and Research, is an agency of the Department of Communities and Justice responsible for research into crime and criminal justice and evaluation of the initiatives designed to reduce crime and reoffending in the state of New South Wales, Australia.
Management and functions
BOCSAR was established in 1969.
The executive director of BOCSAR since July 2019 is Jackie Fitzgerald. She took over from Don Weatherburn PSM, who spent over 30 years in the position.
The Bureau is responsible for identifying factors affecting the distribution and frequency of crime and the effectiveness of the NSW criminal justice system, and for making this information available to its clients.
It develops and maintains statistical databases on crime and criminal justice in NSW, monitors trends in crime and criminal justice, and also conducts research on crime and criminal justice issues and problems.
Statistical information publicly available
Statistical information and various publications of the Bureau are accessible by the public.
Information about crime that is typically stored in the databases includes:
The type of offence committed
Time and location of the offence
The age, gender, plea, outcome of court appearance and penalty (in the cases of persons charged with criminal offences who appear before the courts)
Aggregated data can answer questions such as which areas have high reported crime rates, how many people are charged with a specific offence, or what penalties are imposed for specific offences.
In the news
In September 2018, then director Weatherburn admitted the Bureau was at fault for releasing misleading drug detection results. Figures in some cases were doubled, after BOCSAR had mistakenly added positive results to searches by NSW Police. However errors like this one have been extremely rare.
In early 2019, Weatherburn announced a new BOCSAR review of circle sentencing (a process which puts Aboriginal adult offenders before a circle of elders, members of the community, police and the judiciary, rather than a traditional courtroom), with results due in 2020. The previous one had been published in 2008.
See also
Government of New South Wales
List of New South Wales government agencies
References
External links
Government agencies of New South Wales
Crime in New South Wales
|
https://en.wikipedia.org/wiki/Manipulative%20%28mathematics%20education%29
|
In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience.
The use of manipulatives in mathematics classrooms throughout the world grew considerably in popularity throughout the second half of the 20th century. Mathematical manipulatives are frequently used in the first step of teaching mathematical concepts, that of concrete representation. The second and third steps are representational and abstract, respectively.
Mathematical manipulatives can be purchased or constructed by the teacher. Examples of common manipulatives include number lines, Cuisenaire rods; fraction strips, blocks, or stacks; base ten blocks (also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix); construction sets (such as Polydron and Zometool); colored tiles or tangrams; pattern blocks; colored counting chips; numicon tiles; chainable links; abaci such as "rekenreks", and geoboards. Improvised teacher-made manipulatives used in teaching place value include beans and bean sticks, or single popsicle sticks and bundles of ten popsicle sticks.
Virtual manipulatives for mathematics are computer models of these objects. Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the Ubersketch.
Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the NCTM.
Some of the manipulatives are now used in other subjects in addition to mathematics. For example, Cuisenaire rods are now used in language arts and grammar, and pattern blocks are used in fine arts.
In teaching and learning
Mathematical manipulatives play a key role in young children's mathematics understanding and development. These concrete objects facilitate children's understanding of important math concepts, then later help them link these ideas to representations and abstract ideas. For example, there are manipulatives specifically designed to help students learn fractions, geometry and algebra. Here we will look at pattern blocks, interlocking cubes, and tiles and the various concepts taught through using them. This is by no means an exhaustive list (there are so many possibilities!), rather, these descriptions will provide just a few ideas for how these manipulatives can be used.
Base ten blocks
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and divis
|
https://en.wikipedia.org/wiki/Bitruncation
|
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves.
Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation or
In regular polyhedra and tilings
For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
In regular 4-polytopes and honeycombs
For a regular 4-polytope, a bitruncated form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, and will have double the symmetry if the original 4-polytope is self-dual.
A regular polytope (or honeycomb) {p, q, r} will have its {p, q} cells bitruncated into truncated {q, p} cells, and the vertices are replaced by truncated {q, r} cells.
Self-dual {p,q,p} 4-polytope/honeycombs
An interesting result of this operation is that self-dual 4-polytope {p,q,p} (and honeycombs) remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra: t{q,p}. Two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.
See also
uniform polyhedron
uniform 4-polytope
Rectification (geometry)
Truncation (geometry)
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
External links
Polytopes
|
https://en.wikipedia.org/wiki/Rotation%20of%20axes%20in%20two%20dimensions
|
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle . A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation.
Motivation
Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola, ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates.
The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.
Derivation
The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle into the x′y′ axes, are derived as follows.
In the xy system, let the point P have polar coordinates . Then, in the x′y′ system, P will have polar coordinates .
Using trigonometric functions, we have
and using the standard trigonometric formulae for differences, we have
Substituting equations () and () into equations () and (), we obtain
Equations () and () can be represented in matrix form as
which is the standard matrix equation of a rotation of axes in two dimensions.
The inverse transformation is
or
Examples in two dimensions
Example 1
Find the coordinates of the point after the axes have been rotated through the angle , or 30°.
Solution:
The axes have been rotated counterclockwise through an angle of and the new coordinates are . Note that the point appears to have been rotated clockwise through with respect to fixed axes so it now coincides with the (new) x′ axis.
Example 2
Find the coordinates of the point after the axes have been rotated clockwise 90°, that is, through the angle , or −90°.
Solution:
The axes have been rotated through an angle of , which is in the clockwise direction and the new coordinates are . Again, note that the point appears to have been rotated counterclockwise through with respect to fixed axes.
Rotation of conic sections
The most general equation
|
https://en.wikipedia.org/wiki/Runcination
|
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.
It is a higher order truncation operation, following cantellation, and truncation.
It is represented by an extended Schläfli symbol t0,3{p,q,...}. This operation only exists for 4-polytopes {p,q,r} or higher.
This operation is dual-symmetric for regular uniform 4-polytopes and 3-space convex uniform honeycombs.
For a regular {p,q,r} 4-polytope, the original {p,q} cells remain, but become separated. The gaps at the separated faces become p-gonal prisms. The gaps between the separated edges become r-gonal prisms. The gaps between the separated vertices become {r,q} cells. The vertex figure for a regular 4-polytope {p,q,r} is an q-gonal antiprism (called an antipodium if p and r are different).
For regular 4-polytopes/honeycombs, this operation is also called expansion by Alicia Boole Stott, as imagined by moving the cells of the regular form away from the center, and filling in new faces in the gaps for each opened vertex and edge.
Runcinated 4-polytopes/honeycombs forms:
See also
Uniform polyhedron
Uniform 4-polytope
Rectification (geometry)
Truncation (geometry)
Cantellation (geometry)
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation, p 210 Expansion)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
External links
Polytopes
|
https://en.wikipedia.org/wiki/Pointwise%20mutual%20information
|
In statistics, probability theory and information theory, pointwise mutual information (PMI), or point mutual information, is a measure of association. It compares the probability of two events occurring together to what this probability would be if the events were independent.
PMI (especially in its positive pointwise mutual information variant) has been described as "one of the most important concepts in NLP", where it "draws on the intuition that the best way to weigh the association between two words is to ask how much more the two words co-occur in [a] corpus than we would have a priori expected them to appear by chance."
The concept was introduced in 1961 by Robert Fano under the name of "mutual information", but today that term is instead used for a related measure of dependence between random variables: The mutual information (MI) of two discrete random variables refers to the average PMI of all possible events.
Definition
The PMI of a pair of outcomes x and y belonging to discrete random variables X and Y quantifies the discrepancy between the probability of their coincidence given their joint distribution and their individual distributions, assuming independence. Mathematically:
(with the latter two expressions being equal to the first by Bayes' theorem). The mutual information (MI) of the random variables X and Y is the expected value of the PMI (over all possible outcomes).
The measure is symmetric (). It can take positive or negative values, but is zero if X and Y are independent. Note that even though PMI may be negative or positive, its expected outcome over all joint events (MI) is non-negative. PMI maximizes when X and Y are perfectly associated (i.e. or ), yielding the following bounds:
Finally, will increase if is fixed but decreases.
Here is an example to illustrate:
Using this table we can marginalize to get the following additional table for the individual distributions:
With this example, we can compute four values for . Using base-2 logarithms:
(For reference, the mutual information would then be 0.2141709.)
Similarities to mutual information
Pointwise Mutual Information has many of the same relationships as the mutual information. In particular,
Where is the self-information, or .
Variants
Several variations of PMI have been proposed, in particular to address what has been described as its "two main limitations":
PMI can take both positive and negative values and has no fixed bounds, which makes it harder to interpret.
PMI has "a well-known tendency to give higher scores to low-frequency events", but in applications such as measuring word similarity, it is preferable to have "a higher score for pairs of words whose relatedness is supported by more evidence."
Positive PMI
The positive pointwise mutual information (PPMI) measure is defined by setting negative values of PMI to zero:
This definition is motivated by the observation that "negative PMI values (which imply things are co-occur
|
https://en.wikipedia.org/wiki/Atiyah%E2%80%93Hirzebruch%20spectral%20sequence
|
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex and a generalized cohomology theory , it relates the generalized cohomology groups
with 'ordinary' cohomology groups with coefficients in the generalized cohomology of a point. More precisely, the term of the spectral sequence is , and the spectral sequence converges conditionally to .
Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where . It can be derived from an exact couple that gives the page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with .
In detail, assume to be the total space of a Serre fibration with fibre and base space . The filtration of by its -skeletons gives rise to a filtration of . There is a corresponding spectral sequence with term
and converging to the associated graded ring of the filtered ring
.
This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre is a point.
Examples
Topological K-theory
For example, the complex topological -theory of a point is
where is in degree
By definition, the terms on the -page of a finite CW-complex look like
Since the -theory of a point is
we can always guarantee that
This implies that the spectral sequence collapses on for many spaces. This can be checked on every , algebraic curves, or spaces with non-zero cohomology in even degrees. Therefore, it collapses for all (complex) even dimensional smooth complete intersections in .
Cotangent bundle on a circle
For example, consider the cotangent bundle of . This is a fiber bundle with fiber so the -page reads as
Differentials
The odd-dimensional differentials of the AHSS for complex topological K-theory can be readily computed. For it is the Steenrod square where we take it as the composition
where is reduction mod and is the Bockstein homomorphism (connecting morphism) from the short exact sequence
Complete intersection 3-fold
Consider a smooth complete intersection 3-fold (such as a complete intersection Calabi-Yau 3-fold). If we look at the -page of the spectral sequence
we can see immediately that the only potentially non-trivial differentials are
It turns out that these differentials vanish in both cases, hence . In the first case, since is trivial for we have the first set of differentials are zero. The second set are trivial because sends the identification shows the differential is trivial.
Twisted K-theory
The Atiyah–Hirzebruch spectral sequence can be used to compute twisted K-theory groups as well. In short, twisted K-theory is the group completion of the isomorphism classes of vector bundles defined by gluing data where
for some cohomology class . Then, the spectral sequence reads as
but with different differentials. Fo
|
https://en.wikipedia.org/wiki/Regular%20semigroup
|
In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such that . Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.
History
Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of regularity in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann. It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees.
The term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the papers of Gabriel Thierrin (a student of Paul Dubreil) in the 1950s, and it is still used occasionally.
The basics
There are two equivalent ways in which to define a regular semigroup S:
(1) for each a in S, there is an x in S, which is called a pseudoinverse, with axa = a;
(2) every element a has at least one inverse b, in the sense that aba = a and bab = b.
To see the equivalence of these definitions, first suppose that S is defined by (2). Then b serves as the required x in (1). Conversely, if S is defined by (1), then xax is an inverse for a, since a(xax)a = axa(xa) = axa = a and (xax)a(xax) = x(axa)(xax) = xa(xax) = x(axa)x = xax.
The set of inverses (in the above sense) of an element a in an arbitrary semigroup S is denoted by V(a). Thus, another way of expressing definition (2) above is to say that in a regular semigroup, V(a) is nonempty, for every a in S. The product of any element a with any b in V(a) is always idempotent: abab = ab, since aba = a.
Examples of regular semigroups
Every group is a regular semigroup.
Every band (idempotent semigroup) is regular in the sense of this article, though this is not what is meant by a regular band.
The bicyclic semigroup is regular.
Any full transformation semigroup is regular.
A Rees matrix semigroup is regular.
The homomorphic image of a regular semigroup is regular.
Unique inverses and unique pseudoinverses
A regular semigroup in which idempotents commute (with idempotents) is an inverse semigroup, or equivalently, every element has a unique inverse. To see this, let S be a regular semigroup in which idempotents commute. Then every element of S has at least one inverse. Suppose that a in S has two inverses b and c, i.e.,
aba = a, bab = b, aca = a and cac = c. Also ab, ba, ac and ca are idempotents as above.
Then
b = bab = b(aca)b = bac(a)b = bac(aca)b = bac(ac)(ab) = bac(ab)(ac) = ba(ca)bac = ca(ba)bac = c(aba)bac = cabac = cac = c.
So, by commuting the pairs of idempotents ab & ac and ba & ca, the inverse of a is shown to be un
|
https://en.wikipedia.org/wiki/L-group
|
In mathematics, L-group or l-group may refer to the following groups:
The Langlands dual, LG, of a reductive algebraic group G
A group in L-theory, L(G)
Lattice-ordered groups
|
https://en.wikipedia.org/wiki/Ribbon%20Hopf%20algebra
|
A ribbon Hopf algebra is a quasitriangular Hopf algebra which possess an invertible central element more commonly known as the ribbon element, such that the following conditions hold:
where . Note that the element u exists for any quasitriangular Hopf algebra, and
must always be central and satisfies , so that all that is required is that it have a central square root with the above properties.
Here
is a vector space
is the multiplication map
is the co-product map
is the unit operator
is the co-unit operator
is the antipode
is a universal R matrix
We assume that the underlying field is
If is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.
See also
Quasitriangular Hopf algebra
Quasi-triangular quasi-Hopf algebra
References
Hopf algebras
|
https://en.wikipedia.org/wiki/Calculus%20of%20predispositions
|
Calculus of predispositions is a basic part of predispositioning theory and belongs to the indeterministic procedures.
Overview
"The key component of any indeterministic procedure is the evaluation of a position. Since it is impossible to devise a deterministic chain linking the inter-mediate state with the outcome of the game, the most complex component of any indeterministic method is assessing these intermediate stages. It is precisely the function of predispositions to assess the impact of an intermediate state upon the future course of development."
According to Aron Katsenelinboigen, calculus of predispositions is another method of computing probability. Both methods may lead to the same results and, thus, can be interchangeable. However, it is not always possible to interchange them since computing via frequencies requires availability of statistics, possibility to gather the data as well as having the knowledge of the extent to which one can interlink the system’s constituent elements. Also, no statistics can be obtained on unique events and, naturally, in such cases the calculus of predispositions becomes the only option.
The procedure of calculating predispositions is linked to two steps – dissection of the system on its constituent elements and integration of the analyzed parts in a new whole. According to Katsenelinboigen, the system is structured by two basic types of parameters – material and positional. The material parameters constitute the skeleton of the system. Relationships between them form positional parameters. The calculus of predispositions primarily deals with
analyzing the system’s material and positional parameters as independent variables and
measuring them in unconditional valuations.
"In order to quantify the evaluation of a position we need new techniques, which I have grouped under the heading of calculus of predispositions. This calculus is based on a weight function, which represents a variation on the well-known criterion of optimality for local extremum. This criterion incorporates material parameters and their conditional valuations.
The following key elements distinguish the modified weight function from the criterion of optimality:
First and foremost, the weight function includes not only material parameters as independent (controlling) variables, but also positional (relational) parameters. The valuations of material and positional parameters comprising the weight function are, to a certain extent, unconditional; that is, they are independent of the specific conditions, but do take into account the rules of the game and statistics (experience)." (The Concept of Indeterminism 35)
Relation to frequency probability
There are some differences between frequency-based and predispositions-based methods of computing probability.
The frequency-based method is grounded in statistics and frequencies of events.
The predispositions-based method approaches a system from the point of view of its predisp
|
https://en.wikipedia.org/wiki/Classical%20Wiener%20space
|
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
Definition
Consider E ⊆ Rn and a metric space (M, d). The classical Wiener space C(E; M) is the space of all continuous functions f : E → M. I.e. for every fixed t in E,
as
In almost all applications, one takes E = [0, T ] or [0, +∞) and M = Rn for some n in N. For brevity, write C for C([0, T ]; Rn); this is a vector space. Write C0 for the linear subspace consisting only of those functions that take the value zero at the infimum of the set E. Many authors refer to C0 as "classical Wiener space".
For a stochastic process and the space of all functions from to , one looks at the map . One can then define the coordinate maps or canonical versions defined by . The form another process. The Wiener measure is then the unique measure on such that the coordinate process is a Brownian motion.
Properties of classical Wiener space
Uniform topology
The vector space C can be equipped with the uniform norm
turning it into a normed vector space (in fact a Banach space). This norm induces a metric on C in the usual way: . The topology generated by the open sets in this metric is the topology of uniform convergence on [0, T ], or the uniform topology.
Thinking of the domain [0, T ] as "time" and the range Rn as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space slightly" and get the graph of f to lie on top of the graph of g, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.
Separability and completeness
With respect to the uniform metric, C is both a separable and a complete space:
separability is a consequence of the Stone–Weierstrass theorem;
completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.
Since it is both separable and complete, C is a Polish space.
Tightness in classical Wiener space
Recall that the modulus of continuity for a function f : [0, T ] → Rn is defined by
This definition makes sense even if f is not continuous, and it can be shown that f is continuous if and only if its modulus of continuity tends to zero as δ → 0:
.
By an application of the Arzelà-Ascoli theorem, one can show that a sequence of probability measures on classical Wiener space C is tight if and only if both the following conditions are met:
and
for all ε > 0.
Classical Wiener measure
There is a "standard" measure on C0, known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least) two equivalent characterizations:
If one defines Br
|
https://en.wikipedia.org/wiki/Equivalence%20of%20metrics
|
In mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. Equivalence is a weaker notion than isometry; equivalent metrics do not have to be literally the same. Instead, it is one of several ways of generalizing equivalence of norms to general metric spaces.
Throughout the article, will denote a non-empty set and and will denote two metrics on .
Topological equivalence
The two metrics and are said to be topologically equivalent if they generate the same topology on . The adverb topologically is often dropped. There are multiple ways of expressing this condition:
a subset is -open if and only if it is -open;
the open balls "nest": for any point and any radius , there exist radii such that
the identity function is continuous with continuous inverse; that is, it is a homeomorphism.
The following are sufficient but not necessary conditions for topological equivalence:
there exists a strictly increasing, continuous, and subadditive such that .
for each , there exist positive constants and such that, for every point ,
Strong equivalence
Two metrics and on are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants and such that, for every ,
In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different constants associated with each point of .
Strong equivalence of two metrics implies topological equivalence, but not vice versa. For example, the metrics and on the interval are topologically equivalent, but not strongly equivalent. In fact, this interval is bounded under one of these metrics but not the other. On the other hand, strong equivalences always take bounded sets to bounded sets.
Relation with equivalence of norms
When is a vector space and the two metrics and are those induced by norms and , respectively, then strong equivalence is equivalent to the condition that, for all ,
For linear operators between normed vector spaces, Lipschitz continuity is equivalent to continuity—an operator satisfying either of these conditions is called bounded. Therefore, in this case, and are topologically equivalent if and only if they are strongly equivalent; the norms and are simply said to be equivalent.
In finite dimensional vector spaces, all metrics induced by a norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are equivalent.
Properties preserved by equivalence
The continuity of a function is preserved if either the domain or range is remetrized by an equivalent metric, but uniform continuity is preserved only by strongly equivalent metrics.
The differentiability of a function , for a normed space and a subset of a normed space, is preserved if either the domain or range is reno
|
https://en.wikipedia.org/wiki/Chia-Hsiung%20Tze
|
Chia-Hsiung Tze (often H.C. Tze) is a professor emeritus at Virginia Tech. He is a theoretical particle physicist focusing on group theory, string theory, supersymmetry, octonions and other topics in theoretical physics.
He was a colleague of the Feza Gürsey.
Publications
Articles
Books
References
Living people
Particle physicists
Year of birth missing (living people)
Virginia Tech faculty
|
https://en.wikipedia.org/wiki/Poncelet%20point
|
In geometry, the Poncelet point of four given points is defined as follows:
Let be four points in the plane that do not form an orthocentric system and such that no three of them are collinear. The nine-point circles of triangles meet at one point, the Poncelet point of the points . (If do form an orthocentric system, then triangles all share the same nine-point circle, and the Poncelet point is undefined.)
Properties
If do not lie on a circle, the Poncelet point of lies on the circumcircle of the pedal triangle of with respect to triangle and lies on the other analogous circles. (If they do lie on a circle, then those pedal triangles will be lines; namely, the Simson line of with respect to triangle , and the other analogous Simson lines. In that case, those lines still concur at the Poncelet point, which will also be the anticenter of the cyclic quadrilateral whose vertices are .)
The Poncelet point of lies on the circle through the intersection of lines and , the intersection of lines and , and the intersection of lines and (assuming all these intersections exist).
The Poncelet point of is the center of the unique rectangular hyperbola through .
References
Euclidean plane geometry
|
https://en.wikipedia.org/wiki/Convergence%20of%20measures
|
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
Three of the most common notions of convergence are described below.
Informal descriptions
This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that the measure of some sets could be infinite, or that the underlying space could exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if is a sequence of probability measures on a Polish space.
The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge:
To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.
The notion of weak convergence requires this convergence to take place for every continuous bounded function .
This notion treats convergence for different functions f independently of one another, i.e., different functions f may require different values of N ≤ n to be approximated equally well (thus, convergence is non-uniform in ).
The notion of setwise convergence formalizes the assertion that the measure of each measurable set should converge:
Again, no uniformity over the set is required.
Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence. As a matter of fact, when considering sequences of measures with uniformly bounded
variation on a Polish space, setwise convergence implies the convergence for any bounded measurable function .
As before, this convergence is non-uniform in
The notion of total variation convergence formalizes the assertion that the measure of all measurable sets should converge uniformly, i.e. for every there exists N such that for every n > N and for every measurable set . As before, this implies convergence of
|
https://en.wikipedia.org/wiki/Product%20metric
|
In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces which metrizes the product topology. The most prominent product metrics are the p product metrics for a fixed :
It is defined as the p norm of the n-vector of the distances measured in n subspaces:
For this metric is also called the sup metric:
Choice of norm
For Euclidean spaces, using the L2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. In the category of metric spaces (with Lipschitz maps having Lipschitz constant 1), the product (in the category theory sense) uses the sup metric.
The case of Riemannian manifolds
For Riemannian manifolds and , the product metric on is defined by
for under the natural identification .
References
.
.
Metric geometry
|
https://en.wikipedia.org/wiki/Malliavin%20derivative
|
In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense.
Definition
Let be the Cameron–Martin space, and denote classical Wiener space:
;
By the Sobolev embedding theorem, . Let
denote the inclusion map.
Suppose that is Fréchet differentiable. Then the Fréchet derivative is a map
i.e., for paths , is an element of , the dual space to . Denote by the continuous linear map defined by
sometimes known as the H-derivative. Now define to be the adjoint of in the sense that
Then the Malliavin derivative is defined by
The domain of is the set of all Fréchet differentiable real-valued functions on ; the codomain is .
The Skorokhod integral is defined to be the adjoint of the Malliavin derivative:
See also
H-derivative
References
Generalizations of the derivative
Stochastic calculus
Malliavin calculus
|
https://en.wikipedia.org/wiki/John%20Hogan%20%28mathematician%29
|
S. John Hogan is a professor of Applied Mathematics and leader of the "Applied Nonlinear Mathematics Group" in the Department of Engineering Mathematics, University of Bristol. He is known for his work in numerous applications of non-linear dynamics including water waves liquid crystals.
Hogan is principal investigator on several large EPSRC grants, in 2008 totalling around £6M – an unusually high total for a UK mathematician. These include the "Bristol Centre for Complexity Sciences", the "Bristol Centre for Applied Nonlinear Mathematics", "Applied Nonlinear Mathematics: Making it Real" and
Recent publications
2006 Impact dynamics of large dimensional systems Homer ME and Hogan SJ
2005 Real-time dynamic sub structuring in a coupled oscillator-pendulum system Kyrychko YN, Blyuss KB, Gonzalez-Buelga A, Hogan SJ and Wagg DJ
2005 Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems Kowalczyk PS, di Bernardo M, Champneys AR, Hogan SJ, Homer ME, Kuznetsov YA, Nordmark A and Piiroinen PT
2004 Global dynamics of low immersion high-speed milling Szalai R, Stepan G and Hogan SJ
References
External links
Page at University of Bristol
Year of birth missing (living people)
Living people
English mathematicians
Academics of the University of Bristol
|
https://en.wikipedia.org/wiki/Regulated%20integral
|
In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné.
Definition
Definition on step functions
Let [a, b] be a fixed closed, bounded interval in the real line R. A real-valued function φ : [a, b] → R is called a step function if there exists a finite partition
of [a, b] such that φ is constant on each open interval (ti, ti+1) of Π; suppose that this constant value is ci ∈ R. Then, define the integral of a step function φ to be
It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of [a, b] such that φ is constant on the open intervals of Π1, then the numerical value of the integral of φ is the same for Π1 as for Π.
Extension to regulated functions
A function f : [a, b] → R is called a regulated function if it is the uniform limit of a sequence of step functions on [a, b]:
there is a sequence of step functions (φn)n∈N such that as n → ∞; or, equivalently,
for all ε > 0, there exists a step function φε such that || φε − f ||∞ < ε; or, equivalently,
f lies in the closure of the space of step functions, where the closure is taken in the space of all bounded functions [a, b] → R and with respect to the supremum norm || ⋅ ||∞; or equivalently,
for every , the right-sided limit
exists, and, for every , the left-sided limit
exists as well.
Define the integral of a regulated function f to be
where (φn)n∈N is any sequence of step functions that converges uniformly to f.
One must check that this limit exists and is independent of the chosen sequence, but this
is an immediate consequence of the continuous linear extension theorem of elementary
functional analysis: a bounded linear operator T0 defined on a dense linear subspace E0 of a normed linear space E and taking values in a Banach space F extends uniquely to a bounded linear operator T : E → F with the same (finite) operator norm.
Properties of the regulated integral
The integral is a linear operator: for any regulated functions f and g and constants α and β,
The integral is also a bounded operator: every regulated function f is bounded, and if m ≤ f(t) ≤ M for all t ∈ [a, b], then
In particular:
Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
Extension to functions defined on the whole real line
It is possible to extend the definitions of step function and regulated function and the associated integrals to functions defined on the whole real line. However, care must be taken with certain technical points:
the partition on whose open intervals a step function is required to be constant is allowed to be a countable set, but must be a discrete set, i.e. ha
|
https://en.wikipedia.org/wiki/Connecticut%20Mastery%20Test
|
The Connecticut Mastery Test, or CMT, is a test administered to students in grades 3 through 8. The CMT tests students in mathematics, reading comprehension, writing, and science (science was administered in March 2008). The other major standardized test administered to schoolchildren in Connecticut is the Connecticut Academic Performance Test, or CAPT, which is given in grade 10. Until the 2005–2006 school year, the CMT was administered in the fall; now it is given in the spring.
The CMT is graded on a scale from 1 to 5 in each area, on this scale:
5 - "Advanced"
4 - "Goal"
3 - "Proficient"
2 - "Basic"
1 - "Below basic."
Structure
Editing and Revising
This is the first portion of the CMT writing test. Students read passages that contain numerous spelling and grammar errors. After reading, they will answer multiple choice questions to correct the errors. This test is sixty minutes long and it is scored by a computer.
Direct Assessment of Writing
In this test, students have 45 minutes to write a paper on a designated topic. In third and fourth grade, the essay is a fictional narrative; in fifth and sixth it is an expository piece; in seventh and eighth grade it is a persuasive essay. It is scored by two trained professionals. Each reader scores it from one to six. The two scores are combined to make one total score, the state target goal is 8.0 out of 12.
Degrees of Reading Power
Also known as the DRP, this is the first portion of the reading section. Students must read through passages which have blanks in them. They must then choose the correct answer to fill in the blank from a choice of options. Students must fill in 49 answers (seven questions per passage, seven passages) in the DRP section of the test booklets. The questions gradually get harder as the students go on. The test is 45 minutes long and is put through a machine.
Reading Comprehension
This test requires students to read four passages and answer questions about what they just read. There are multiple choice questions as well as written responses, in which the students are given lines to write their answers on. These questions often involve personal connections, the reader's opinion on a topic, and other questions that do not have a definite correct answer. The multiple choice questions are machine-scored while the written responses are scored by professional readers who score it with a 0, 1, or 2, depending on how well the question was answered. The test is divided into 2 sessions (2 passages, 15 questions per session). Each session is 45 minutes long.
Mathematics
The mathematics portion of the CMT assesses students on skills and concepts they are expected to have learned by the time of the test. In grades three and four, there are two test sessions, and in grades five through eight there are three. Each test session is 60 minutes long.
The test consists of three formats: multiple choice, open-ended, and grid-in.
Multiple choice questions are where students are provid
|
https://en.wikipedia.org/wiki/5-polytope
|
In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
Definition
A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met:
Each cell must join exactly two 4-faces.
Adjacent 4-faces are not in the same four-dimensional hyperplane.
The figure is not a compound of other figures which meet the requirements.
Characteristics
The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Classification
5-polytopes may be classified based on properties like "convexity" and "symmetry".
A 5-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 5-polytope is contained in the 5-polytope or its interior; otherwise, it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.
A uniform 5-polytope has a symmetry group under which all vertices are equivalent, and its facets are uniform 4-polytopes. The faces of a uniform polytope must be regular.
A semi-regular 5-polytope contains two or more types of regular 4-polytope facets. There is only one such figure, called a demipenteract.
A regular 5-polytope has all identical regular 4-polytope facets. All regular 5-polytopes are convex.
A prismatic 5-polytope is constructed by a Cartesian product of two lower-dimensional polytopes. A prismatic 5-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of a square and a cube), but is considered separately because it has symmetries other than those inherited from its factors.
A 4-space tessellation is the division of four-dimensional Euclidean space into a regular grid of polychoral facets. Strictly speaking, tessellations are not polytopes as they do not bound a "5D" volume, but we include them here for the sake of completeness because they are similar in many ways to polytopes. A uniform 4-space tessellation is one whose vertices are related by a space group and wh
|
https://en.wikipedia.org/wiki/Uniform%208-polytope
|
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets.
Regular 8-polytopes
Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.
There are exactly three such convex regular 8-polytopes:
{3,3,3,3,3,3,3} - 8-simplex
{4,3,3,3,3,3,3} - 8-cube
{3,3,3,3,3,3,4} - 8-orthoplex
There are no nonconvex regular 8-polytopes.
Characteristics
The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Uniform 8-polytopes by fundamental Coxeter groups
Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
Selected regular and uniform 8-polytopes from each family include:
Simplex family: A8 [37] -
135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
{37} - 8-simplex or ennea-9-tope or enneazetton -
Hypercube/orthoplex family: B8 [4,36] -
255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
{4,36} - 8-cube or octeract-
{36,4} - 8-orthoplex or octacross -
Demihypercube D8 family: [35,1,1] -
191 uniform 8-polytopes as permutations of rings in the group diagram, including:
{3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
{3,3,3,3,3,31,1} - 8-orthoplex, 511 -
E-polytope family E8 family: [34,1,1] -
255 uniform 8-polytopes as permutations of rings in the group diagram, including:
{3,3,3,3,32,1} - Thorold Gosset's semiregular 421,
{3,34,2} - the uniform 142, ,
{3,3,34,1} - the uniform 241,
Uniform prismatic forms
There are many uniform prismatic families, including:
The A8 family
The A8 family has symmetry of order 362880 (9 factorial).
There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (128+8-1 cases) These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.
See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.
The B8 family
The B8 family has symmetry of order 10321920 (8 factorial x 28)
|
https://en.wikipedia.org/wiki/Uniform%207-polytope
|
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.
Regular 7-polytopes
Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.
There are exactly three such convex regular 7-polytopes:
{3,3,3,3,3,3} - 7-simplex
{4,3,3,3,3,3} - 7-cube
{3,3,3,3,3,4} - 7-orthoplex
There are no nonconvex regular 7-polytopes.
Characteristics
The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Uniform 7-polytopes by fundamental Coxeter groups
Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
The A7 family
The A7 family has symmetry of order 40320 (8 factorial).
There are 71 (64+8-1) forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing.
See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes.
The B7 family
The B7 family has symmetry of order 645120 (7 factorial x 27).
There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Johnson and Bowers names.
See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.
The D7 family
The D7 family has symmetry of order 322560 (7 factorial x 26).
This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these, 63 (2×32−1) are repeated from the B7 family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.
See also list of D7 polytopes for Coxeter plane graphs of these polytopes.
The E7 family
The E7 Coxeter group has order 2,903,040.
There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.
Regular and uniform honeycombs
There are five fundamental affine Coxeter groups and sixteen prismatic groups that generate regular and u
|
https://en.wikipedia.org/wiki/Uniform%209-polytope
|
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets.
Regular 9-polytopes
Regular 9-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w}, with w {p,q,r,s,t,u,v} 8-polytope facets around each peak.
There are exactly three such convex regular 9-polytopes:
{3,3,3,3,3,3,3,3} - 9-simplex
{4,3,3,3,3,3,3,3} - 9-cube
{3,3,3,3,3,3,3,4} - 9-orthoplex
There are no nonconvex regular 9-polytopes.
Euler characteristic
The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Uniform 9-polytopes by fundamental Coxeter groups
Uniform 9-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
Selected regular and uniform 9-polytopes from each family include:
Simplex family: A9 [38] -
271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
{38} - 9-simplex or deca-9-tope or decayotton -
Hypercube/orthoplex family: B9 [4,38] -
511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
{4,37} - 9-cube or enneract -
{37,4} - 9-orthoplex or enneacross -
Demihypercube D9 family: [36,1,1] -
383 uniform 9-polytope as permutations of rings in the group diagram, including:
{31,6,1} - 9-demicube or demienneract, 161 - ; also as h{4,38} .
{36,1,1} - 9-orthoplex, 611 -
The A9 family
The A9 family has symmetry of order 3628800 (10 factorial).
There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.
The B9 family
There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing. Bowers-style acronym names are given in parentheses for cross-referencing.
The D9 family
The D9 family has symmetry of order 92,897,280 (9 factorial × 28).
This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9
|
https://en.wikipedia.org/wiki/Uniform%206-polytope
|
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.
The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}.
History of discovery
Regular polytopes: (convex faces)
1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
Convex uniform polytopes:
1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra)
Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete.
Uniform 6-polytopes by fundamental Coxeter groups
Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.
There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes.
Uniform prismatic families
Uniform prism
There are 6 categorical uniform prisms based on the uniform 5-polytopes.
Uniform duoprism
There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Five are formed as the product of a uniform 4-polytope with a regular polygon, and six are formed by the product of two uniform polyhedra:
Uniform triaprism
There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.
Enumerating the convex uniform 6-polytopes
Simplex family: A6 [34] -
35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
{34} - 6-simplex -
Hypercube/orthoplex family: B6 [4,34] -
63 uniform 6-polytopes as permutations
|
https://en.wikipedia.org/wiki/Additive%20basis
|
In additive number theory, an additive basis is a set of natural numbers with the property that, for some finite number , every natural number can be expressed as a sum of or fewer elements of . That is, the sumset of copies of consists of all natural numbers. The order or degree of an additive basis is the number . When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set for which all but finitely many natural numbers can be expressed as a sum of or fewer elements of .
For example, by Lagrange's four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal numbers for -sided polygons form an additive basis of order . Similarly, the solutions to Waring's problem imply that the th powers are an additive basis, although their order is more than . By Vinogradov's theorem, the prime numbers are an asymptotic additive basis of order at most four, and Goldbach's conjecture would imply that their order is three.
The unproven Erdős–Turán conjecture on additive bases states that, for any additive basis of order , the number of representations of the number as a sum of elements of the basis tends to infinity in the limit as goes to infinity. (More precisely, the number of representations has no finite supremum.) The related Erdős–Fuchs theorem states that the number of representations cannot be close to a linear function. The Erdős–Tetali theorem states that, for every , there exists an additive basis of order whose number of representations of each is .
A theorem of Lev Schnirelmann states that any sequence with positive Schnirelmann density is an additive basis. This follows from a stronger theorem of Henry Mann according to which the Schnirelmann density of a sum of two sequences is at least the sum of their Schnirelmann densities, unless their sum consists of all natural numbers. Thus, any sequence of Schnirelmann density is an additive basis of order at most .
References
Additive number theory
|
https://en.wikipedia.org/wiki/Root%20datum
|
In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.
Definition
A root datum consists of a quadruple
,
where
and are free abelian groups of finite rank together with a perfect pairing between them with values in which we denote by ( , ) (in other words, each is identified with the dual of the other).
is a finite subset of and is a finite subset of and there is a bijection from onto , denoted by .
For each , .
For each , the map induces an automorphism of the root datum (in other words it maps to and the induced action on maps to )
The elements of are called the roots of the root datum, and the elements of are called the coroots.
If does not contain for any , then the root datum is called reduced.
The root datum of an algebraic group
If is a reductive algebraic group over an algebraically closed field with a split maximal torus then its root datum is a quadruple
,
where
is the lattice of characters of the maximal torus,
is the dual lattice (given by the 1-parameter subgroups),
is a set of roots,
is the corresponding set of coroots.
A connected split reductive algebraic group over is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.
For any root datum , we can define a dual root datum by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If is a connected reductive algebraic group over the algebraically closed field , then its Langlands dual group is the complex connected reductive group whose root datum is dual to that of .
References
Michel Demazure, Exp. XXI in SGA 3 vol 3
T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1
Representation theory
Algebraic groups
|
https://en.wikipedia.org/wiki/Elm%C4%81rs%20Zemgalis
|
Elmārs Zemgalis (9 September 1923 – 8 December 2014) was a Latvian-American chess master and mathematics professor at Highline College. He was awarded an Honorary Grandmaster title in 2003.
Biography
Zemgalis started to play chess when he was eleven, eventually winning the championships of Riga and Jelgava. After the Soviet Union invaded his native Latvia for the second time in 1944, Zemgalis fled to Germany. As a Displaced Person after World War II, he played in twelve international tournaments. In 1946, he took second place, behind Wolfgang Unzicker, in Augsburg, with 13/16. In 1946, he took second place, behind Fedor Bohatirchuk, in Regensburg (Klaus Junge Memorial), with 6.5/9. In 1947, he took second place, behind Lūcijs Endzelīns in Hanau (Hermanis Matisons Memorial). In 1948, he won in Esslingen (Württemberg-ch), with 7/9. In 1949, he won in Rujtā (Württemberg-ch). In 1949, he tied for first place with Efim Bogoljubow in Oldenburg. In 1949, he tied for first place with Leonids Dreibergs in Esslingen.
In 1951, he emigrated to the United States, where he became a mathematics professor. By 1952, Zemgalis had settled in Seattle. He was arguably the top player in the Pacific Northwest for the next fifteen years. In 1952, he won (3:1) a match against Olaf Ulvestad in Seattle. In 1953 and 1959, he won the Washington state championships. His 9–0 win in the 1953 Championship and his 6–0 win in the 1959 Championship are the only perfect score in the history of the tournament. In 1962, he won (4.5: 3.5) a match against Viktors Pupols.
William John Donaldson wrote a book on his chess career: Elmars Zemgalis: Grandmaster without the title (2001). Zemgalis was awarded the Honorary Grandmaster title by FIDE in 2003.
References
External links
De.chessbase.com
1923 births
2014 deaths
Latvian chess players
Chess grandmasters
Sportspeople from Riga
American chess players
20th-century American mathematicians
Latvian World War II refugees
Latvian emigrants to the United States
21st-century American mathematicians
|
https://en.wikipedia.org/wiki/CUTEr
|
CUTEr (Constrained and Unconstrained Testing Environment, revisited) is an open source testing environment for optimization and linear algebra solvers. CUTEr provides a collection of test problems along with a set of tools to help developers design, compare, and improve new and existing test problem solvers.
CUTEr is the successor of the original Constrained and Unconstrained Testing Environment (CUTE) of Bongartz, Conn, Gould and Toint. It provides support for a larger number of platforms and operating systems as well as a more convenient optimization toolbox.
The test problems provided in CUTEr are written in Standard Input Format (SIF). A decoder to convert from this format into well-defined subroutines and data files is available as a separate package. Once translated, these files may be manipulated to provide tools suitable for testing optimization packages. Ready-to-use interfaces to existing packages, such as IPOPT, MINOS, SNOPT, filterSQP, Knitro and more are provided. The problems in the CUTE subset are also available in the AMPL format.
More than 1000 problems are available in the collection, including problems in:
linear programming,
convex and nonconvex quadratic programming,
linear and nonlinear least squares, and
more general convex and nonconvex large-scale and sparse equality and inequality-constrained nonlinear programming.
Over time, the CUTEr test set has become the de facto standard benchmark for research and production-level optimization solvers, and is used and cited in numerous published research articles.
The SIF is a superset of the original MPS format for linear programming and of its extension QPS for quadratic programming. Therefore, access to problem collections such as the Netlib linear programs and the Maros and Meszaros convex quadratic programs is possible. Moreover, the collection covers the Argonne test set, the Hock and Schittkowski collection, the Dembo network problems, the Gould QPs, and others.
CUTEr is available on a variety of UNIX platforms, including Linux and Mac OS X, and is designed to be accessible and easily manageable on heterogeneous networks.
References
Notes
N. I. M. Gould, D. Orban and Ph. L. Toint, CUTEr (and SifDec): a Constrained and Unconstrained Testing Environment, revisited, ACM Transactions on Mathematical Software, 29:4, pp 373–394, 2003.
External links
The official CUTEr website
CUTEr license
Numerical software
Mathematical optimization software
|
https://en.wikipedia.org/wiki/Wieferich%20pair
|
In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy
pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)
Wieferich pairs are named after German mathematician Arthur Wieferich.
Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).
Known Wieferich pairs
There are only 7 Wieferich pairs known:
(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence and in OEIS)
Wieferich triple
A Wieferich triple is a triple of prime numbers p, q and r that satisfy
pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2).
There are 17 known Wieferich triples:
(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 3188089), (5, 20771, 18043), (5, 20771, 950507), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (41, 138200401, 2953), (83, 13691, 821), (199, 1843757, 2251), (431, 2393, 54787), and (1657, 2281, 1667). (sequences , and in OEIS)
Barker sequence
Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that
p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12).
For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple.
For the smallest Wieferich n-tuple, see , for the ordered set of all Wieferich tuples, see .
Wieferich sequence
Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (a Wieferich triple)
The Wieferich sequence of 83:
83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. (a Wieferich pair)
The Wieferich sequence of 59: (this sequence needs more terms to be periodic)
59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ... it also gets 5.
However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:
3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).
The Wieferich sequence of 14:
14, 29, ? (There are no known Wieferich primes in base 29 except 2, but 22 = 4 divides 29 - 1 = 28)
The Wieferich sequence of 39:
39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)
It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that valu
|
https://en.wikipedia.org/wiki/Geometria
|
The term geometria may refer to:
Geometry, a branch of mathematics
Geometria (film), a 1987 short film by Guillermo del Toro
376 Geometria, a main belt asteroid
|
https://en.wikipedia.org/wiki/Demihypercube
|
In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n−1)-demicubes, and 2n (n−1)-simplex facets are formed in place of the deleted vertices.
They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms do not have all regular facets but are all uniform polytopes.
The vertices and edges of a demihypercube form two copies of the halved cube graph.
An n-demicube has inversion symmetry if n is even.
Discovery
Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above three. He called it a 5-ic semi-regular. It also exists within the semiregular k21 polytope family.
The demihypercubes can be represented by extended Schläfli symbols of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes.
Constructions
They are represented by Coxeter-Dynkin diagrams of three constructive forms:
... (As an alternated orthotope) s{21,1,...,1}
... (As an alternated hypercube) h{4,3n−1}
.... (As a demihypercube) {31,n−3,1}
H.S.M. Coxeter also labeled the third bifurcating diagrams as 1k1 representing the lengths of the three branches and led by the ringed branch.
An n-demicube, n greater than 2, has n(n−1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.
In general, a demicube's elements can be determined from the original n-cube: (with Cn,m = mth-face count in n-cube = 2n−m n!/(m!(n−m)!))
Vertices: Dn,0 = 1/2 Cn,0 = 2n−1 (Half the n-cube vertices remain)
Edges: Dn,1 = Cn,2 = 1/2 n(n−1) 2n−2 (All original edges lost, each square faces create a new edge)
Faces: Dn,2 = 4 * Cn,3 = 2/3 n(n−1)(n−2) 2n−3 (All original faces lost, each cube creates 4 new triangular faces)
Cells: Dn,3 = Cn,3 + 23 Cn,4 (tetrahedra from original cells plus new ones)
Hypercells: Dn,4 = Cn,4 + 24 Cn,5 (16-cells and 5-cells respectively)
...
[For m = 3,...,n−1]: Dn,m = Cn,m + 2m Cn,m+1 (m-demicubes and m-simplexes respectively)
...
Facets: Dn,n−1 = 2n + 2n−1 ((n−1)-demicubes and (n−1)-simplices respectively)
Symmetry group
The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group [4,3n−1]) has index 2. It is the Coxeter group [3n−3,1,1] of order , and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.
Orthotopic constructions
Constructions as alternated orthotopes have the same top
|
https://en.wikipedia.org/wiki/Jan%20Denef
|
Jan Denef (born 4 September 1951) is a Belgian mathematician. He is an Emeritus Professor of Mathematics at the Katholieke Universiteit Leuven (KU Leuven).
Denef obtained his PhD from KU Leuven in 1975 with a thesis on Hilbert's tenth problem; his advisors were Louis Philippe Bouckaert and Willem Kuijk.
He is a specialist of model theory, number theory and algebraic geometry. He is well known for his early work on Hilbert's tenth problem and for developing the theory of motivic integration in a series of papers with François Loeser. He has also worked on computational number theory.
Recently he proved a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem.
In 2002 Denef was an Invited Speaker at the International Congresses of Mathematicians in Beijing. His Hirsch-index is 24.
References
Publications
External links
1951 births
Living people
20th-century Belgian mathematicians
21st-century Belgian mathematicians
Model theorists
Academic staff of KU Leuven
KU Leuven alumni
|
https://en.wikipedia.org/wiki/Motivic%20integration
|
Motivic integration is a notion in algebraic geometry that was introduced by Maxim Kontsevich in 1995 and was developed by Jan Denef and François Loeser. Since its introduction it has proved to be quite useful in various branches of algebraic geometry, most notably birational geometry and singularity theory. Roughly speaking, motivic integration assigns to subsets of the arc space of an algebraic variety, a volume living in the Grothendieck ring of algebraic varieties. The naming 'motivic' mirrors the fact that unlike ordinary integration, for which the values are real numbers, in motivic integration the values are geometric in nature.
References
External links
AMS Bulletin Vol. 42 Tom Hales
What is motivic measure?
Lecture Notes (2019) Devlin Mallory
Motivic Integration
math.AG/9911179 A.Craw
An introduction to motivic integration
Lecture Notes (version of 2008) François Loeser
Seattle lecture notes on motivic integration
Lecture Notes W.Veys
Arc spaces, motivic integration and stringy invariants
Algebraic geometry
Definitions of mathematical integration
|
https://en.wikipedia.org/wiki/Fran%C3%A7ois%20Loeser
|
François Loeser (born August 25, 1958) is a French mathematician. He is Professor of Mathematics at the Pierre-and-Marie-Curie University in Paris. From 2000 to 2010 he was Professor at École Normale Supérieure. Since 2015, he is a senior member of the Institut Universitaire de France.
He was awarded the CNRS Silver Medal in 2011 and the Charles-Louis de Saulces de Freycinet Prize of the French Academy of Sciences in 2007. He was awarded an ERC Advanced Investigator Grant in 2010 and has been a Plenary Speaker at the European Congress of Mathematics in Amsterdam in 2008. In 2014 Loeser was an Invited Speaker at the International Congresses of Mathematicians in Seoul. In 2015 he was elected as a fellow of the American Mathematical Society "for contributions to algebraic and arithmetic geometry and to model theory".
He was elected member of Academia Europaea in 2019.
He is a specialist of algebraic geometry and is best known for his work on motivic integration, part of it in collaboration with Jan Denef.
References
Publications
External links
Loeser's home page
1958 births
Living people
Scientists from Mulhouse
20th-century French mathematicians
21st-century French mathematicians
École Normale Supérieure alumni
Fellows of the American Mathematical Society
Academic staff of Pierre and Marie Curie University
Model theorists
|
https://en.wikipedia.org/wiki/Metric%20derivative
|
In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as vector spaces).
Definition
Let be a metric space. Let have a limit point at . Let be a path. Then the metric derivative of at , denoted , is defined by
if this limit exists.
Properties
Recall that ACp(I; X) is the space of curves γ : I → X such that
for some m in the Lp space Lp(I; R). For γ ∈ ACp(I; X), the metric derivative of γ exists for Lebesgue-almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such that the above inequality holds.
If Euclidean space is equipped with its usual Euclidean norm , and is the usual Fréchet derivative with respect to time, then
where is the Euclidean metric.
References
Differential calculus
Metric geometry
|
https://en.wikipedia.org/wiki/List%20of%20World%20Rally%20Championship%20records
|
The list of records in the World Rally Championship includes records and statistics set in the World Rally Championship (WRC) from the 1973 season to present.
Drivers
Wins
Statistics
Age
Manufacturers
Co-drivers
Rallies
Fastest rallies
Closest wins
Nationalities
Drivers
Driver wins per nationalities
Co-drivers
See also
Power Stage (Power Stage statistics)
List of World Rally Championship Drivers' champions
List of World Rally Championship Co-Drivers' champions
List of World Rally Championship Manufacturers' champions
List of World Rally Championship event winners
Notes
References
External links
Statistics at World Rally Archive
RallyBase
World Rally Championship
Records
|
https://en.wikipedia.org/wiki/Uniform%20polytope
|
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges).
This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.
Operations
Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombicosidodecahedron in three dimensions and the grand antiprism in four dimensions. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.
Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation.
Rectification operators
Regular n-polytopes have n orders of rectification. The zeroth rectification is the original form. The (n−1)-th rectification is the dual. A rectification reduces edges to vertices, a birectification reduces faces to vertices, a trirectification reduces cells to vertices, a quadirectification reduces 4-faces to vertices, a quintirectification reduced 5-faces to vertices, and so on.
An extended Schläfli symbol can be used for representing rectified forms, with a single subscript:
k-th rectification = tk{p1, p2, ..., pn-1} = kr.
Truncation operators
Truncation operations that can be applied to regular n-polytopes in any combination. The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, sterication cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.
t0,1 or t: Truncation - applied to polygons and higher. A truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges. (The term, coined by Kepler, comes from Latin truncare 'to cut off'.)
There are higher truncations also: bitruncation t1,2 or 2t, tritruncation t2,3 or 3t, quadritruncation t3,4 or 4t, quintitruncation t4,5 or 5t, etc.
t0,2 or rr: Cantellation - applied to polyhedra and higher. It can be seen as rectifying its rect
|
https://en.wikipedia.org/wiki/Markus%20Rost
|
Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld.
He is known for his work on norm varieties (a key part in the proof of the Bloch–Kato conjecture) and for the Rost invariant (a cohomological invariant with values in Galois cohomology of degree 3). Together with J.-P. Serre he is one of the cofounders of the theory of cohomological invariants of linear algebraic groups. He has also made numerous contributions to the theory of torsors, quadratic forms, central simple algebras, Jordan algebras (the Rost-Serre invariant), exceptional groups, and essential dimension. Most of his results are available only on his webpage.
In 2012 he became a fellow of the American Mathematical Society.
References
Further reading
External links
Markus Rost's homepage
Living people
21st-century German mathematicians
Fellows of the American Mathematical Society
Year of birth missing (living people)
|
https://en.wikipedia.org/wiki/Formation%20matrix
|
In statistics and information theory, the expected formation matrix of a likelihood function is the matrix inverse of the Fisher information matrix of , while the observed formation matrix of is the inverse of the observed information matrix of .
Currently, no notation for dealing with formation matrices is widely used, but in books and articles by Ole E. Barndorff-Nielsen and Peter McCullagh, the symbol is used to denote the element of the i-th line and j-th column of the observed formation matrix. The geometric interpretation of the Fisher information matrix (metric) leads to a notation of following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by so that using Einstein notation we have .
These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.
See also
Fisher information
Shannon entropy
Notes
References
Barndorff-Nielsen, O.E., Cox, D.R. (1989), Asymptotic Techniques for Use in Statistics, Chapman and Hall, London.
Barndorff-Nielsen, O.E., Cox, D.R., (1994). Inference and Asymptotics. Chapman & Hall, London.
P. McCullagh, "Tensor Methods in Statistics", Monographs on Statistics and Applied Probability, Chapman and Hall, 1987.
Edwards, A.W.F. (1984) Likelihood. CUP.
Estimation theory
Information theory
|
https://en.wikipedia.org/wiki/Pure%20spinor
|
In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space with respect to a scalar product .
They were introduced by Élie Cartan in the 1930s and further developed by Claude Chevalley.
They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory, introduced by Roger Penrose in the 1960s.
They have been applied to the study of supersymmetric Yang-Mills theory in 10D, superstrings, generalized complex structures
and parametrizing solutions of integrable hierarchies.
Clifford algebra and pure spinors
Consider a complex vector space , with either even dimension or odd dimension , and a nondegenerate complex scalar product
, with values on pairs of vectors .
The Clifford algebra is the quotient of the full tensor algebra
on by the ideal generated by the relations
Spinors are modules of the Clifford algebra, and so in particular there is an action of the
elements of on the space of spinors. The complex subspace that annihilates
a given nonzero spinor has dimension . If then is said to be a pure spinor. In terms of stratification of spinor modules by orbits of the spin group , pure spinors correspond to the smallest orbits, which are the Shilov boundary of the stratification by the orbit types of the spinor representation on the irreducible spinor (or half-spinor) modules.
Pure spinors, defined up to projectivization, are called projective pure spinors. For of even dimension , the space of projective pure spinors is the homogeneous space
; for of odd dimension , it is .
Irreducible Clifford module, spinors, pure spinors and the Cartan map
The irreducible Clifford/spinor module
Following Cartan and Chevalley,
we may view as a direct sum
where is a totally isotropic subspace of dimension , and is its dual space, with scalar product defined as
or
respectively.
The Clifford algebra representation as endomorphisms of the irreducible Clifford/spinor module , is generated by the linear elements , which act as
for either or , and
for , when is homogeneous of degree .
Pure spinors and the Cartan map
A pure spinor is defined to be any element that is annihilated by a maximal isotropic subspace with respect to the scalar product . Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that annihilates it, up to multiplication by a complex number, as follows.
Denote the Grassmannian of maximal isotropic (-dimensional) subspaces of as . The Cartan map
is defined, for any element , with basis , to have value
i.e. the image of under the endomorphism formed from taking the product of the Clifford representation endomorphisms
, which is independent of the choice of basis .
This is a -dimensional subspace, due to the isotropy conditions,
which imp
|
https://en.wikipedia.org/wiki/List%20of%20Newcastle%20United%20F.C.%20records%20and%20statistics
|
This article lists the records of Newcastle United Football Club.
Honours and achievements
Source:
League
First Division (level 1)
Champions (4): 1904–05, 1906–07, 1908–09, 1926–27
Runners-up: 1995–96, 1996–97
Second Division / First Division / Championship (level 2)
Champions (4): 1964–65, 1992–93, 2009–10, 2016–17
Runners-up: 1897–98, 1947–48
Cup
FA Cup
Winners (6): 1909–10, 1923–24, 1931–32, 1950–51, 1951–52, 1954–55
Runners-up (7): 1904–05, 1905–06, 1907–08, 1910–11, 1973–74, 1997–98, 1998–99
Football League Cup / EFL Cup
Runners-up: 1975–76, 2022–23
FA Charity Shield
Winners: 1909
Runners-up (5): 1932, 1951, 1952, 1955, 1996
Inter-Cities Fairs Cup
Winners: 1968–69
Minor titles
Sheriff of London Charity Shield
Winners: 1907
Texaco Cup
Winners: 1973–74, 1974–75
UEFA Intertoto Cup
Winners: 2006
Anglo-Italian Cup
Winners: 1973
Club records
Attendances
Highest attendance – 68,386 (v. Chelsea, First Division, 3 September 1930)
Highest average attendance – 56,299, Second Division, 1947–48
Wins
Record victory: 13–0 v. Newport County, Second Division, 5 October 1946
Record away league victory: 8–0 v. Sheffield United, Premier League, 24 September 2023
Record away FA Cup victory: 9–0 v. Southport, FA Cup, 1 February 1932
Record UEFA Champions League victory: 4–1 v. PSG, UEFA Champions League, 4 October 2023
Defeats
Record defeat: 0–9 v. Burton Wanderers, Second Division, 15 April 1895
Goals
Most League goals scored in a season – 98 in 42 matches, First Division, 1951–52
Fewest League goals scored in a season – 30 in 42 matches, Second Division, 1980–81
Most League goals conceded in a season – 109 in 42 matches, First Division, 1960–61
Fewest League goals conceded in a season – 33 in 38 matches Premier League, 2022–23 & 33 in 34 matches, First Division, 1904–05
Most different scorers in a single Premier League match – 8, Sean Longstaff, Dan Burn, Sven Botman, Callum Wilson, Anthony Gordon, Miguel Almirón, Bruno Guimarães and Alexander Isak (v. Sheffield United, Premier League, 24 September 2023)
Top 10 record transfer fees paid
Top 10 record transfer fees received
Player records
Appearances
Youngest player – Steve Watson, 16 years 233 days (v. Wolves, Second Division, 10 November 1990)
Youngest player in European competition – Adam Campbell, 17 years 236 days (v. Atromitos, UEFA Europa League, 23 August 2012)
Oldest player – Billy Hampson, 44 years 225 days (v. Birmingham City, First Division, 9 April 1927)
Most appearances
As of 25 November 2012. (Competitive matches only, includes appearances as substitute):
Current player with most appearances – Jamaal Lascelles, 225 (as of 23 April 2023)
Goalscorers
Most goals in a season – 41, Andy Cole , (1993/94)
Most League goals in a season – 36, Hughie Gallacher, (1926–27)
Most goals in a single match – 6, Len Shackleton (v. Newport County, Second Division, 5 October 1946)
Most goals in the League – 178, Jackie Milburn (1946 to 1957)
Most goals in Euro
|
https://en.wikipedia.org/wiki/Disintegration%20theorem
|
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.
Motivation
Consider the unit square in the Euclidean plane , . Consider the probability measure defined on by the restriction of two-dimensional Lebesgue measure to . That is, the probability of an event is simply the area of . We assume is a measurable subset of .
Consider a one-dimensional subset of such as the line segment . has -measure zero; every subset of is a -null set; since the Lebesgue measure space is a complete measure space,
While true, this is somewhat unsatisfying. It would be nice to say that "restricted to" is the one-dimensional Lebesgue measure , rather than the zero measure. The probability of a "two-dimensional" event could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" : more formally, if denotes one-dimensional Lebesgue measure on , then
for any "nice" . The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.
Statement of the theorem
(Hereafter, will denote the collection of Borel probability measures on a topological space .)
The assumptions of the theorem are as follows:
Let and be two Radon spaces (i.e. a topological space such that every Borel probability measure on it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon measure).
Let .
Let be a Borel-measurable function. Here one should think of as a function to "disintegrate" , in the sense of partitioning into . For example, for the motivating example above, one can define , , which gives that , a slice we want to capture.
Let be the pushforward measure . This measure provides the distribution of (which corresponds to the events ).
The conclusion of the theorem: There exists a -almost everywhere uniquely determined family of probability measures , which provides a "disintegration" of into such that:
the function is Borel measurable, in the sense that is a Borel-measurable function for each Borel-measurable set ;
"lives on" the fiber : for -almost all , and so ;
for every Borel-measurable function , In particular, for any event , taking to be the indicator function of ,
Applications
Product spaces
The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.
When is written as a Cartesian product and is the natural projection, then each fibre can be canonically identified with and there exists a Borel family of probability measures in (which is -almost everywhere uniquely determined) such that
which is in particular
|
https://en.wikipedia.org/wiki/Cusp%20%28singularity%29
|
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.
For a plane curve defined by an analytic, parametric equation
a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope ). Cusps are local singularities in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point.
For a curve defined by an implicit equation
which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of are a power of a linear polynomial; however, not all singular points that have this property are cusps. The theory of Puiseux series implies that, if is an analytic function (for example a polynomial), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as
where is a real number, is a positive even integer, and is a power series of order (degree of the nonzero term of the lowest degree) larger than . The number is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of . In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where .
The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.
Classification in differential geometry
Consider a smooth real-valued function of two variables, say where and are real numbers. So is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.
One such family of equivalence classes is denoted by where is a non-negative integer. A function is said to be of type if it lies in the orbit of i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms are said to give normal forms for the type -singularities. Notice that the are the same as the since the diffeomorphic change of coordinate in the source takes to So we can drop the ± from notation.
The
|
https://en.wikipedia.org/wiki/Courant%20bracket
|
In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of p-forms.
The case p = 1 was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges Poisson geometry and pre-symplectic geometry, based on work with his advisor Alan Weinstein. The twisted version of the Courant bracket was introduced in 2001 by Pavol Severa, and studied in collaboration with Weinstein.
Today a complex version of the p=1 Courant bracket plays a central role in the field of generalized complex geometry, introduced by Nigel Hitchin in 2002. Closure under the Courant bracket is the integrability condition of a generalized almost complex structure.
Definition
Let X and Y be vector fields on an N-dimensional real manifold M and let ξ and η be p-forms. Then X+ξ and Y+η are sections of the direct sum of the tangent bundle and the bundle of p-forms. The Courant bracket of X+ξ and Y+η is defined to be
where is the Lie derivative along the vector field X, d is the exterior derivative and i is the interior product.
Properties
The Courant bracket is antisymmetric but it does not satisfy the Jacobi identity for p greater than zero.
The Jacobi identity
However, at least in the case p=1, the Jacobiator, which measures a bracket's failure to satisfy the Jacobi identity, is an exact form. It is the exterior derivative of a form which plays the role of the Nijenhuis tensor in generalized complex geometry.
The Courant bracket is the antisymmetrization of the Dorfman bracket, which does satisfy a kind of Jacobi identity.
Symmetries
Like the Lie bracket, the Courant bracket is invariant under diffeomorphisms of the manifold M. It also enjoys an additional symmetry under the vector bundle automorphism
where α is a closed p+1-form. In the p=1 case, which is the relevant case for the geometry of flux compactifications in string theory, this transformation is known in the physics literature as a shift in the B field.
Dirac and generalized complex structures
The cotangent bundle, of M is the bundle of differential one-forms. In the case p=1 the Courant bracket maps two sections of , the direct sum of the tangent and cotangent bundles, to another section of . The fibers of admit inner products with signature (N,N) given by
A linear subspace of in which all pairs of vectors have zero inner product is said to be an isotropic subspace. The fibers of are 2N-dimensional and the maximal dimension of an isotropic subspace is N. An N-dimensional isotropic subspace is called a maximal isotropic subspace.
A Dirac structure is a maximally isotropic subbundle of whose sections are closed under the Courant bracket. Dirac structures include as special cases symplectic structures, Poisson structures and foliated geometries.
A generalized complex structure is def
|
https://en.wikipedia.org/wiki/More%20or%20Less
|
More or Less may refer to:
More or Less (radio programme), a UK programme focusing on numbers and statistics
More or Less (puzzle), an alternate name for the logic puzzle Futoshiki
More or Less (pricing game), a pricing game on the game show The Price Is Right
More or Less (album), a 2018 album by Dan Mangan
"More or Less", a song by Screaming Trees from Sweet Oblivion
"More or Less", a song by Talib Kweli from Eardrum
More or Les, stage name for Leslie Seaforth, a Canadian rapper, DJ and producer
See also
Approximation
|
https://en.wikipedia.org/wiki/Acta%20Mathematica
|
Acta Mathematica is a peer-reviewed open-access scientific journal covering research in all fields of mathematics.
According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals". According to the Journal Citation Reports, the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics".
Publication history
The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees.
Poincaré episode
The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered in 1887 by Oscar II of Sweden for the best mathematical work concerning the stability of the Solar System by purporting to prove the stability of a special case of the three-body problem. This episode was rediscovered in the 1990s by Daniel Goroff, in his preface to the English translation of "Les méthodes nouvelles de la mécanique céleste" by June Barrow-Green and K.G. Andersson.
The prized or lauded paper was to be published in Acta Mathematica, but after the issue containing the paper was printed, Poincaré found an error that invalidated his proof. He paid more than the prize money to destroy the print run and reprint the issue without his paper, and instead published a corrected paper a year later in the same journal that demonstrated that the system could be unstable. This paper later became one of the foundational works of chaos theory.
References
Literature
External links
Mathematics journals
Publications established in 1882
Royal Swedish Academy of Sciences
Multilingual journals
Springer Science+Business Media academic journals
Quarterly journals
|
https://en.wikipedia.org/wiki/End%20%28topology%29
|
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification.
The notion of an end of a topological space was introduced by .
Definition
Let X be a topological space, and suppose that
is an ascending sequence of compact subsets of X whose interiors cover X. Then X has one end for every sequence
where each Un is a connected component of X \ Kn. The number of ends does not depend on the specific sequence {Ki} of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences.
Using this definition, a neighborhood of an end {Ui} is an open set V such that V ⊇ Un for some n. Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the end compactification (this "compactification" is not always compact; the topological space X has to be connected and locally connected).
The definition of ends given above applies only to spaces X that possess an exhaustion by compact sets (that is, X must be hemicompact). However, it can be generalized as follows: let X be any topological space, and consider the direct system {K} of compact subsets of X and inclusion maps. There is a corresponding inverse system { 0( X \ K ) }, where 0(Y) denotes the set of connected components of a space Y, and each inclusion map Y → Z induces a function 0(Y) → 0(Z). Then set of ends of X is defined to be the inverse limit of this inverse system.
Under this definition, the set of ends is a functor from the category of topological spaces, where morphisms are only proper continuous maps, to the category of sets. Explicitly, if φ : X → Y is a proper map and x = (xK)K is an end of X (i.e. each element xK in the family is a connected component of X ∖ K and they are compatible with maps induced by inclusions) then φ(x) is the family where ranges over compact subsets of Y and φ* is the map induced by φ from to . Properness of φ is used to ensure that each φ−1(K) is compact in X.
The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence.
Examples
The set of ends of any compact space is the empty set.
The real line has two ends. For example, if we let Kn be the closed interval [−n, n], then the two ends are the sequences of open sets Un = (n, ∞) and Vn = (−∞, −n). These ends are usually referred to as "infinity" and "minus infinity", respectively.
If n > 1, then Euclidean space has only one end. This is because has only one unbounded component for any compact set K.
More generally, if M is a compact manifold with boundary, then the number of ends of the interior of M is equal to the number of connected components of the boundary of
|
https://en.wikipedia.org/wiki/John%20Michael%20Cullen
|
John Michael Cullen (14 December 1927 – 23 March 2001) was an Australian ornithologist, of English origin. Mike Cullen began his academic career by studying mathematics at Wadham College, Oxford, but later switched to zoology, spending time at the Edward Grey Institute of Field Ornithology while investigating the ecology of marsh tits. He subsequently achieved his PhD with Niko Tinbergen with a study of the behaviour of the common tern on the Farne Islands off the coast of Northumberland.
In 1976 he moved to Australia, to Monash University in Melbourne, Victoria. There he was involved in an investigation of Abbott's booby on Christmas Island which was threatened by phosphate mining. He served on the Field Investigation Committee of the Royal Australasian Ornithologists Union (RAOU) for which he organised the Rolling Bird Survey project. However, he is best known for long-term studies of the little penguin at Phillip Island and in Port Phillip Bay at St Kilda, in collaboration with Pauline Reilly and others.
References
Dann, Peter. (2002). Obituary. Professor J. Michael (Mike) Cullen, 14 December 1927 - 23 March 2001. VWSG Bulletin 25: 92–93.
Robin, Libby. (2001). The Flight of the Emu: a hundred years of Australian ornithology 1901-2001. Carlton, Vic. Melbourne University Press.
External links
For a Eulogy by Richard Dawkins see http://richarddawkins.net/article,2623,Tribute-to-a-Beloved-Mentor,Richard-Dawkins
John Krebs and Richard Dawkins, Obituary in Guardian
1927 births
2001 deaths
Cullen, Mike
Cullen, Mike
Academic staff of Monash University
20th-century British zoologists
Alumni of Wadham College, Oxford
British emigrants to Australia
|
https://en.wikipedia.org/wiki/Vertex%20%28curve%29
|
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature, and some authors define a vertex to be more specifically a local extremum of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes.
Examples
A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form:
it can be found by completing the square or by differentiation. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.
For a circle, which has constant curvature, every point is a vertex.
Cusps and osculation
Vertices are points where the curve has 4-point contact with the osculating circle at that point. In contrast, generic points on a curve typically only have 3-point contact with their osculating circle. The evolute of a curve will generically have a cusp when the curve has a vertex; other, more degenerate and non-stable singularities may occur at higher-order vertices, at which the osculating circle has contact of higher order than four. Although a single generic curve will not have any higher-order vertices, they will generically occur within a one-parameter family of curves, at the curve in the family for which two ordinary vertices coalesce to form a higher vertex and then annihilate.
The symmetry set of a curve has endpoints at the cusps corresponding to the vertices, and the medial axis, a subset of the symmetry set, also has its endpoints in the cusps.
Other properties
According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices. A more general fact is that every simple closed space curve which lies on the boundary of a convex body, or even bounds a locally convex disk, must have four vertices. Every curve of constant width must have at least six vertices.
If a planar curve is bilaterally symmetric, it will have a vertex at the point or points where the axis of symmetry crosses the curve. Thus, the notion of a vertex for a curve is closely related to that of an optical vertex, the point where an optical axis crosses a lens surface.
Notes
References
.
.
.
.
Curves
|
https://en.wikipedia.org/wiki/Zahorski%20theorem
|
In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a set of zero measure.
This result was proved by in 1939 and first published in 1941.
References
.
.
Theorems in real analysis
|
https://en.wikipedia.org/wiki/Generalized%20complex%20structure
|
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.
These structures first arose in Hitchin's program of characterizing geometrical structures via functionals of differential forms, a connection which formed the basis of Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke and Cumrun Vafa's 2004 proposal that topological string theories are special cases of a topological M-theory. Today generalized complex structures also play a leading role in physical string theory, as supersymmetric flux compactifications, which relate 10-dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.
Definition
The generalized tangent bundle
Consider an N-manifold M. The tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers consist of all tangent vectors to M. A section of T is a vector field on M. The cotangent bundle of M, denoted T*, is the vector bundle over M whose sections are one-forms on M.
In complex geometry one considers structures on the tangent bundles of manifolds. In symplectic geometry one is instead interested in exterior powers of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the generalized tangent bundle, which is the direct sum of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.
The fibers are endowed with a natural inner product with signature (N, N). If X and Y are vector fields and ξ and η are one-forms then the inner product of X+ξ and Y+η is defined as
A generalized almost complex structure is just an almost complex structure of the generalized tangent bundle which preserves the natural inner product:
such that and
Like in the case of an ordinary almost complex structure, a generalized almost complex structure is uniquely determined by its -eigenbundle, i.e. a subbundle of the complexified generalized tangent bundle
given by
Such subbundle L satisfies the following properties:
Vice versa, any subbundle L satisfying (i), (ii) is the -eigenbundle of a unique generalized almost complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.
Courant bracket
In ordinary complex geometry, an almost complex structure is integrable to a complex structure if and only if the Lie bracket of two sections of the holomorphic subbundle is another section of the holomorphic subbundle.
In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.