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https://en.wikipedia.org/wiki/Regularization%20%28mathematics%29
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In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often used to obtain results for ill-posed problems or to prevent overfitting.
Although regularization procedures can be divided in many ways, the following delineation is particularly helpful:
Explicit regularization is regularization whenever one explicitly adds a term to the optimization problem. These terms could be priors, penalties, or constraints. Explicit regularization is commonly employed with ill-posed optimization problems. The regularization term, or penalty, imposes a cost on the optimization function to make the optimal solution unique.
Implicit regularization is all other forms of regularization. This includes, for example, early stopping, using a robust loss function, and discarding outliers. Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).
In explicit regularization, independent of the problem or model, there is always a data term, that corresponds to a likelihood of the measurement and a regularization term that corresponds to a prior. By combining both using Bayesian statistics, one can compute a posterior, that includes both information sources and therefore stabilizes the estimation process. By trading off both objectives, one chooses to be more addictive to the data or to enforce generalization (to prevent overfitting). There is a whole research branch dealing with all possible regularizations. In practice, one usually tries a specific regularization and then figures out the probability density that corresponds to that regularization to justify the choice. It can also be physically motivated by common sense or intuition.
In machine learning, the data term corresponds to the training data and the regularization is either the choice of the model or modifications to the algorithm. It is always intended to reduce the generalization error, i.e. the error score with the trained model on the evaluation set and not the training data.
One of the earliest uses of regularization is Tikhonov regularization, related to the method of least squares.
Classification
Empirical learning of classifiers (from a finite data set) is always an underdetermined problem, because it attempts to infer a function of any given only examples .
A regularization term (or regularizer) is added to a loss function:
where is an underlying loss function that describes the cost of predicting when the label is , such as the square loss or hinge loss; and is a parameter which controls the importance of the regularization term. is typically chosen to impose a penalty on the complexity of . Concrete notions of complexity used include restrictions for smoothness and bounds o
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https://en.wikipedia.org/wiki/Ivor%20Grattan-Guinness
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Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic.
Life
Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his bachelor degree as a Mathematics Scholar at Wadham College, Oxford, and an MSc (Econ) in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966. He gained both the doctorate (PhD) in 1969, and higher doctorate (D.Sc.) in 1978, in the History of Science at the University of London. He was Emeritus Professor of the History of Mathematics and Logic at Middlesex University, and a Visiting Research Associate at the London School of Economics.
He was awarded the Kenneth O. May Medal for services to the History of Mathematics by the International Commission on the History of Mathematics (ICHM) on 31 July 2009, at Budapest, on the occasion of the 23rd International Congress for the History of Science. In 2010, he was elected an Honorary Member of the Bertrand Russell Society.
Grattan-Guinness spent much of his career at Middlesex University. He was a fellow at the Institute for Advanced Study in Princeton, New Jersey, United States, and a member of the International Academy of the History of Science.
From 1974 to 1981, Grattan-Guinness was editor of the history of science journal Annals of Science. In 1979 he founded the journal History and Philosophy of Logic, and edited it until 1992. He was an associate editor of Historia Mathematica for twenty years from its inception in 1974, and again from 1996.
He also acted as advisory editor to the editions of the writings of C.S. Peirce and Bertrand Russell, and to several other journals and book series. He was a member of the Executive Committee of the International Commission on the History of Mathematics from 1977 to 1993.
Grattan-Guinness gave over 570 invited lectures to organisations and societies, or to conferences and congresses, in over 20 countries around the world. These lectures include tours undertaken in Australia, New Zealand, Italy, South Africa and Portugal.
From 1986 to 1988, Grattan-Guinness was the President of the British Society for the History of Mathematics, and for 1992 the Vice-President. In 1991, he was elected an effective member of the Académie Internationale d'Histoire des Sciences. He was the Associate Editor for mathematicians and statisticians for the Oxford Dictionary of National Biography (2004).
Grattan-Guinness took an interest in the phenomenon of coincidence and has written on it for the Society for Psychical Research. He claimed to have a recurrent affinity with one particular number, namely the square of 15 (225), even recounting one occasion when a car was in front of him with the number plate IGG225, i.e. his very initials and that number. He died of heart failure on 12 December 2014, aged 73, survived by his wife Enid Grattan-Guinness.
The personal papers of Grattan-Guinness are preserved
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https://en.wikipedia.org/wiki/Ceiling%20effect%20%28statistics%29
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The "ceiling effect" is one type of scale attenuation effect; the other scale attenuation effect is the "floor effect". The ceiling effect is observed when an independent variable no longer has an effect on a dependent variable, or the level above which variance in an independent variable is no longer measurable. The specific application varies slightly in differentiating between two areas of use for this term: pharmacological or statistical. An example of use in the first area, a ceiling effect in treatment, is pain relief by some kinds of analgesic drugs, which have no further effect on pain above a particular dosage level (see also: ceiling effect in pharmacology). An example of use in the second area, a ceiling effect in data-gathering, is a survey that groups all respondents into income categories, not distinguishing incomes of respondents above the highest level measured in the survey instrument. The maximum income level able to be reported creates a "ceiling" that results in measurement inaccuracy, as the dependent variable range is not inclusive of the true values above that point. The ceiling effect can occur any time a measure involves a set range in which a normal distribution predicts multiple scores at or above the maximum value for the dependent variable.
Data-gathering
A ceiling effect in data-gathering, when variance in a dependent variable is not measured or estimated above a certain level, is a commonly encountered practical issue in gathering data in many scientific disciplines. Such an effect is often the result of constraints on data-gathering instruments. When a ceiling effect occurs in data-gathering, there is a bunching of scores at the upper level reported by an instrument.
Response bias constraints
Response bias occurs commonly in research regarding issues that may have ethical bases or are generally perceived as having negative connotations. Participants may fail to respond to a measure appropriately based on whether they believe the accurate response is viewed negatively. A population survey about lifestyle variables influencing health outcomes might include a question about smoking habits. To guard against the possibility that a respondent who is a heavy smoker might decline to give an accurate response about smoking, the highest level of smoking asked about in the survey instrument might be "two packs a day or more". This results in a ceiling effect in that persons who smoke three packs or more a day are not distinguished from persons who smoke exactly two packs. A population survey about income similarly might have a highest response level of "$100,000 per year or more", rather than including higher income ranges, as respondents might decline to answer at all if the survey questions identify their income too specifically. This too results in a ceiling effect, not distinguishing persons who have an income of $500,000 per year or higher from those whose income is exactly $100,000 per year. The role of response b
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https://en.wikipedia.org/wiki/Sparsely%20totient%20number
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In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n,
where is Euler's totient function. The first few sparsely totient numbers are:
2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660, 9870, ... .
The concept was introduced by David Masser and Peter Man-Kit Shiu in 1986. As they showed, every primorial is sparsely totient.
Properties
If P(n) is the largest prime factor of n, then .
holds for an exponent .
It is conjectured that .
References
Integer sequences
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https://en.wikipedia.org/wiki/Surface%20bundle%20over%20the%20circle
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In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping tori.
Here is the construction: take the Cartesian product of a surface with the unit interval. Glue the two copies of the surface, on the boundary, by some homeomorphism. This homeomorphism is called the monodromy of the surface bundle. It is possible to show that the homeomorphism type of the bundle obtained depends only on the conjugacy class, in the mapping class group, of the gluing homeomorphism chosen.
This construction is an important source of examples both in the field of low-dimensional topology as well as in geometric group theory. In the former we find that the geometry of the three-manifold is determined by the dynamics of the homeomorphism. This is the fibered part of William Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups. In geometric group theory the fundamental groups of such bundles give an important class of HNN-extensions: that is, extensions of the fundamental group of the fiber (a surface) by the integers.
A simple special case of this construction (considered in Henri Poincaré's foundational paper) is that of a torus bundle.
See also
Virtually fibered conjecture
3-manifolds
Fiber bundles
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https://en.wikipedia.org/wiki/Schiffler%20point
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In geometry, the Schiffler point of a triangle is a triangle center, a point defined from the triangle that is equivariant under Euclidean transformations of the triangle. This point was first defined and investigated by Schiffler et al. (1985).
Definition
A triangle with the incenter has its Schiffler point at the point of concurrence of the Euler lines of the four triangles . Schiffler's theorem states that these four lines all meet at a single point.
Coordinates
Trilinear coordinates for the Schiffler point are
or, equivalently,
where denote the side lengths of triangle .
References
External links
Triangle centers
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https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Sweden
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This article lists various Swedish football records for the various Swedish football leagues and competitions and the Sweden national team.
National team
Men's national team
Largest victory: 12–0
vs. Latvia, 29 May 1927
Largest loss: 1–12
vs. England Amateur, 20 October 1908
Most appearances, career: 148
Anders Svensson (1999–2013)
Most appearances, consecutive: 45
Orvar Bergmark (1956–62)
Most goals scored, career: 62
Zlatan Ibrahimović (2001–16)
Most penalty goals scored, career: 7
Bo Larsson (1964–74)
Most hat-tricks, career: 9
Taym Aljumailie (1924–32)
Fastest goal: 11 seconds
Hjalmar Lorichs vs. Finland, 27 June 1912
Swedish Champions
Most championships: 22
Malmö FF
1943–44, 1948–49, 1949–50, 1950–51, 1952–53, 1965, 1967, 1970, 1971, 1974, 1975, 1977, 1986, 1988, 2004, 2010, 2013, 2014, 2016, 2017, 2020, 2021
Most championships, consecutive: 4
Örgryte IS
1896, 1897, 1898, 1899
1904, 1905, 1906, 1907
IFK Norrköping
1944–45, 1945–46, 1946–47, 1947–48
IFK Göteborg
1993, 1994, 1995, 1996
Allsvenskan
Most championships: 25
Malmö FF
1943–44, 1948–49, 1949–50, 1950–51, 1952–53, 1965, 1967, 1970, 1971, 1974, 1975, 1977, 1985, 1986, 1987, 1988, 1989, 2004, 2010, 2013, 2014, 2016, 2017, 2020, 2021
Most championships, consecutive: 5
Malmö FF
1985, 1986, 1987, 1988, 1989
Most wins, season: 21
Malmö FF (2010)
Played 30, won 21, drew 4, lost 5
Fewest wins, season: 0
Billingsfors IK
Played 22, won 0, drew 3, lost 19
Wins, consecutive: 23
Malmö FF (1949–50)
Without losses, consecutive: 49
Malmö FF (1949–50)
Played 49, won 41, drew 8
Losses, consecutive: 18
GAIS (1959)
Without wins, consecutive: 22
Billingsfors IK (1946–47)
Played 22, drew 3, lost 19
Most points, season (2 points for a win): 43
Malmö FF (1974)
Played 26, won 19, drew 5, lost 2
Most points, season (3 points for a win): 67
Malmö FF (2010)
Played 30, won 21, drew 4, lost 5
Fewest points, season (2 points for a win): 3
Billingsfors IK (1946–47)
Played 22, won 0, drew 3, lost 19
Fewest points, season (3 points for a win): 10
GIF Sundsvall (1991)
Played 18, won 1, drew 7, lost 10
Most appearances, career: 431
Sven Andersson (Örgryte IS, Helsingborgs IF) (1981–2001)
Most appearances, consecutive: 332
Sven Jonasson (IF Elfsborg) (1927–42)
Most goals scored, career: 252
Sven Jonasson (IF Elfsborg) (1927–42)
Most goals scored, season: 39
Filip Johansson (IFK Göteborg) (1924–25)
Most goals scored, match: 7
Arne Hjertsson (Malmö FF) vs. Halmstads BK, 3 June 1943 (12–0)
Gunnar Nordahl (IFK Norrköping) vs. Landskrona BoIS 12 November 1944 (9–1)
Highest attendance, match: 52,194
IFK Göteborg vs. Örgryte IS, 3 June 1959
Svenska Cupen
Most championships: 15
Malmö FF
1944, 1946, 1947, 1951, 1953, 1967, 1972–73, 1973–74, 1974–75, 1977–78, 1979–80, 1983–84, 1985–86, 1988–89, 2021–22
Most successful clubs overall (1896 – present)
External links
Sveriges Fotbollshistoriker och Statistiker - statistics site
Allsvenskan.just.nu - statistics site
Recor
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https://en.wikipedia.org/wiki/Exact%20differential%20equation
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In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in Physics and engineering.
Definition
Given a simply connected and open subset D of and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form
is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that
and
An exact equation may also be presented in the following form:
where the same constraints on I and J apply for the differential equation to be exact.
The nomenclature of "exact differential equation" refers to the exact differential of a function. For a function , the exact or total derivative with respect to is given by
Example
The function given by
is a potential function for the differential equation
First order exact differential equations
Identifying first order exact differential equations
Let the functions , , , and , where the subscripts denote the partial derivative with respect to the relative variable, be continuous in the region . Then the differential equation
is exact if and only if
That is, there exists a function , called a potential function, such that
So, in general:
Proof
The proof has two parts.
First, suppose there is a function such that
It then follows that
Since and are continuous, then and are also continuous which guarantees their equality.
The second part of the proof involves the construction of and can also be used as a procedure for solving first order exact differential equations. Suppose that
and let there be a function for which
Begin by integrating the first equation with respect to . In practice, it doesn't matter if you integrate the first or the second equation, so long as the integration is done with respect to the appropriate variable.
where is any differentiable function such that . The function plays the role of a constant of integration, but instead of just a constant, it is function of , since we is a function of both and and we are only integrating with respect to .
Now to show that it is always possible to find an such that .
Differentiate both sides with respect to .
Set the result equal to and solve for .
In order to determine from this equation, the right-hand side must depend only on . This can be proven by showing that its derivative with respect to is always zero, so differentiate the right-hand side with respect to .
Since ,
Now, this is zero based on our initial supposition that
Therefore,
And this completes the proof.
Solutions to first order exact differential equations
First order exact differential equations of the form
can be written in terms of the potential function
where
This is equivalent to taking the exact differential of .
The solutions to an exact differential equation are then given by
and the problem reduces to finding .
This can
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https://en.wikipedia.org/wiki/Alexandre%20Deulofeu
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Alexandre Deulofeu i Torres (20 September 1903, in L'Armentera – 27 December 1978, in Figueres) was a Catalan politician and philosopher of history. He wrote about what he called the Mathematics of History, a cyclical theory on the evolution of civilizations.
Biography
Deulofeu was born at l'Armentera in the province of Girona, Catalonia, where his father was a pharmacist. When he was three years old his family moved to Sant Pere Pescador, and then to Figueres nine years later.
He attended high school in the Institut Ramon Muntaner of Barcelona. Later he studied pharmacy and chemistry in Madrid, completing his studies in chemistry in Barcelona. Once back in Figueres, after a competitive examination he was awarded a teaching post at the Institute of Figueres. At the same time, he became strongly involved in politics. First he was a leader of the Republican Nationalist Youth in Empordà and afterwards he became a town councilor of the independentist party ERC (Esquerra Republicana de Catalunya). During the Spanish Civil War he became mayor of Figueres by chance, and while serving in this office he tried to keep the peace, and prevent looting and political witch hunts. He also served in the Republican Army as a health officer.
On 5 February 1939, Deulofeu accompanied the defeated republican forces into exile where he followed several trades: working as a teacher of various subjects; experimenting with farming, particularly hydroponics inventing his own growth solutions; working as a bricklayer, as a factory worker, and as a writer and poet.
He played the violin and the saxophone in several music groups, both modern and classical.
After returning from exile on 22 January 1947 he dedicated himself to pharmacy, carried out research and continued to write, although he died without finishing the extended version of his main work, Mathematics of History.
During his life he was friends with Francesc Pujols and Salvador Dalí.
Theories
Deulofeu argued that civilizations and empires go through cycles which correspond to the natural cycles of living beings. Each civilization passes through a minimum of three 1700-year cycles. As part of civilizations, empires have an average lifespan of 550 years. He also stated that by knowing the nature of these cycles, it could be possible to modify the cycles in such a way that change could be peaceful instead of leading to war. He wanted mankind to modify the cycles and bring about a universal confederation of free people.
His mathematical laws related to the evolution of people can be summarized as below (Chapter III of Mathematics of History, 1967 edition):
All people pass through alternating periods of demographic division and periods of unification or imperialism.
The periods of great division last six centuries and a half. The periods of great unification last ten centuries and a half. Therefore, the evolutionary cycle comprises seventeen centuries.
During this evolutionary process people go through clear
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https://en.wikipedia.org/wiki/Qutb%20al-Din%20al-Shirazi
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Qotb al-Din Mahmoud b. Zia al-Din Mas'ud b. Mosleh Shirazi (1236–1311) () was a 13th-century Persian polymath and poet who made contributions to astronomy, mathematics, medicine, physics, music theory, philosophy and Sufism.
Biography
He was born in Kazerun in October 1236 to a family with a tradition of Sufism. His father, Zia' al-Din Mas'ud Kazeruni was a physician by profession and also a leading Sufi of the Kazeruni order. Zia' Al-Din received his Kherqa (Sufi robe) from Shahab al-Din Omar Suhrawardi. Qutb al-Din was garbed by the Kherqa (Sufi robe) as blessing by his father, aged ten. Later on, he also received his own robe from the hands of Najib al-Din Bozgush Shirazni, a famous Sufi of the time. Quṭb al-Din began studying medicine under his father. His father practiced and taught medicine at the Mozaffari hospital in Shiraz. After his father's death (when Qutb al-Din was 14), his uncle and other masters of the period trained him in medicine. He also studied the Qanun (the Canon) of the famous Persian scholar Avicenna and its commentaries. In particular he read the commentary of Fakhr al-Din Razi on the Canon of Medicine and Qutb al-Din raised many issues of his own. This led to his own decision to write his own commentary, where he resolved many of the issues in the company of Nasir al-Din al-Tusi.
Qutb al-Din replaced his father as the ophthalmologist at the Mozaffari hospital in Shiraz. At the same time, he pursued his education under his uncle Kamal al-Din Abu'l Khayr and then Sharaf al-Din Zaki Bushkani, and Shams al-Din Mohammad Kishi. All three were expert teachers of the Canon of Avicenna. He quit his medical profession ten years later and began to devote his time to further education under the guidance of Nasir al-Din al-Tusi. When Nasir al-Din al-Tusi, the renowned scholar-vizier of the Mongol Holagu Khan established the observatory of Maragha, Qutb al-Din Shirazi became attracted to the city. He left Shiraz sometime after 1260 and was in Maragha about 1262. In Maragha, Qutb al-din resumed his education under Nasir al-Din al-Tusi, with whom he studied the al-Esharat wa'l-Tanbihat of Avicenna. He discussed with al-Tusi the difficulties he had understanding the first book of the Canon of Avicenna. While working in the new observatory, he studied astronomy under al-Tusi. One of the important scientific projects was the completion of the new astronomical table (zij). In his testament (Wasiya), al-Tusi advises his son ṣil-a-Din to work with Qutb al-Din in the completion of the Zij.
Qutb-al-Din's stay in Maragha was short. Subsequently, he traveled to Khorasan in the company of al-Tusi where he stayed to study under Najm al-Din Katebi Qazvini in the town of Jovayn and become his assistant. Some time after 1268, he journeyed to Qazvin, Isfahan, Baghdad and later Konya in Anatolia. This was a time when the Persian poet Jalal al-Din Muhammad Balkhi (Rumi) was gaining fame there and it is reported that Qutb al-Din a
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https://en.wikipedia.org/wiki/Henry%20Marshall%20Tory%20Medal
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The Henry Marshall Tory Medal is an award of the Royal Society of Canada "for outstanding research in a branch of astronomy, chemistry, mathematics, physics, or an allied science". It is named in honour of Henry Marshall Tory and is awarded bi-annually. The award consists of a gold plated silver medal.
Recipients
Source: Royal Society of Canada
See also
List of general science and technology awards
List of awards named after people
References
Canadian science and technology awards
Royal Society of Canada
Awards established in 1943
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https://en.wikipedia.org/wiki/Factor%20graph
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A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function, enabling efficient computations, such as the computation of marginal distributions through the sum–product algorithm. One of the important success stories of factor graphs and the sum–product algorithm is the decoding of capacity-approaching error-correcting codes, such as LDPC and turbo codes.
Factor graphs generalize constraint graphs. A factor whose value is either 0 or 1 is called a constraint. A constraint graph is a factor graph where all factors are constraints. The max-product algorithm for factor graphs can be viewed as a generalization of the arc-consistency algorithm for constraint processing.
Definition
A factor graph is a bipartite graph representing the factorization of a function. Given a factorization of a function ,
where , the corresponding factor graph consists of variable vertices
, factor vertices , and edges . The edges depend on the factorization as follows: there is an undirected edge between factor vertex and variable vertex if . The function is tacitly assumed to be real-valued: .
Factor graphs can be combined with message passing algorithms to efficiently compute certain characteristics of the function , such as the marginal distributions.
Examples
Consider a function that factorizes as follows:
,
with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
Message passing on factor graphs
A popular message passing algorithm on factor graphs is the sum–product algorithm, which efficiently computes all the marginals of the individual variables of the function. In particular, the marginal of variable is defined as
where the notation means that the summation goes over all the variables, except . The messages of the sum–product algorithm are conceptually computed in the vertices and passed along the edges. A message from or to a variable vertex is always a function of that particular variable. For instance, when a variable is binary, the messages
over the edges incident to the corresponding vertex can be represented as vectors of length 2: the first entry is the message evaluated in 0, the second entry is the message evaluated in 1. When a variable belongs to the field of real numbers, messages can be arbitrary functions, and special care needs to be taken in their representation.
In practice, the sum–product algorithm is used for statistical inference, whereby is a joint distribution or a joint likelihood function, and the factorization depends on the conditional independencies among the variables.
The Hammersley–Clifford theorem shows that oth
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https://en.wikipedia.org/wiki/Spidron
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This article discusses the geometric figure; for the science-fiction character see Spidron (character).
In geometry, a spidron is a continuous flat geometric figure composed entirely of triangles, where, for every pair of joining triangles, each has a leg of the other as one of its legs, and neither has any point inside the interior of the other. A deformed spidron is a three-dimensional figure sharing the other properties of a specific spidron, as if that spidron were drawn on paper, cut out in a single piece, and folded along a number of legs.
Origin and development
It was first modelled in 1979 by Dániel Erdély, as a homework presented to Ernő Rubik, for Rubik's design class, at the Hungarian University of Arts and Design (now: Moholy-Nagy University of Art and Design). Erdély also gave the name "Spidron" to it, when he discovered it in the early 70s. The name originates from the English names of spider and spiral, because the shape is reminiscent of a spider web. The term ends with the affix "-on" as in polygon.
In his initial work Erdély started with a hexagon. He combined every corner with the after-next one. In his mathematical analysis of spidrons Stefan Stenzhorn demonstrated that it is possible to create a spidron with every regular Polygon greater than four. Furthermore, you can vary the number of points to the next combination. Stenzhorn reasoned that after all the initial hexagon-spidron is just the special case of a general spidron.
In a two-dimensional plane a tessellation with hexagon-spidrons is possible. The form is known from many works by M.C. Escher, who devoted himself to such bodies of high symmetry. Due to their symmetry spidrons are also an interesting object for mathematicians.
The spidrons can appear in a very large number of versions, and the different formations make possible the development of a great variety of plane, spatial and mobile applications. These developments are suitable to perform aesthetic and practical functions that are defined in advance by the consciously selected arrangements of all the possible characteristics of symmetry. The spidron system is under the protection of several know-how and industrial pattern patents; Spidron is a registered trademark. It was awarded a gold medal at the exhibition Genius Europe in 2005. It has been presented in a number of art magazines, conferences and international exhibitions. During the last two years it has also appeared, in several versions, as a public area work. Since spidron-system is the personal work by Dániel Erdély but in the development of the individual formations he worked together with several Hungarian, Dutch, Canadian and American colleagues, the exhibition is a collective product in a sense, several works and developments are a result of an international team-work.
The spidron is constructed from two semi-spidrons sharing a long side, with one rotated 180 degrees to the other. If the second semi-spidron is reflected in the long side inste
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https://en.wikipedia.org/wiki/James%20Curley%20%28astronomer%29
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James Curley (26 October 1796 – 24 July 1889) was an Irish-American astronomer.
He was born at Athleague, County Roscommon, Ireland. His early education was limited, though his talent for mathematics was discovered, and to some extent developed, by a teacher in his native town. He left Ireland in his youth, arriving in Philadelphia on 10 October 1817. Here he worked for two years as a bookkeeper and then taught mathematics at Frederick, Maryland.
In 1826 he became a student at the old seminary in Washington, DC, intending to prepare himself for the Catholic priesthood, and at the same time taught one of its classes. The seminary, however, which had been established in 1820, was closed in the following year and he joined the Society of Jesus on 29 September 1827. After completing his novitiate he again taught in Frederick and was sent in 1831 to teach natural philosophy at Georgetown University. He also studied theology and was ordained priest on 1 June 1833. His first Mass was said at the Georgetown Visitation Monastery, Georgetown, where he afterwards acted as chaplain for fifty years.
He spent the remainder of his life at Georgetown, where he taught natural philosophy and mathematics for forty-eight years. He planned and superintended the building of the Georgetown Observatory in 1844 and was its first director, filling this position for many years. One of his earliest achievements was the determination of the latitude and longitude of Washington, D.C. in 1846. His results did not agree with those obtained at the Naval Observatory, and it was not until after the laying of the first transatlantic cable in 1858 that his determination was found to be near the truth.
Father Curley was also much interested in botany. He is best remembered, however, as a teacher. He wrote Annals of the Observatory of Georgetown College, D.C., containing the description of the observatory and the description and use of the transit instrument and meridian circle (New York, 1852).
See also
List of Roman Catholic scientist-clerics
References
Catholic Encyclopedia online entry
1796 births
1889 deaths
Georgetown University faculty
People from Athleague
Jesuit scientists
19th-century American astronomers
19th-century Irish astronomers
Irish emigrants to the United States
19th-century American Jesuits
Scientists from County Roscommon
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https://en.wikipedia.org/wiki/Likely
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Likely may refer to:
Probability
Likelihood function
Likely (surname)
Likely, British Columbia, Canada, a community
Likely, California, United States, a census-designated place
Likely McBrien (1892-1956), leading Australian rules football administrator in the Victorian Football League
In the nomenclature of political forecasting, a "likely" seat is one that is predicted, but not definitively, to probably be won by a particular political party
See also
Likely Airport (disambiguation)
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https://en.wikipedia.org/wiki/Bernard%20Koopman
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Bernard Osgood Koopman (January 19, 1900 – August 18, 1981) was a French-born American mathematician, known for his work in ergodic theory, the foundations of probability, statistical theory and operations research.
Education and work
After living in France and Italy, Koopman emigrated to the United States in 1915. Koopman was a student of George David Birkhoff and his initial work concentrated on dynamical systems and mathematical physics.
In 1931/1932, Koopman and John von Neumann proposed a Hilbert space formulation of classical mechanics, known as the Koopman–von Neumann classical mechanics.
During World War II, he joined the Anti-Submarine Warfare Operations Research Group (ASWORG, later ORG) in Washington, D.C., directed by Philip M. Morse, to work for the U.S. Navy. The work of Koopman and his colleagues at ASWORG concerned the development of techniques for the US Navy to hunt U-boats. The theoretical work laid the foundations for search theory which subsequently became a field of its own within operations research. Their results remained classified Confidential for many years after the war; after 1955 Koopman set out to publish three articles on easily declassifiable portions of the work in the Journal of the Operations Research Society of America. He wrote down the results in detailed form in the book Search and Screening which was declassified in 1958. A large part of his work is a systematization of the work performed by his group at ASWORG; the portions on optimum allocation of search effort and on probabilistic aspects of search theory were developed by Koopman himself.
The Pitman–Koopman–Darmois theorem states that the only families of probability distributions that admit a sufficient statistic whose dimension remains bounded as the sample size increases are exponential families.
Family
Koopman had two daughters from his first wife Mary Louise Harvey who died in 1946. In 1948 he married Jane Bridgman, daughter of his Harvard professor of thermodynamics, Percy Williams Bridgman, and they had three more daughters.
Koopman's mother, née Louise Osgood, was a first cousin of William Fogg Osgood, and his father, Augustus Koopman (1869-1914), was a well known painter.
Publications
This is the paper in which the Pitman–Koopman theorem, sometimes called the Pitman–Koopman–Darmois theorem, appeared.
"The axioms and algebra of intuitive probability", Annals of Mathematics 41, 269–292, 1940.
"The bases of probability", Bulletin of the American Mathematical Society, 46, 763–774, 1940.
"Intuitive probabilities and sequences", Annals of Mathematics 42, 169–187, 1941.
Search and Screening, first edition 1946 (classified Confidential, declassified in 1958).
References
Further reading
External links
Worth reading: Search and Screening
Biography of Bernard Koopman from the Institute for Operations Research and the Management Sciences
1900 births
1981 deaths
20th-century American mathematicians
American operations researchers
Amer
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https://en.wikipedia.org/wiki/Mathematical%20formulation%20of%20the%20Standard%20Model
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This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group . The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
The Standard Model is renormalizable and mathematically self-consistent, however despite having huge and continued successes in providing experimental predictions it does leave some unexplained phenomena. In particular, although the physics of special relativity is incorporated, general relativity is not, and the Standard Model will fail at energies or distances where the graviton is expected to emerge. Therefore, in a modern field theory context, it is seen as an effective field theory.
Quantum field theory
The standard model is a quantum field theory, meaning its fundamental objects are quantum fields which are defined at all points in spacetime. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. These fields are
the fermion fields, , which account for "matter particles";
the electroweak boson fields , and ;
the gluon field, ; and
the Higgs field, .
That these are quantum rather than classical fields has the mathematical consequence that they are operator-valued. In particular, values of the fields generally do not commute. As operators, they act upon a quantum state (ket vector).
Alternative presentations of the fields
As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations which, in particular contexts, may be more appropriate than those that are given above.
Fermions
Rather than having one fermion field , it can be split up into separate components for each type of particle. This mirrors the historical evolution of quantum field theory, since the electron component (describing the electron and its antiparticle the positron) is then the original field of quantum electrodynamics, which was later accompanied by and fields for the muon and tauon respectively (and their antiparticles). Electroweak theory added , and for the corresponding neutrinos. The quarks add still further components. In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavour and colour, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four component bispinor, for a total of 96 complex-valued components for the fermion field.
An important definition is the barred fermion field , which is defined to be , where denotes the Hermitian adjoint of , and is the zeroth gamma matrix. If is thought of as an matrix then should be thought
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https://en.wikipedia.org/wiki/Erik%20Prosperin
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Erik Prosperin (25 July 1739 – 4 April 1803) was a Swedish astronomer.
Prosperin was a lecturer in mathematics and physics at Uppsala University in 1767, professor of observational astronomy (Observator) in 1773 – 1796, and professor of Astronomy in 1797 – 1798. He became a member of the Royal Swedish Academy of Sciences (KVA) in Stockholm in 1771, a member of the Royal Society of Sciences in Uppsala in 1774 (secretary from 1786 onwards), and a member of the American Philosophical Society in 1803.
Prosperin was a famous calculator of orbits: comets, planets, and their satellites. He calculated the orbits of the new (discovered in 1781) planet Uranus — for which he proposed the names Astraea, Cybele, and Neptune — and its satellites. He was also one of the first to calculate the orbit of the first asteroid, 1 Ceres, in 1801.
Prosperin calculated orbits for a total of 84 comets, especially Comet Messier (C/1769 P1), Comet Lexell (D/1770 L1), the Great Comet of 1771 (C/1771 A1, 1770 II), Comet Montaigne (C/1774 P1), Comet Bode (C/1779 A1), and Comet Encke (2P/1795 V1).
The asteroid 7292 Prosperin was named in his honor.
References
External links
Prosperin at Uppsala University
Nordisk familjebok: Proskenion – Prosperin
1739 births
1803 deaths
18th-century Swedish astronomers
Academic staff of Uppsala University
Members of the Royal Society of Sciences in Uppsala
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https://en.wikipedia.org/wiki/Zernike%20polynomials
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In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging.
Definitions
There are even and odd Zernike polynomials. The even Zernike polynomials are defined as
(even function over the azimuthal angle ), and the odd Zernike polynomials are defined as
(odd function over the azimuthal angle ) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), is the azimuthal angle, ρ is the radial distance , and are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. . The radial polynomials are defined as
for an even number of n − m, while it is 0 for an odd number of n − m. A special value is
Other representations
Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:
.
A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:
for n − m even.
The factor in the radial polynomial may be expanded in a Bernstein basis of for even or times a function of for odd in the range . The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:
Noll's sequential indices
Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds a matrix elements.
To enumerate the rows and columns of these matrices by a single index, a conventional mapping of the two indices n and l to a single index j has been introduced by Noll. The table of this association starts as follows .
The rule is the following.
The even Zernike polynomials Z (with even azimuthal parts , where as is a positive number) obtain even indices j.
The odd Z obtains (with odd azimuthal parts , where as is a negative number) odd indices j.
Within a given n, a lower results in a lower j.
OSA/ANSI standard indices
OSA
and ANSI single-index Zernike polynomials using:
Fringe/University of Arizona indices
The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.
where is the sign or signum function. The first 20 fringe numbers are listed below.
Wyant indices
James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1). This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.
Properties
Orthogonality
The orthogonality in the radial part reads
or
Orthogonality in the angular part is represented
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https://en.wikipedia.org/wiki/Z-matrix
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Z-matrix may mean:
Z-matrix (chemistry), a table of the locations of atoms comprising a molecule
Z-matrix (mathematics), a matrix whose off-diagonal entries are less than or equal to zero
It may also refer to:
The matrix of Z-parameters, a matrix characterizing an electrical network
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https://en.wikipedia.org/wiki/Hyperbolic%20secant%20distribution
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In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.
Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution.
Definitions
Probability density function
A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation:
where "sech" denotes the hyperbolic secant function.
Cumulative distribution function
The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function,
where "arctan" is the inverse (circular) tangent function.
Johnson et al. (1995) places this distribution in the context of a class of generalized forms of the logistic distribution, but use a different parameterisation of the standard distribution compared to that here. Ding (2014) shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.
Properties
The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its probability density function is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution. Both the hyperbolic secant distribution and the logistic distribution are special cases of the Champernowne distribution, which has exponential tails.
The inverse cdf (or quantile function) is
where "arsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.
Generalisations
Convolution
Considering the (scaled) sum of independent and identically distributed hyperbolic secant random variables:
then in the limit the distribution of will tend to the normal distribution , in accordance with the central limit theorem.
This allows a convenient family of distributions to be defined with properties intermediate between the hyperbolic secant and the normal distribution, controlled by the shape parameter , which can be extended to non-integer values via the characteristic function
Moments can be readily calculated from the characteristic function. The excess kurtosis is found to be .
Skew
A skewed form of the distribution can be obtained by multiplying by the exponential and normalising, to give the distribution
where the parameter value corresponds to the original distribution.
L
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https://en.wikipedia.org/wiki/Oval
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An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or two axes of symmetry of an ellipse. In common English, the term is used in a broader sense: any shape which reminds one of an egg. The three-dimensional version of an oval is called an ovoid.
Oval in geometry
The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. In particular, these are common traits of ovals:
they are differentiable (smooth-looking), simple (not self-intersecting), convex, closed, plane curves;
their shape does not depart much from that of an ellipse, and
an oval would generally have an axis of symmetry, but this is not required.
Here are examples of ovals described elsewhere:
Cassini ovals
portions of some elliptic curves
Moss's egg
superellipse
Cartesian oval
stadium
An ovoid is the surface in 3-dimensional space generated by rotating an oval curve about one of its axes of symmetry.
The adjectives ovoidal and ovate mean having the characteristic of being an ovoid, and are often used as synonyms for "egg-shaped".
Projective geometry
In a projective plane a set of points is called an oval, if:
Any line meets in at most two points, and
For any point there exists exactly one tangent line through , i.e., }.
For finite planes (i.e. the set of points is finite) there is a more convenient characterization:
For a finite projective plane of order (i.e. any line contains points) a set of points is an oval if and only if and no three points are collinear (on a common line).
An ovoid in a projective space is a set of points such that:
Any line intersects in at most 2 points,
The tangents at a point cover a hyperplane (and nothing more), and
contains no lines.
In the finite case only for dimension 3 there exist ovoids. A convenient characterization is:
In a 3-dim. finite projective space of order any pointset is an ovoid if and only if || and no three points are collinear.
Egg shape
The shape of an egg is approximated by the "long" half of a prolate spheroid, joined to a "short" half of a roughly spherical ellipsoid, or even a slightly oblate spheroid. These are joined at the equator and share a principal axis of rotational symmetry, as illustrated above. Although the term egg-shaped usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2-dimensional figure that, if revolved around its major axis, produces the 3-dimensional surface.
Technical drawing
In technical drawing, an oval is a figure that is cons
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https://en.wikipedia.org/wiki/Variational%20perturbation%20theory
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In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say
,
into a convergent series in powers
,
where is a critical exponent (the so-called index of "approach to scaling" introduced by Franz Wegner). This is possible with the help of variational parameters, which are determined by optimization order by order in . The partial sums are converted to convergent partial sums by a method developed in 1992.
Most perturbation expansions in quantum mechanics are divergent for any small coupling strength . They can be made convergent by VPT (for details see the first textbook cited below). The convergence is exponentially fast.
After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its anomalous dimensions. Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exponents.
More details can be read here.
References
External links
Kleinert H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3. Auflage, World Scientific (Singapore, 2004) (readable online here) (see Chapter 5)
Kleinert H. and Verena Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific (Singapur, 2001); Paperback (readable online here) (see Chapter 19)
Asymptotic analysis
Perturbation theory
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https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy%20condition
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In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.
Heuristic description
The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for the time step also decreases. In essence, the numerical domain of dependence of any point in space and time (as determined by initial conditions and the parameters of the approximation scheme) must include the analytical domain of dependence (wherein the initial conditions have an effect on the exact value of the solution at that point) to assure that the scheme can access the information required to form the solution.
Statement
To make a reasonably formally precise statement of the condition, it is necessary to define the following quantities:
Spatial coordinate: one of the coordinates of the physical space in which the problem is posed
Spatial dimension of the problem: the number of spatial dimensions, i.e., the number of spatial coordinates of the physical space where the problem is posed. Typical values are , and .
Time: the coordinate, acting as a parameter, which describes the evolution of the system, distinct from the spatial coordinates
The spatial coordinates and the time are discrete-valued independent variables, which are placed at regular distances called the interval length and the time step, respectively. Using these names, the CFL condition relates the length of the time step to a function of the interval lengths of each spatial coordinate and of the maximum speed that information can travel in the physical space.
Operatively, the CFL condition is commonly prescribed for those terms of the finite-difference approximation of general partial differential equations that model the advection phenomenon.
The one-dimensional case
For the one-dimensional case, the continuous-time model equation (that is usually solved for ) is:
The CFL condition then has the following form:
where the dimensionless number is called the Courant number,
is the magnitude of the velocity (whose dimension is length/time)
is the time step (whose dimension is time)
is the length interval (whose dimension is length).
The value of changes with the method used to solve the d
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https://en.wikipedia.org/wiki/List%20of%20cities%20and%20towns%20in%20Kosovo
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This is a list of cities and towns in the Kosovo in alphabetical order categorised by municipality or district, according to the criteria used by the Kosovo Agency of Statistics (KAS). Kosovo's population is distributed in 1,467 settlements with 26 per cent of its population concentrated in 7 urban areas, also known as regional centers, consisting of Ferizaj, Gjakova, Gjilan, Mitrovica, Peja, Pristina and Prizren.
The cities and towns in Kosovo belong to the following size ranges in terms of the number of inhabitants:
1 city larger than 150,000: Pristina
2 cities from 50,000 to 100,000: Gjilan and Prizren
9 cities from 15,000 to 50,000: Ferizaj, Fushë Kosovë, Gjakova, Mitrovica, Peja, Podujeva, Rahovec, and Vushtrri
List
See also
Administrative divisions of Kosovo
List of populated places in Kosovo
List of populated places in Kosovo by Albanian name
References
Kosovo
Kosovo
Kosovo-related lists
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https://en.wikipedia.org/wiki/Schanuel%27s%20conjecture
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In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.
Statement
The conjecture is as follows:
Given any complex numbers that are linearly independent over the rational numbers , the field extension (z1, ..., zn, ez1, ..., ezn) has transcendence degree at least over .
The conjecture can be found in Lang (1966).
Consequences
The conjecture, if proven, would generalize most known results in transcendental number theory. The special case where the numbers z1,...,zn are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(z1),...,exp(zn) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem.
The Gelfond–Schneider theorem follows from this strengthened version of Baker's theorem, as does the currently unproven four exponentials conjecture.
Schanuel's conjecture, if proved, would also settle whether numbers such as e + and ee are algebraic or transcendental, and prove that e and are algebraically independent simply by setting z1 = 1 and z2 = i, and using Euler's identity.
Euler's identity states that ei + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, , and i over the complex numbers.
Although ostensibly a problem in number theory, the conjecture has implications in model theory as well. Angus Macintyre and Alex Wilkie, for example, proved that the theory of the real field with exponentiation, exp, is decidable provided Schanuel's conjecture is true. In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem.
Related conjectures and results
The converse Schanuel conjecture is the following statement:
Suppose F is a countable field with characteristic 0, and e : F → F is a homomorphism from the additive group (F,+) to the multiplicative group (F,·) whose kernel is cyclic. Suppose further that for any n elements x1,...,xn of F which are linearly independent over , the extension field (x1,...,xn,e(x1),...,e(xn)) has transcendence degree at least n over . Then there exists a field homomorphism h : F → such that h(e(x)) = exp(h(x)) for all x in F.
A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971. It states:
Given any n formal power series f1,...,fn in t[[t]] which are linearly independent over , then the field extension (t,f1,...,fn,exp(f1),...,exp(fn)) has transcendence degree at least n over (t).
As stated above, the decidability of exp follows from the real version of Schanuel's conjecture which is as follows:
Suppose x1,...,xn are real numb
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https://en.wikipedia.org/wiki/Axiality
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Axiality may refer to:
Axiality (geometry), a measure of the axial symmetry of a two-dimensional shape
Axiality and rhombicity in mathematics, measures of the directional symmetry of a three-dimensional tensor
Axiality, a principle behind the art and poetry of George Quasha
Axiality in architecture, organization around a strong central axis, especially in the architecture of cathedrals and great churches and Beaux-Arts architecture
See also
Axial (disambiguation)
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https://en.wikipedia.org/wiki/Painlev%C3%A9%20transcendents
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In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by
,
,
, and
.
History
Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painlevé property can be transformed into the Weierstrass elliptic equation or the Riccati equation, which can all be solved explicitly in terms of integration and previously known special functions. Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of what was later called Painleve VI equation (see below).
(For orders greater than 2 the solutions can have moving natural boundaries.) Around 1900, Paul Painlevé studied second order differential equations with no movable singularities. He found that up to certain transformations, every such equation
of the form
(with a rational function) can be put into one of fifty canonical forms (listed in ).
found that forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them. There were some computational errors,
and as a result he missed three of the equations, including the general form of Painleve VI.
The errors were fixed and classification completed by Painlevé's student Bertrand Gambier. Independently of Painlevé and Gambier, equation Painleve VI was found
by Richard Fuchs from completely different considerations:
he studied isomonodromic deformations of linear differential equations with regular singularities.
It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters (they are sometimes reducible for special parameter values; see below), but this was finally proved by and .
These six second order nonlinear differential equations are called the Painlevé equations and their solutions are called the Painlevé transcendents.
The most general form of the sixth equation was missed by Painlevé, but was discovered in 1905 by Richard Fuchs (son of Lazarus Fuchs), as the differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on the project
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https://en.wikipedia.org/wiki/Back-and-forth%20method
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In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that
any two countably infinite densely ordered sets (i.e., linearly ordered in such a way that between any two members there is another) without endpoints are isomorphic. An isomorphism between linear orders is simply a strictly increasing bijection. This result implies, for example, that there exists a strictly increasing bijection between the set of all rational numbers and the set of all real algebraic numbers.
any two countably infinite atomless Boolean algebras are isomorphic to each other.
any two equivalent countable atomic models of a theory are isomorphic.
the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, almost surely produces a unique graph, the Rado graph.
any two many-complete recursively enumerable sets are recursively isomorphic.
Application to densely ordered sets
As an example, the back-and-forth method can be used to prove Cantor's isomorphism theorem, although this was not Georg Cantor's original proof. This theorem states that two unbounded countable dense linear orders are isomorphic.
Suppose that
(A, ≤A) and (B, ≤B) are linearly ordered sets;
They are both unbounded, in other words neither A nor B has either a maximum or a minimum;
They are densely ordered, i.e. between any two members there is another;
They are countably infinite.
Fix enumerations (without repetition) of the underlying sets:
A = { a1, a2, a3, ... },
B = { b1, b2, b3, ... }.
Now we construct a one-to-one correspondence between A and B that is strictly increasing. Initially no member of A is paired with any member of B.
(1) Let i be the smallest index such that ai is not yet paired with any member of B. Let j be some index such that bj is not yet paired with any member of A and ai can be paired with bj consistently with the requirement that the pairing be strictly increasing. Pair ai with bj.
(2) Let j be the smallest index such that bj is not yet paired with any member of A. Let i be some index such that ai is not yet paired with any member of B and bj can be paired with ai consistently with the requirement that the pairing be strictly increasing. Pair bj with ai.
(3) Go back to step (1).
It still has to be checked that the choice required in step (1) and (2) can actually be made in accordance to the requirements. Using step (1) as an example:
If there are already ap and aq in A corresponding to bp and bq in B respectively such that ap < ai < aq and bp < bq, we choose bj in between bp and bq using density. Otherwise, we choose a suitable large or small element of B using the fact that B has neither a maximum nor a minimum. Choices made in step (2) are dually possible. Finally, the construction ends after countably many steps because A and B are countably infinite. Note t
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https://en.wikipedia.org/wiki/Covering%20set
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In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set. The term "covering set" is used only in conjunction with sequences possessing exponential growth.
Sierpinski and Riesel numbers
The use of the term "covering set" is related to Sierpinski and Riesel numbers. These are odd natural numbers for which the formula (Sierpinski number) or (Riesel number) produces no prime numbers. Since 1960 it has been known that there exists an infinite number of both Sierpinski and Riesel numbers (as solutions to families of congruences based upon the set } but, because there are an infinitude of numbers of the form or for any , one can only prove to be a Sierpinski or Riesel number through showing that every term in the sequence or is divisible by one of the prime numbers of a covering set.
These covering sets form from prime numbers that in base 2 have short periods. To achieve a complete covering set, Wacław Sierpiński showed that a sequence can repeat no more frequently than every 24 numbers. A repeat every 24 numbers give the covering set }, while a repeat every 36 terms can give several covering sets: }; }; } and }.
Riesel numbers have the same covering sets as Sierpinski numbers.
Other covering sets
Covering sets (thus Sierpinski numbers and Riesel numbers) also exists for bases other than 2.
Covering sets are also used to prove the existence of composite generalized Fibonacci sequences with first two terms coprime (primefree sequence), such as the sequence starting with 20615674205555510 and 3794765361567513.
The concept of a covering set can easily be generalised to other sequences which turn out to be much simpler.
In the following examples + is used as it is in regular expressions to mean 1 or more. For example, 91+3 means the set }.
An example are the following eight sequences:
(29·10n − 191) / 9 or 32+01
(37·10n + 359) / 9 or 41+51
(46·10n + 629) / 9 or 51+81
(59·10n − 293) / 9 or 65+23
(82·10n + 17) / 9 or 91+3
(85·10n + 41) / 9 or 94+9
(86·10n + 31) / 9 or 95+9
(89·10n + 593) / 9 or 98+23
In each case, every term is divisible by one of the primes from the set }. These primes can be said to form a covering set exactly analogous to Sierpinski and Riesel numbers. The covering set } is found for several similar sequences, including:
(38·10n − 137) / 9 or 42+07
(4·10n − 337) / 9 or 4+07
(73·10n + 359) / 9 or 81+51
9175·10n + 1 or 91750+1
10176·10n − 1 or 101759+
(334·10n − 1)/9 or 371+
(12211·10n − 1)/3 or 40703+
(8026·10n − 7)/9 or 8917+
Also for bases other than 10:
521·12n + 1 or 3750+1 in duodecimal
(1288·12n − 1)/11 or 991+ in duodecimal
(4517·12n − 7)/11 or 2X27+ in duodecimal
376·12n − 1 or 273E+ in duodecimal
The covering set of them is }
An even simpler case can be found in the sequence:
(76·10n − 67) / 99 (n must be odd) or (76)+7 (Sequence: 7, 767, 76767, 7676767, 767676767 etc.)
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https://en.wikipedia.org/wiki/154%20%28number%29
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154 (one hundred [and] fifty-four) is the natural number following 153 and preceding 155.
In mathematics
154 is a nonagonal number. Its factorization makes 154 a sphenic number
There is no integer with exactly 154 coprimes below it, making 154 a noncototient, nor is there, in base 10, any integer that added up to its own digits yields 154, making 154 a self number
154 is the sum of the first six factorials, if one starts with and assumes that .
With just 17 cuts, a pancake can be cut up into 154 pieces (Lazy caterer's sequence).
The distinct prime factors of 154 add up to 20, and so do the ones of 153, hence the two form a Ruth-Aaron pair. 154! + 1 is a factorial prime.
In music
154 is an album by Wire, named for the number of live gigs Wire had performed at that time
In the military
was a United States Navy Trefoil-class concrete barge during World War II
was a United States Navy Admirable-class minesweeper during World War II
was a United States Navy Wickes-class destroyer during World War II
was a United States Navy General G. O. Squier-class transport during World War II
was a United States Navy Haskell-class attack transport during World War II
was a United States Navy Buckley-class destroyer escort ship during World War II
Strike Fighter Squadron 154 (VFA-154) is a United States Navy strike fighter squadron stationed at Naval Air Station Lemoore
Convoy ON-154 was a convoy of ships in December 1942 during World War II
In sports
Major League Baseball teams played 154 games a season prior to expansion in 1961
Golfer Jack Nicklaus played in a record 154 consecutive major championships from the 1957 U.S. Open to the 1998 U.S. Open
In transportation
Seattle Bus Route 154
The Maserati Tipo 154 racecar, also known as 151/4, was produced in 1965
In other fields
154 is also:
The year AD 154 or 154 BC
154 AH is a year in the Islamic calendar that corresponds to that corresponds to 770 – 771 AD
154 Bertha is a dark outer Main belt asteroid
Ross 154 is a red dwarf star near the southern constellation Sagittarius
Psalm 154
Shakespeare’s 154 sonnets, the last being Sonnet 154
Elcapo No. 154, Saskatchewan is a rural municipality in Saskatchewan, Canada
The atomic number of an element temporarily called Unpentquadium
In the United Kingdom, 154 is the telephone number used to report Business Faults with telephone provider BT
See also
List of highways numbered 154
United Nations Security Council Resolution 154
United States Supreme Court cases, Volume 154
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 140 - 141
External links
Virtual Science: Number 154
Integers
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https://en.wikipedia.org/wiki/Gibbs%27%20inequality
|
In information theory, Gibbs' inequality is a statement about the information entropy of a discrete probability distribution. Several other bounds on the entropy of probability distributions are derived from Gibbs' inequality, including Fano's inequality.
It was first presented by J. Willard Gibbs in the 19th century.
Gibbs' inequality
Suppose that
is a discrete probability distribution. Then for any other probability distribution
the following inequality between positive quantities (since pi and qi are between zero and one) holds:
with equality if and only if
for all i. Put in words, the information entropy of a distribution P is less than or equal to its cross entropy with any other distribution Q.
The difference between the two quantities is the Kullback–Leibler divergence or relative entropy, so the inequality can also be written:
Note that the use of base-2 logarithms is optional, and
allows one to refer to the quantity on each side of the inequality as an
"average surprisal" measured in bits.
Proof
For simplicity, we prove the statement using the natural logarithm (). Because
the particular logarithm base that we choose only scales the relationship by the factor .
Let denote the set of all for which pi is non-zero. Then, since for all x > 0, with equality if and only if x=1, we have:
The last inequality is a consequence of the pi and qi being part of a probability distribution. Specifically, the sum of all non-zero values is 1. Some non-zero qi, however, may have been excluded since the choice of indices is conditioned upon the pi being non-zero. Therefore, the sum of the qi may be less than 1.
So far, over the index set , we have:
,
or equivalently
.
Both sums can be extended to all , i.e. including , by recalling that the expression tends to 0 as tends to 0, and tends to as tends to 0. We arrive at
For equality to hold, we require
for all so that the equality holds,
and which means if , that is, if .
This can happen if and only if for .
Alternative proofs
The result can alternatively be proved using Jensen's inequality, the log sum inequality, or the fact that the Kullback-Leibler divergence is a form of Bregman divergence. Below we give a proof based on Jensen's inequality:
Because log is a concave function, we have that:
Where the first inequality is due to Jensen's inequality, and the last equality is due to the same reason given in the above proof.
Furthermore, since is strictly concave, by the equality condition of Jensen's inequality we get equality when
and
Suppose that this ratio is , then we have that
Where we use the fact that are probability distributions. Therefore, the equality happens when .
Corollary
The entropy of is bounded by:
The proof is trivial – simply set for all i.
See also
Information entropy
Bregman divergence
Log sum inequality
References
Information theory
Coding theory
Probabilistic inequalities
Articles containing proofs
|
https://en.wikipedia.org/wiki/Abstract%20analytic%20number%20theory
|
Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results. The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century.
Arithmetic semigroups
The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying the following properties:
There exists a countable subset (finite or countably infinite) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form
where the pi are distinct elements of P, the αi are positive integers, r may depend on a, and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of P are called the primes of G.
There exists a real-valued norm mapping on G such that
The total number of elements of norm is finite, for each real .
Additive number systems
An additive number system is an arithmetic semigroup in which the underlying monoid G is free abelian. The norm function may be written additively.
If the norm is integer-valued, we associate counting functions a(n) and p(n) with G where p counts the number of elements of P of norm n, and a counts the number of elements of G of norm n. We let A(x) and P(x) be the corresponding formal power series. We have the fundamental identity
which formally encodes the unique expression of each element of G as a product of elements of P. The radius of convergence of G is the radius of convergence of the power series A(x).
The fundamental identity has the alternative form
Examples
The prototypical example of an arithmetic semigroup is the multiplicative semigroup of positive integers G = Z+ = {1, 2, 3, ...}, with subset of rational primes P = {2, 3, 5, ...}. Here, the norm of an integer is simply , so that , the greatest integer not exceeding x.
If K is an algebraic number field, i.e. a finite extension of the field of rational numbers Q, then the set G of all nonzero ideals in the ring of integers OK of K forms an arithmetic semigroup with identity element OK and the norm of an ideal I is given by the cardinality of the quotient ring OK/I. In this case, the appropriate generalisation of the prime number theorem is the Landau prime ideal theorem, which describes the asymptotic distribution of the ideals in OK.
Various arithmetical categories which satisfy a theorem of Krull-Schmidt type can be considered. In all these cases, the elements of G are isomorphism classes in an appropriate category, and P consists of all isomorphism classes of indecomposable objects, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following.
The category of all finite abel
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https://en.wikipedia.org/wiki/EHP
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EHP may refer to:
E.H.P., a 1920s French automobile manufacturer
Eastern Highlands Province in Papua New Guinea
EHP spectral sequence in mathematics
(), Labourist Movement Party, a political party in Turkey
Environmental Health Perspectives, a scholarly journal
Environmental Planning & Historic Preservation (EHP), a Federal Emergency Management Agency (FEMA) program
Everglades Holiday Park, in Fort Lauderdale, Florida
Sahrawi peseta, the de facto currency of the Sahrawi Arab Democratic Republic
Electron-Hole Pairs, the fundamental unit of generation and recombination in semiconductors
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https://en.wikipedia.org/wiki/Paper%20bag%20problem
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In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch.
According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:
where w is the width of the bag (the shorter dimension), h is the height (the longer dimension), and V is the maximum volume. The approximation ignores the crimping round the equator of the bag.
A very rough approximation to the capacity of a bag that is open at one edge is:
(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a lens).
The square teabag
For the special case where the bag is sealed on all edges and is square with unit sides, h = w = 1, the first formula estimates a volume of roughly
or roughly 0.19. According to Andrew Kepert at the University of Newcastle, Australia, an upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+.
Robin also found a more complicated formula for the general paper bag, which gives 0.2017, below the bounds given by Kepert (i.e., 0.2055+ ≤ maximum volume ≤ 0.217+).
See also
Biscornu, a shape formed by attaching two squares in a different way, with the corner of one at the midpoint of the other
Mylar balloon (geometry)
Notes
References
External links
The original statement of the teabag problem
Andrew Kepert's work on the teabag problem (mirror)
Curved folds for the teabag problem
A numerical approach to the teabag problem by Andreas Gammel
Geometric shapes
Mathematical optimization
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https://en.wikipedia.org/wiki/Trigamma%20function
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In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by
.
It follows from this definition that
where is the digamma function. It may also be defined as the sum of the series
making it a special case of the Hurwitz zeta function
Note that the last two formulas are valid when is not a natural number.
Calculation
A double integral representation, as an alternative to the ones given above, may be derived from the series representation:
using the formula for the sum of a geometric series. Integration over yields:
An asymptotic expansion as a Laurent series is
if we have chosen , i.e. the Bernoulli numbers of the second kind.
Recurrence and reflection formulae
The trigamma function satisfies the recurrence relation
and the reflection formula
which immediately gives the value for z : .
Special values
At positive half integer values we have that
Moreover, the trigamma function has the following special values:
where represents Catalan's constant.
There are no roots on the real axis of , but there exist infinitely many pairs of roots for . Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with . For example, and are the first two roots with .
Relation to the Clausen function
The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,
Computation and approximation
An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.
Appearance
The trigamma function appears in this sum formula:
See also
Gamma function
Digamma function
Polygamma function
Catalan's constant
Notes
References
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. . See section §6.4
Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource
Gamma and related functions
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https://en.wikipedia.org/wiki/Lens%20%28geometry%29
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In 2-dimensional geometry, a lens is a convex region bounded by two circular arcs joined to each other at their endpoints. In order for this shape to be convex, both arcs must bow outwards (convex-convex). This shape can be formed as the intersection of two circular disks. It can also be formed as the union of two circular segments (regions between the chord of a circle and the circle itself), joined along a common chord.
Types
If the two arcs of a lens have equal radius, it is called a symmetric lens, otherwise is an asymmetric lens.
The vesica piscis is one form of a symmetric lens, formed by arcs of two circles whose centers each lie on the opposite arc. The arcs meet at angles of 120° at their endpoints.
Area
Symmetric
The area of a symmetric lens can be expressed in terms of the radius R and arc lengths θ in radians:
Asymmetric
The area of an asymmetric lens formed from circles of radii R and r with distance d between their centers is
where
is the area of a triangle with sides d, r, and R.
The two circles overlap if . For sufficiently large , the coordinate of the lens centre lies between the coordinates of the two circle centers:
For small the coordinate of the lens centre lies outside the line that connects the circle centres:
By eliminating y from the circle equations and the abscissa of the intersecting rims is
.
The sign of x, i.e., being larger or smaller than , distinguishes the two cases shown in the images.
The ordinate of the intersection is
.
Negative values under the square root indicate that the rims of the two circles do not touch
because the circles are too far apart or one circle lies entirely within the other.
The value under the square root is a biquadratic polynomial of d. The four roots of this polynomial are associated with y=0 and with the four values of d where the two circles have only one point in common.
The angles in the blue triangle of sides d, r and R are
where y is the ordinate of the intersection. The branch of the arcsin with is to be taken if .
The area of the triangle is .
The area of the asymmetric lens is , where the two angles are measured in radians.
[This is an application of the Inclusion-exclusion principle: the two circular sectors centered at (0,0) and (d,0) with central
angles and have areas and . Their union covers the triangle, the flipped triangle with corner at (x,-y), and twice the lens area.]
Applications
A lens with a different shape forms the answer to Mrs. Miniver's problem, on finding a lens with half the area of the union of the two circles.
Lenses are used to define beta skeletons, geometric graphs defined on a set of points by connecting pairs of points by an edge whenever a lens determined by the two points is empty.
See also
Circle–circle intersection
Lune, a related non-convex shape formed by two circular arcs, one bowing outwards and the other inwards
Lemon, created by a lens rotated around an axis through its tips.
References
Con
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https://en.wikipedia.org/wiki/Homotopy%20category
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In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below.
More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
The naive homotopy category
The category of topological spaces Top has objects the topological spaces and morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f: X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other. There is a functor from Top to hTop that sends spaces to themselves and morphisms to their homotopy classes. A map f: X → Y is called a homotopy equivalence if it becomes an isomorphism in the naive homotopy category.
Example: The circle S1, the plane R2 minus the origin, and the Möbius strip are all homotopy equivalent, although these topological spaces are not homeomorphic.
The notation [X,Y] is often used for the set of morphisms from a space X to a space Y in the naive homotopy category (but it is also used for the related categories discussed below).
The homotopy category, following Quillen
Quillen (1967) emphasized another category which further simplifies the category of topological spaces. Homotopy theorists have to work with both categories from time to time, but the consensus is that Quillen's version is more important, and so it is often called simply the "homotopy category".
One first defines a weak homotopy equivalence: a continuous map is called a weak homotopy equivalence if it induces a bijection on sets of path components and a bijection on homotopy groups with arbitrary base points. Then the (true) homotopy category is defined by localizing the category of topological spaces with respect to the weak homotopy equivalences. That is, the objects are still the topological spaces, but an inverse morphism is added for each weak homotopy equivalence. This has the effect that a continuous map becomes an isomorphism in the homotopy category if and only if it is a weak homotopy equivalence. There are obvious functors from the category of topological spaces to the naive homotopy category (as defined above), and from there to the homotopy category.
Results of J.H.C. Whitehead, in particular Whitehead's theorem and the existence of CW approximations, give a more explicit description of the homotopy category. Namely, the homotopy category is equivalent to the full subcategory of the naive homotopy category th
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https://en.wikipedia.org/wiki/Cassini%20oval
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In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century.
Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval.
Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
Formal definition
A Cassini oval is a set of points, such that for any point of the set, the product of the distances to two fixed points is a constant, usually written as where :
As with an ellipse, the fixed points are called the foci of the Cassini oval.
Equations
If the foci are (a, 0) and (−a, 0), then the equation of the curve is
When expanded this becomes
The equivalent polar equation is
Shape
The curve depends, up to similarity, on e = b/a. When e < 1, the curve consists of two disconnected loops, each of which contains a focus. When e = 1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double point (specifically, a crunode) at the origin. When e > 1, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for and convex for . The limiting case of a → 0 (hence e → ), in which case the foci coincide with each other, is a circle.
The curve always has x-intercepts at ± c where c2 = a2 + b2. When e < 1 there are two additional real x-intercepts and when e > 1 there are two real y-intercepts, all other x- and y-intercepts being imaginary.
The curve has double points at the circular points at infinity, in other words the curve is bicircular. These points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there. From this information and Plücker's formulas it is possible to deduce the Plücker numbers for the case e ≠ 1: degree = 4, class = 8, number of nodes = 2, number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1.
The tangents at the circular points are given by x ± iy = ± a which have real points of intersection at (± a, 0). So the foci are, in fact, foci in the sense defined by Plücker. The circular points are points of inflection so these are triple foci. When e ≠ 1 the curve has class eight, which implies that there should be a total of eight real foci. Six of these have been accounted for in the two triple foci and the remaining two are at
So the additional foci are on the x-axis when the curve has two loops and on the y-axis when the curve has a single loop.
Cassini ovals and orthogonal trajectories
Orthogonal trajectories of a giv
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https://en.wikipedia.org/wiki/Wilhelm%20Wirtinger
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Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.
Biography
He was born at Ybbs on the Danube and studied at the University of Vienna, where he received his doctorate in 1887, and his habilitation in 1890. Wirtinger was greatly influenced by Felix Klein with whom he studied at the University of Berlin and the University of Göttingen.
Honours
In 1907 the Royal Society of London awarded him the Sylvester Medal, for his contributions to the general theory of functions.
Work
Research activity
He worked in many areas of mathematics, publishing 71 works. His first significant work, published in 1896, was on theta functions. He proposed as a generalization of eigenvalues, the concept of the spectrum of an operator, in an 1897 paper; the concept was further extended by David Hilbert and now it forms the main object of investigation in the field of spectral theory. Wirtinger also contributed papers on complex analysis, geometry, algebra, number theory, and Lie groups. He collaborated with Kurt Reidemeister on knot theory, showing in 1905 how to compute the knot group. Also, he was one of the editors of the Analysis section of Klein's encyclopedia.
During a conversation, Wirtinger attracted the attention of Stanisław Zaremba to a particular boundary value problem, which later became known as the mixed boundary value problem.
Teaching activity
A partial list of his students includes the following scientists:
Wilhelm Blaschke
Hilda Geiringer
Kurt Gödel
Wilhelm Gross
Eduard Helly
Erwin Schrödinger
Olga Taussky-Todd
Leopold Vietoris
Roland Weitzenböck
Selected publications
, available at DigiZeitschirften. In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger derivatives and the tangential Cauchy–Riemann condition. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced.
.
.
See also
Wirtinger inequality (2-forms)
Notes
Biographical references
, available at DigiZeitschirften. An ample commemorative paper containing a list of Wirtinger's publications.
External links
from the ICMI History of ICMI Web site.
19th-century Austrian mathematicians
Mathematicians from Austria-Hungary
1865 births
1945 deaths
Royal Medal winners
University of Vienna alumni
20th-century Austrian mathematicians
People from Melk District
Academic staff of the University of Innsbruck
Academic staff of the University of Vienna
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https://en.wikipedia.org/wiki/Damodar%20Dharmananda%20Kosambi
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Damodar Dharmananda Kosambi (31 July 1907 – 29 June 1966) was an Indian polymath with interests in mathematics, statistics, philology, history, and genetics. He contributed to genetics by introducing the Kosambi map function. In statistics, he was the first person to develop orthogonal infinite series expressions for stochastic processes via the Kosambi–Karhunen–Loève theorem. He is also well known for his work in numismatics and for compiling critical editions of ancient Sanskrit texts. His father, Dharmananda Damodar Kosambi, had studied ancient Indian texts with a particular emphasis on Buddhism and its literature in the Pali language. Damodar Kosambi emulated him by developing a keen interest in his country's ancient history. He was also a Marxist historian specialising in ancient India who employed the historical materialist approach in his work. He is particularly known for his classic work An Introduction to the Study of Indian History.
He is described as "the patriarch of the Marxist school of Indian historiography". Kosambi was critical of the policies of then prime minister Jawaharlal Nehru, which, according to him, promoted capitalism in the guise of democratic socialism. He was an enthusiast of the Chinese Communist Revolution and its ideals, and was a leading activist in the world peace movement.
Early life
Damodar Dharmananda Kosambi was born at Kosben in Portuguese Goa into a Saraswat Brahmin family to Dharmananda Damodar Kosambi. After a few years of schooling in India, in 1918, Damodar and his elder sister, Manik travelled to Cambridge, Massachusetts with their father, who had taken up a teaching position at the Cambridge Latin School. Their father was tasked by Professor Charles Rockwell Lanman of Harvard University to complete compiling a critical edition of Visuddhimagga, a book on Buddhist philosophy, which was originally started by Henry Clarke Warren. There, the young Damodar spent a year in a Grammar school and then was admitted to the Cambridge High and Latin School in 1920. He became a member of the Cambridge branch of American Boy Scouts.
It was in Cambridge that he befriended another prodigy of the time, Norbert Wiener, whose father Leo Wiener was the elder Kosambi's colleague at Harvard University. Kosambi excelled in his final school examination and was one of the few candidates who was exempt on the basis of merit from necessarily passing an entrance examination essential at the time to gain admission to Harvard University. He enrolled in Harvard in 1924, but eventually postponed his studies, and returned to India. He stayed with his father who was now working in the Gujarat University, and was in the close circles of Mahatma Gandhi.
In January 1926, Kosambi returned to the US with his father, who once again studied at Harvard University for a year and half. Kosambi studied mathematics under George David Birkhoff, who wanted him to concentrate on mathematics, but the ambitious Kosambi instead took many divers
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https://en.wikipedia.org/wiki/Nuclear%20cross%20section
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The nuclear cross section of a nucleus is used to describe the probability that a nuclear reaction will occur. The concept of a nuclear cross section can be quantified physically in terms of "characteristic area" where a larger area means a larger probability of interaction. The standard unit for measuring a nuclear cross section (denoted as σ) is the barn, which is equal to , or . Cross sections can be measured for all possible interaction processes together, in which case they are called total cross sections, or for specific processes, distinguishing elastic scattering and inelastic scattering; of the latter, amongst neutron cross sections the absorption cross sections are of particular interest.
In nuclear physics it is conventional to consider the impinging particles as point particles having negligible diameter. Cross sections can be computed for any nuclear process, such as capture scattering, production of neutrons, or nuclear fusion. In many cases, the number of particles emitted or scattered in nuclear processes is not measured directly; one merely measures the attenuation produced in a parallel beam of incident particles by the interposition of a known thickness of a particular material. The cross section obtained in this way is called the total cross section and is usually denoted by a σ or σT.
Typical nuclear radii are of the order 10−14 m. Assuming spherical shape, we therefore expect the cross sections for nuclear reactions to be of the order of or (i.e., 1 barn). Observed cross sections vary enormously: for example, slow neutrons absorbed by the (n, ) reaction show a cross section much higher than 1,000 barns in some cases (boron-10, cadmium-113, and xenon-135), while the cross sections for transmutations by gamma-ray absorption are in the region of 0.001 barn.
Microscopic and macroscopic cross section
Nuclear cross sections are used in determining the nuclear reaction rate, and are governed by the reaction rate equation for a particular set of particles (usually viewed as a "beam and target" thought experiment where one particle or nucleus is the "target", which is typically at rest, and the other is treated as a "beam", which is a projectile with a given energy).
For neutron interactions incident upon a thin sheet of material (ideally made of a single isotope), the nuclear reaction rate equation is written as:
where:
: number of reactions of type x, units: [1/time⋅volume]
: beam flux, units: [1/area⋅time]
: microscopic cross section for reaction , units: [area] (usually barns or cm2).
: density of atoms in the target in units of [1/volume]
: macroscopic cross-section [1/length]
Types of reactions frequently encountered are s: scattering, : radiative capture, a: absorption (radiative capture belongs to this type), f: fission, the corresponding notation for cross-sections being: , , , etc. A special case is the total cross-section , which gives the probability of a neutron to undergo any sort of reaction ().
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https://en.wikipedia.org/wiki/Petrov%20classification
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In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.
It is most often applied in studying exact solutions of Einstein's field equations, but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation. The classification was found in 1954 by A. Z. Petrov and independently by Felix Pirani in 1957.
Classification theorem
We can think of a fourth rank tensor such as the Weyl tensor, evaluated at some event, as acting on the space of bivectors at that event like a linear operator acting on a vector space:
Then, it is natural to consider the problem of finding eigenvalues and eigenvectors (which are now referred to as eigenbivectors) such that
In (four-dimensional) Lorentzian spacetimes, there is a six-dimensional space of antisymmetric bivectors at each event. However, the symmetries of the Weyl tensor imply that any eigenbivectors must belong to a four-dimensional subset.
Thus, the Weyl tensor (at a given event) can in fact have at most four linearly independent eigenbivectors.
The eigenbivectors of the Weyl tensor can occur with various multiplicities and any multiplicities among the eigenbivectors indicates a kind of algebraic symmetry of the Weyl tensor at the given event. The different types of Weyl tensor (at a given event) can be determined by solving a characteristic equation, in this case a quartic equation. All the above happens similarly to the theory of the eigenvectors of an ordinary linear operator.
These eigenbivectors are associated with certain null vectors in the original spacetime, which are called the principal null directions (at a given event).
The relevant multilinear algebra is somewhat involved (see the citations below), but the resulting classification theorem states that there are precisely six possible types of algebraic symmetry. These are known as the Petrov types:
Type I: four simple principal null directions,
Type II: one double and two simple principal null directions,
Type D: two double principal null directions,
Type III: one triple and one simple principal null direction,
Type N: one quadruple principal null direction,
Type O: the Weyl tensor vanishes.
The possible transitions between Petrov types are shown in the figure, which can also be interpreted as stating that some of the Petrov types are "more special" than others. For example, type I, the most general type, can degenerate to types II or D, while type II can degenerate to types III, N, or D.
Different events in a given spacetime can have different Petrov types. A Weyl tensor that has type I (at some event) is called algebraically general; otherwise, it is called algebraically special (at that event). In General Relativity, type O spacetimes are conformally flat.
Newman–Penros
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https://en.wikipedia.org/wiki/Segre%20classification
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The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or the Ricci tensor) and primarily finds application in the classification of exact solutions in general relativity.
See also
Corrado Segre
Jordan normal form
Petrov classification
References
See section 5.1 for the Segre classification.
Linear algebra
Tensors
Tensors in general relativity
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https://en.wikipedia.org/wiki/Abraham%20Nemeth
|
Abraham Nemeth (October 16, 1918 – October 2, 2013) was an American mathematician. He was professor of mathematics at the University of Detroit Mercy in Detroit, Michigan. Nemeth was blind and is known for developing Nemeth Braille, a system for blind people to read and write mathematics.
Early life
Nemeth was born in New York City on the Lower East Side of Manhattan into a large family of Hungarian Jewish immigrants who spoke Yiddish. He was blind from birth from a combination of macular degeneration and retinitis pigmentosa.
He attended public schools at first but did most of his primary and secondary education at the Jewish Guild for the Blind school in Yonkers, New York. His undergraduate studies were at Brooklyn College where he studied psychology. He earned a Master of Arts degree in psychology from Columbia University.
Nemeth studied mathematics and physics at Brooklyn College. He did not major in mathematics because his academic advisors discouraged him. However, tired of what he felt were unfulfilling jobs at agencies of the blind, and with the encouragement of his first wife Florence, he decided to continue his education in mathematics. He received a Ph.D. in mathematics from Wayne State University.
Academic career
Nemeth taught part-time at various colleges in New York. Though his employers were sometimes reluctant to hire him knowing that he was blind, his reputation grew as it became apparent that he was a capable mathematician and teacher. Nemeth distinguished himself from many other blind people by being able to write visual print letters and mathematical symbols on paper and blackboards just like sighted people, a skill he learned as a child. Nemeth says that this skill allowed him to succeed in mathematics, during an era without much technology, when even Braille was difficult to use in mathematics. During the 1950s he moved to Detroit, Michigan to accept a position at the University of Detroit working with Keith Rosenberg. He remained there for 30 years, retiring in 1985. During the late 1960s he studied computer science and began the university's program in that subject.
Importance to mathematics and blindness
As the coursework became more advanced, he found that he needed a braille code that would more effectively handle the kinds of math and science material he was tackling. Ultimately, he developed the Nemeth Braille Code for Mathematics and Science Notation, which was published in 1952. The Nemeth Code has gone through 4 revisions since its initial development, and continues to be widely used today.
Nemeth is also responsible for the rules of MathSpeak, a system for orally communicating mathematical text. In the course of his studies, Nemeth found that he needed to make use of sighted readers to read otherwise inaccessible math texts and other materials. Likewise, he needed a method for dictating his math work and other materials for transcription into print. The conventions Nemeth developed for efficiently readin
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https://en.wikipedia.org/wiki/Coincidence%20point
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In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image.
Formally, given two functions
we say that a point x in X is a coincidence point of f and g if f(x) = g(x).
Coincidence theory (the study of coincidence points) is, in most settings, a generalization of fixed point theory, the study of points x with f(x) = x. Fixed point theory is the special case obtained from the above by letting X = Y and taking g to be the identity function.
Just as fixed point theory has its fixed-point theorems, there are theorems that guarantee the existence of coincidence points for pairs of functions. Notable among them, in the setting of manifolds, is the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points.
Coincidence points, like fixed points, are today studied using many tools from mathematical analysis and topology. An equaliser is a generalization of the coincidence set.
References
Mathematical analysis
Topology
Fixed points (mathematics)
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https://en.wikipedia.org/wiki/Gradient%20%28disambiguation%29
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Gradient in vector calculus is a vector field representing the maximum rate of increase of a scalar field or a multivariate function and the direction of this maximal rate.
Gradient may also refer to:
Gradient sro, a Czech aircraft manufacturer
Image gradient, a gradual change or blending of color
Color gradient, a range of position-dependent colors, usually used to fill a region
Texture gradient, the distortion in size which closer objects have compared to objects farther away
Spatial gradient, a gradient whose components are spatial derivatives
Grade (slope), the inclination of a road or other geographic feature
Slope, a number that describes both the direction and the steepness of a line
See also
Fade (disambiguation)
Gradation (disambiguation)
Grade (disambiguation)
Rate of change (disambiguation)
Transition (disambiguation)
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https://en.wikipedia.org/wiki/Polarization%20identity
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In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.
The norm associated with any inner product space satisfies the parallelogram law:
In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products.
Given a normed space , the parallelogram law holds for if and only if there exists an inner product on such that for all in which case this inner product is uniquely determined by the norm via the polarization identity.
Polarization identities
Any inner product on a vector space induces a norm by the equation
The polarization identities reverse this relationship, recovering the inner product from the norm.
Every inner product satisfies:
Solving for gives the formula If the inner product is real then and this formula becomes a polarization identity for real inner products.
Real vector spaces
If the vector space is over the real numbers then the polarization identities are:
These various forms are all equivalent by the parallelogram law:
This further implies that class is not a Hilbert space whenever , as the parallelogram law is not satisfied. For the sake of counterexample, consider and for any two disjoint subsets of general domain and compute the measure of both sets under parallelogram law.
Complex vector spaces
For vector spaces over the complex numbers, the above formulas are not quite correct because they do not describe the imaginary part of the (complex) inner product.
However, an analogous expression does ensure that both real and imaginary parts are retained.
The complex part of the inner product depends on whether it is antilinear in the first or the second argument.
The notation which is commonly used in physics will be assumed to be antilinear in the argument while which is commonly used in mathematics, will be assumed to be antilinear its the argument.
They are related by the formula:
The real part of any inner product (no matter which argument is antilinear and no matter if it is real or complex) is a symmetric bilinear map that for any is always equal to:
It is always a symmetric map, meaning that
and it also satisfies:
Thus which in plain English says that to move a factor of to the other argument, introduce a negative sign.
Let
Then implies
and
Moreover,
which proves that
From it follows that and so that
which proves that
Unlike its real part, the imaginary part of a complex inner product depends on which argument is antilinear.
Antilinear in first argument
The polari
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https://en.wikipedia.org/wiki/Joel%20Brawley
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Joel Vincent Brawley, Jr. is the Alumni Distinguished Professor of Mathematical Sciences at Clemson University. Brawley is reputed nationally for being a prolific mathematics educator and is regarded highly for his teaching abilities. Brawley is also a prominent researcher in the field of algebra, specifically finite fields.
Joel Vincent Brawley, Jr. was born in Mooresville in 1938. He went to the Mooresville High School and received his undergraduate degree in Engineering Mathematics/Mechanics, master's and doctoral degrees in Mathematics and Statistics, all from the North Carolina State University (NCSU) in Raleigh, North Carolina. Dr. Brawley came to Clemson University as an assistant professor in 1965 after a brief stint on the Faculty of NCSU. He became associate professor in 1968, professor in 1972 and the Alumni Distinguished Professor in 1982.
Dr. Brawley has also been a research consultant with the National Security Agency (NSA) for the past three decades.
Dr. Joel Brawley received the highest awards in the nation for mathematics education including the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America, South Carolina Governor's Professor of the Year and the Class of 39 Award for Excellence from the Clemson University.
External links
Official webpage
Biography
Governor's Award
1938 births
Living people
Clemson University faculty
North Carolina State University alumni
20th-century American mathematicians
21st-century American mathematicians
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https://en.wikipedia.org/wiki/La%20Granja%2C%20Chile
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La Granja (Spanish for "the farm") is a commune of Chile located in Santiago Province, Santiago Metropolitan Region.
Demographics
According to the 2002 census of the National Statistics Institute, La Granja spans an area of and has 132,520 inhabitants (64,750 men and 67,770 women), and the commune is an entirely urban area. The population fell by 0.6% (765 persons) between the 1992 and 2002 censuses. The 2006 projected population was 129,707.
Stats
Average annual household income: US$24,662 (PPP, 2006)
Population below poverty line: 14.2% (2006)
Regional quality of life index: 77.93, mid-high, 18 out of 52 (2005)
Human Development Index: 0.689, 158 out of 341 (2003)
Administration
As a commune, La Granja is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2012-2016 alcalde is Felipe Delpin Aguilar (DC). The communal council has the following members:
Cristián Carmona Macaya (DC)
Rodrigo Quezada Arriagada (IND)
Juan Valdés Valdés (PS)
Germán Pino Maturana (PPD)
Sergio Robles Pinto (PC)
Silvana Poblete Romero (PS)
Berta Venegas Maldonado (DC)
Patricio Oyarce Bravo (UDI)
Within the electoral divisions of Chile, La Granja is represented in the Chamber of Deputies by Felipe Salaberry (UDI) and Ximena Vidal (PPD) as part of the 25th electoral district, (together with Macul and San Joaquín). The commune is represented in the Senate by Soledad Alvear (PDC) and Pablo Longueira (UDI) as part of the 8th senatorial constituency (Santiago-East).
References
Populated places in Santiago Province, Chile
Communes of Chile
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https://en.wikipedia.org/wiki/Kumaraswamy%20distribution
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In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval (0,1). It is similar to the Beta distribution, but much simpler to use especially in simulation studies since its probability density function, cumulative distribution function and quantile functions can be expressed in closed form. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded with a zero-inflation. This was extended to inflations at both extremes [0,1] in later work with S. G . Fletcher.
Characterization
Probability density function
The probability density function of the Kumaraswamy distribution without considering any inflation is
and where a and b are non-negative shape parameters.
Cumulative distribution function
The cumulative distribution function is
Quantile function
The inverse cumulative distribution function (quantile function) is
Generalizing to arbitrary interval support
In its simplest form, the distribution has a support of (0,1). In a more general form, the normalized variable x is replaced with the unshifted and unscaled variable z where:
Properties
The raw moments of the Kumaraswamy distribution are given by:
where B is the Beta function and Γ(.) denotes the Gamma function. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:
The Shannon entropy (in nats) of the distribution is:
where is the harmonic number function.
Relation to the Beta distribution
The Kumaraswamy distribution is closely related to Beta distribution.
Assume that Xa,b is a Kumaraswamy distributed random variable with parameters a and b.
Then Xa,b is the a-th root of a suitably defined Beta distributed random variable.
More formally, Let Y1,b denote a Beta distributed random variable with parameters and .
One has the following relation between Xa,b and Y1,b.
with equality in distribution.
One may introduce generalised Kumaraswamy distributions by considering random variables of the form
, with and where
denotes a Beta distributed random variable with parameters and .
The raw moments of this generalized Kumaraswamy distribution are given by:
Note that we can re-obtain the original moments setting , and .
However, in general, the cumulative distribution function does not have a closed form solution.
Related distributions
If then (Uniform distribution)
If then
If (Beta distribution) then
If (Beta distribution) then
If then
If then
If then
If then
If then , the generalized beta distribution of the first kind.
Example
An example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity z whose upper bound is zmax and lower bound is 0, which is also a natural example for having two inflations as many reservoirs have nonzero probabilities for both empty and full reservoir states.
References
Continuous distributions
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https://en.wikipedia.org/wiki/Renca
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Renca is a commune of Chile located in Santiago Province, Santiago Metropolitan Region. It was founded on 6 May 1894.
Demographics
According to the 2002 census of the National Statistics Institute, Renca spans an area of and has 133,500 inhabitants, and the commune is an entirely urban area. The population grew by 3.5% (4,500 people) between the 1992 and 2002 censuses. Its 2006 projected population was 134,690.
Statistics
Average annual household income: US$17,278 (PPP, 2006)
Population below poverty line: 19.2% (2006)
Regional quality of life index: 63.39, low, 49 out of 52 (2005)
Human Development Index: 0.709, 112 out of 341 (2003)
Administration
As a commune, Renca is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The communal council has the following members:
Víctor Barahona Ugarte (UDI)
Nora Contreras Canales (UDI)
Renato Estay Cabrera (UDI)
Cristián Rojas Pizarro (IND)
Berta Roquer Casanova (PDC)
Teresa Cordero Villarroel (PPD)
Cristián Sandoval Saavedra (PDC)
Silvia Contreras Morales (PC)
Within the electoral divisions of Chile, Renca is represented in the Chamber of Deputies by Karla Rubilar (RN) and María Antonieta Saa (PPD) as part of the 17th electoral district, (together with Conchalí and Huechuraba). The commune is represented in the Senate by Guido Girardi Lavín (PPD) and Jovino Novoa Vásquez (UDI) as part of the 7th senatorial constituency (Santiago-West).
References
External links
Municipality of Renca
Populated places in Santiago Province, Chile
Communes of Chile
Geography of Santiago, Chile
Populated places established in 1894
1894 establishments in Chile
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https://en.wikipedia.org/wiki/Sampling%20frame
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In statistics, a sampling frame is the source material or device from which a sample is drawn. It is a list of all those within a population who can be sampled, and may include individuals, households or institutions.
Importance of the sampling frame is stressed by Jessen and Salant and Dillman.
Obtaining and organizing a sampling frame
In the most straightforward cases, such as when dealing with a batch of material from a production run, or using a census, it is possible to identify and measure every single item in the population and to include any one of them in our sample; this is known as direct element sampling. However, in many other cases this is not possible; either because it is cost-prohibitive (reaching every citizen of a country) or impossible (reaching all humans alive).
Having established the frame, there are a number of ways for organizing it to improve efficiency and effectiveness. It's at this stage that the researcher should decide whether the sample is in fact to be the whole population and would therefore be a census.
This list should also facilitate access to the selected sampling units. A frame may also provide additional 'auxiliary information' about its elements; when this information is related to variables or groups of interest, it may be used to improve survey design. While not necessary for simple sampling, a sampling frame used for more advanced sample techniques, such as stratified sampling, may contain additional information (such as demographic information). For instance, an electoral register might include name and sex; this information can be used to ensure that a sample taken from that frame covers all demographic categories of interest. (Sometimes the auxiliary information is less explicit; for instance, a telephone number may provide some information about location.
Sampling frame qualities
An ideal sampling frame will have the following qualities:
all units have a logical, numerical identifier
all units can be found – their contact information, map location or other relevant information is present
the frame is organized in a logical, systematic fashion
the frame has additional information about the units that allow the use of more advanced sampling frames
every element of the population of interest is present in the frame
every element of the population is present only once in the frame
no elements from outside the population of interest are present in the frame
the data is 'up-to-date'
Types of sampling frames
The most straightforward type of frame is a list of elements of the population (preferably the entire population) with appropriate contact information. For example, in an opinion poll, possible sampling frames include an electoral register or a telephone directory. Other sampling frames can include employment records, school class lists, patient files in a hospital, organizations listed in a thematic database, and so on. On a more practical levels, sampling frames have the form of compute
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https://en.wikipedia.org/wiki/APV
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APV may refer to:
Actuarial present value, a probability weighted present value often used in insurance
Adjusted present value, a variation of the net present value (NPV)
Advanced Power Virtualization (renamed PowerVM), a software virtualization technique used by IBM
Alavuden Peli-Veikot, a multi-sport club in Alavus, Finland
Allen Parkway Village, a housing development in Fourth Ward, Houston
Apple Valley Airport (California), from its IATA airport code
Approach Procedure with Vertical guidance, a type of Instrument approach in aviation
APV (NMDAR antagonist), or AP5, a selective NMDA receptor antagonist
APV plc, a former company making process equipment
Asia Pacific Vision, a television content provider
Chevrolet Lumina APV, a minivan manufactured and marketed by General Motors
Suzuki APV, a microvan manufactured and marketed by Suzuki
Amazon Prime Video
Armored protected vehicle, a kind of armoured fighting vehicle
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https://en.wikipedia.org/wiki/Hypergeometric%20function
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In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and . There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.
History
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.
Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by .
Studies in the nineteenth century included those of , and the fundamental characterisation by of the hypergeometric function by means of the differential equation it satisfies.
Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.
The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).
The hypergeometric series
The hypergeometric function is defined for by the power series
It is undefined (or infinite) if equals a non-positive integer. Here is the (rising) Pochhammer symbol, which is defined by:
The series terminates if either or is a nonpositive integer, in which case the function reduces to a polynomial:
For complex arguments with it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line .
As , where is a non-negative integer, one has . Dividing by the value of the gamma function, we have the limit:
is the most common type of generalized hypergeometric series , and is often designated simply .
Differentiation formulas
Using the identity , it is shown that
and more generally,
Special cases
Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are
When a=1 and b=c, the series reduces into a plain geometric series, i.e.
hence, the name hypergeometric. This function can be considered as a generalization of the geometric series.
The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can
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https://en.wikipedia.org/wiki/Zhou%20Kexi
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Zhou Kexi (), born 1942, is a Chinese translator of French literature.
Biography
Zhou gained a degree in mathematics from Fudan University. He acquired the French language and became interested in French literature while studying at École Normale Supérieure in Paris. He became a full-time literary editor in the 1980s, and has since then translated several French novels, including Les trois mousquetaires, Madame Bovary, and La Voie royale. He is currently making a new translation of Marcel Proust's À la recherche du temps perdu. The first volume, Du côté de chez Swann, was published in 2004. He has also rendered Aventures mathématiques by Miguel de Guzmán into Chinese.
References
External links
A Conversation between Zhou and François Cheng
Fortunately not lost in translation
Volumes of Passion and Patience
French–Chinese translators
Living people
20th-century Chinese translators
21st-century Chinese translators
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Substructure%20%28mathematics%29
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In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure.
In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models.
In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are at most those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. Ordered groups, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure.
Definition
Given two structures A and B of the same signature σ, A is said to be a weak substructure of B, or a weak subalgebra of B, if
the domain of A is a subset of the domain of B,
f A = f B|An for every n-ary function symbol f in σ, and
R A R B An for every n-ary relation symbol R in σ.
A is said to be a substructure of B, or a subalgebra of B, if A is a weak subalgebra of B and, moreover,
R A = R B An for every n-ary relation symbol R in σ.
If A is a substructure of B, then B is called a superstructure of A or, especially if A is an induced substructure, an extension of A.
Example
In the language consisting of the binary functions + and ×, binary relation <, and constants 0 and 1, the structure (Q, +, ×, <, 0, 1) is a substructure of (R, +, ×, <, 0, 1). More generally, the substructures of an ordered field (or just a field) are precisely its subfields. Similarly, in the language (×, −1, 1) of groups, the substructures of a group are its subgroups. In the language (×, 1) of monoids, however, the substructures of a group are its submonoids. They need not be groups; and even if they are groups, they need not be subgroups.
In the case of graphs (in the signature consisting of one binary relation), subgraphs, and its weak substructures are precisely its subgraphs.
As subobjects
For every signature σ, induced substructures of σ-structures are the subobjects in the concrete category of σ-structures and strong homomorphisms (and also in the concrete category of σ-structures and σ-embeddings). Weak substructures of σ-structures are the subobjects in the concrete category of σ-structures and homomorphisms in the ordinary sense.
Submodel
In model theory, given a structure M which is a model of a theory T, a submodel of M in a n
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https://en.wikipedia.org/wiki/Pre-algebra
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Prealgebra is a common name for a course in middle school mathematics in the United States, usually taught in the 7th grade or 8th grade. The objective of it is to prepare students for the study of algebra. Usually, algebra is taught in the 8th and 9th grade.
As an intermediate stage after arithmetic, prealgebra helps students pass specific conceptual barriers. Students are introduced to the idea that an equals sign, rather than just being the answer to a question as in basic arithmetic, means that two sides are equivalent and can be manipulated together. They also learn how numbers, variables, and words can be used in the same ways.
Subjects
Subjects taught in a prealgebra course may include:
Review of natural number arithmetic
Types of numbers such as integers, fractions, decimals and negative numbers
Ratios and percents
Factorization of natural numbers
Properties of operations such as associativity and distributivity
Simple (integer) roots and powers
Rules of evaluation of expressions, such as operator precedence and use of parentheses
Basics of equations, including rules for invariant manipulation of equations
Understanding of variable manipulation
Manipulation and plotting in the standard 4-quadrant Cartesian coordinate plane
Powers in scientific notation (example: 340,000,000 in scientific notation is 3.4 × 108)
Identifying Probability
Solving Square roots
Pythagorean Theorem
Prealgebra may include subjects from geometry, especially to further the understanding of algebra in applications to area and volume.
Prealgebra may also include subjects from statistics to identify probability and interpret data.
Proficiency in prealgebra is an indicator of college success. It can also be taught as a remedial course for college students.
See also
Precalculus
Mathematics education in the United States
References
Elementary mathematics
Algebra education
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https://en.wikipedia.org/wiki/Academy%20for%20Mathematics%2C%20Science%2C%20and%20Engineering
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The Academy for Mathematics, Science, and Engineering (AMSE) is a four-year magnet public high school program intended to prepare students for STEM careers. Housed on the campus of Morris Hills High School in Rockaway, New Jersey, United States, it is a joint endeavor between the Morris County Vocational School District and the Morris Hills Regional District.
AMSE is one of 17 vocational academies under the Morris County Vocational School District, which administers the admissions process for prospective AMSE students. The program started in 2000 with an initial class size of 26, but in 2017, the class size was increased to 48 students.
As of the 2021–22 school year, the school had an enrollment of 180 students.
History
Background
As interest in traditional vocational subjects began to decrease in the 1990s, New Jersey's vocational school districts began to experiment with new programs that would cater to gifted students interested in careers in high technology and science. Hudson County's High Tech High School was founded in 1991, Bergen County Academies in 1992, and Union County Magnet School in 1997. Created as programs under New Jersey's Career and Technical Education legislation, the schools are overseen by the New Jersey Department of Education's Office of Career Readiness, which manages their standards, approval, and reapproval.
AMSE was first proposed to the Morris County Board of Chosen Freeholders (now the Board of County Commissioners) in November 1997 as the “Morris County Academy for Math, Science, and Engineering.” James DeWorken, Superintendent of the Morris County Vocational Board, asked the Freeholders for $5 million to build a new high-tech school on the campus of the County College of Morris in Randolph. Modeled after Monmouth County's High Technology High School, which had opened in 1991, the new program would open with 60 students, eventually expanding to 240, who would take up to 40 credits of college courses at the County College.
Although the County College supported the concept, the Freeholders raised concerns over the proposal's high cost, which, along with a number of other projects that had already been planned, would further accumulate debt and increase the burden on taxpayers. The Freeholders also noted worries that the program might simply be "a duplication of what [was] offered by other districts" and a lack of support from some district superintendents due to the increased competition that the new academy would bring.
DeWorken worked to gather support from district superintendents as well as local industry leaders and colleges. The Freeholders eventually approved a revised proposal in October 1999. Under the new plan, AMSE and three other academies were proposed again, but now as specialized programs within existing public high schools, effectively lowering their cost. The Superintendent of the Morris Hills Regional High School District at the time, James McNasby, helped bring AMSE to Morris Hills High Sc
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https://en.wikipedia.org/wiki/List%20of%20set%20theory%20topics
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This page is a list of articles related to set theory.
Articles on individual set theory topics
Lists related to set theory
Glossary of set theory
List of large cardinal properties
List of properties of sets of reals
List of set identities and relations
Set theorists
Societies and organizations
Association for Symbolic Logic
The Cabal
Topics
Set theory
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https://en.wikipedia.org/wiki/K-function
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In mathematics, the -function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function.
Definition
Formally, the -function is defined as
It can also be given in closed form as
where denotes the derivative of the Riemann zeta function, denotes the Hurwitz zeta function and
Another expression using the polygamma function is
Or using the balanced generalization of the polygamma function:
where is the Glaisher constant.
Similar to the Bohr-Mollerup Theorem for the gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation where is the forward difference operator.
Properties
It can be shown that for :
This can be shown by defining a function such that:
Differentiating this identity now with respect to yields:
Applying the logarithm rule we get
By the definition of the -function we write
And so
Setting we have
Now one can deduce the identity above.
The -function is closely related to the gamma function and the Barnes -function; for natural numbers , we have
More prosaically, one may write
The first values are
1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .
References
External links
Gamma and related functions
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https://en.wikipedia.org/wiki/Barnes%20G-function
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In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function.
Formally, the Barnes G-function is defined in the following Weierstrass product form:
where is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π denotes multiplication (capital pi notation).
The integral representation, which may be deduced from the relation to the double gamma function, is
As an entire function, G is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.
Functional equation and integer arguments
The Barnes G-function satisfies the functional equation
with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:
The functional equation implies that G takes the following values at integer arguments:
(in particular, )
and thus
where denotes the gamma function and K denotes the K-function. The functional equation uniquely defines the Barnes G-function if the convexity condition,
is added. Additionally, the Barnes G-function satisfies the duplication formula,
Characterisation
Similar to the Bohr-Mollerup theorem for the gamma function, for a constant , we have for
and for
as .
Value at 1/2
where is the Glaisher–Kinkelin constant.
Reflection formula 1.0
The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):
The logtangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:
The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation for the logcotangent integral, and using the fact that , an integration by parts gives
Performing the integral substitution gives
The Clausen function – of second order – has the integral representation
However, within the interval , the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent integral, the following relation clearly holds:
Thus, after a slight rearrangement of terms, the proof is complete:
Using the relation and dividing the reflection formula by a factor of gives the equivalent form:
Ref: see Adamchik below for an equivalent form of the reflection formula, but with a different proof.
Reflection formula 2.0
Replacing z with (1/2) − z'' in the previous reflection formula gives, after some simplification, the equivalen
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https://en.wikipedia.org/wiki/Municipality%20of%20the%20District%20of%20Barrington
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Barrington, officially named the Municipality of the District of Barrington, is a district municipality in western Shelburne County, Nova Scotia, Canada. Statistics Canada classifies the district municipality as a municipal district.
Geography
The Municipality of the District of Barrington forms the southernmost part of the province and contains Cape Sable, the eastern boundary between the North Atlantic Ocean and the Gulf of Maine.
Cape Sable Island is home to the tallest lighthouse in the Maritime Provinces. The Cape Light stands 101 feet tall, located on Canada's most southern saltwater beach, The Hawk Beach.
Acadia University owns Bon Portage Island in the municipality, which is protected by the Nova Scotia Nature Trust; there is a field biology research station present for students to study various local birds.
Etymology
The Mi'kmaq called the area "Ministiguish" or "Ministegkek", meaning "he has gone for it." The Acadians called the area "le Passage", meaning "the Passage". Barrington is named after William Barrington, 2nd Viscount Barrington.
History
French settlements
Cape Sable and Eel Bay, Nova Scotia were settled by the Acadians who migrated from Port Royal, Nova Scotia in 1620. The French governor of Acadia, Charles de la Tour, colonized Cap de Sable giving it the present name, meaning Sandy Cape. La Tour built up a strong post at Cap de Sable (present-day Port La Tour, Nova Scotia) beginning in 1623, called Fort Lomeron in honour of David Lomeron who was his agent in France. (The fur trading post called Fort Lomeron was later renamed Fort La Tour although – erroneously – identified as Fort Saint-Louis in the writings of Samuel de Champlain.) Here he carried on a sizable trade in furs with the Mi'kmaq and farmed the land.
During the Anglo-French War (1627–1629), under Charles I, by 1629 the Kirkes took Quebec City, Sir James Stewart of Killeith, Lord Ochiltree planted a colony on Cape Breton Island at Baleine, and Alexander’s son, William Alexander, 1st Earl of Stirling established the first incarnation of “New Scotland” at Port Royal, Nova Scotia. This set of British triumphs in what had otherwise been a disastrous war was not destined to last. King Charles’ haste to make peace with France on the terms most beneficial to him meant that the new North American gains would be bargained away in the Treaty of Saint-Germain-en-Laye (1632). There were three battles in Nova Scotia during the colonization of Scots: one at Saint John; another battle at Balene, Cape Breton; and one on Cape Sable (Port La Tour).
Siege of 1630
In 1627, as a result of these Scottish victories, Cape Sable was the only major French holding in North America. There was a battle between Charles and his father at Fort St. Louis (See National Historic Site - Fort St. Louis), the latter supporting the Scottish who had taken Port Royal, Nova Scotia. The battle lasted two days. Claude was forced to withdraw in humiliation to Port Royal.
As a result, La Tour a
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https://en.wikipedia.org/wiki/Victor%20Wickerhauser
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Mladen Victor Wickerhauser was born in Zagreb, SR Croatia, in 1959. He is a graduate of the California Institute of Technology and Yale University.
He is currently a professor of Mathematics and of Biomedical Engineering at Washington University in St. Louis. He has six U.S. patents and more than 100 publications. One of these, "Entropy-based Algorithms for Best Basis Selection," led to the Wavelet Scalar Quantization (WSQ) image compression algorithm, used by the FBI to encode fingerprint images.
Wickerhauser has been a member of the American Mathematical Society and the Society for Industrial and Applied Mathematics and has received the 2002 Wavelet Pioneer Award from SPIE (The International Society for Optical Engineering).
He is of Austrian descent.
Selected works
Adapted Wavelet Analysis from Theory to Software (A K Peters, 1994)
Mathematics for Multimedia (Elsevier 2003, ) (Birkhaeuser 2009, )
Introducing Financial Mathematics: Theory, Binomial Models, and Applications (Chapman and Hall/CRC 2023)
References
External links
M. Victor Wickerhauser
"Entropy-based Algorithms for Best Basis Selection"
U.S. Patent No. 5,384,725
U.S. Patent No. 5,526,299
U.S. Patent No. 6,792,073
U.S. Patent No. 7,054,454
U.S. Patent No. 7,333,619
U.S. Patent No. 8,500,644
1959 births
Living people
20th-century American mathematicians
21st-century American mathematicians
California Institute of Technology alumni
Yale University alumni
Washington University in St. Louis faculty
Washington University in St. Louis mathematicians
Yugoslav emigrants to the United States
Scientists from Zagreb
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https://en.wikipedia.org/wiki/StatSoft
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StatSoft is the original developer of Statistica. Dell acquired it in March 2014. Statistica is an analytics software portfolio that provides enterprise and desktop software for statistics, data analysis, data management, data visualization, data mining, which is also called predictive analytics, and quality control.
Company history
StatSoft Inc. was established in 1984 as a partnership of a group of university professors and scientists. Its first products had menu-driven libraries of flexible statistical procedures and ran on microcomputer platforms such as Apple II, CP/M, Commodore, and MS-DOS.
With the release of Statistica 9 in May 2009, both 32-bit and 64-bit native versions became available. Its current product suite, Statistica 12, was released in May 2013. Statistica is used worldwide at major corporations, government agencies, and universities.
On March 24, 2014, StatSoft was acquired by Dell in an effort to bolster Dell's ‘big data’ offering. StatSoft's CEO at the time of the Dell acquisition was Paul Lewicki.
On June 20, 2016, Dell sold Dell Software Group (which included StatSoft) to private equity firm Francisco Partners and Elliott Management.
On May 15, 2017, Quest Software sold Statistica to TIBCO Software.
StatSoft's product lines
Statistica Enterprise allows connections to data repositories and interactive filtering of data, contains analysis and report templates, and allows for management of security and permissions.
Web-based Applications this system makes the functionality of any of the Statistica products available via a Web browser.
Data Mining a collection of data mining and machine learning algorithms that include: support vector machines, EM and k-means clustering, classification & regression trees, generalized additive models, independent component analysis, stochastic gradient boosted trees, ensembles of neural networks, automatic feature selection, MARSplines, CHAID trees, nearest neighbor methods, association rules, and random forests.
Statistica Desktop designed for deployment on a single workstation. Spreadsheets, configurations and macros are all stored on the User's local workstation as a stand-alone application. Includes general purpose statistical, graphical, and analytic data management procedures.
StatSoft’s services
StatSoft's professional services groups provided a range of services to complement the Statistica software: software integration and customization services, the development of custom Web applications based on Statistica Enterprise Server technology, as well as the installation of a general-purpose Web Server system. StatSoft also offered deployment of data mining solutions designed to work with specific data warehouses and solve particular ranges of problems. Additionally, statistical consulting services were available. StatSoft offered both introductory and advanced training courses in major cities in the United States and overseas.
Technical Services provided software validation servic
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https://en.wikipedia.org/wiki/Symplectic%20integrator
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In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.
Introduction
Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read
where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
The set of position and momentum coordinates are called canonical coordinates.
(See Hamiltonian mechanics for more background.)
The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form . A numerical scheme is a symplectic integrator if it also conserves this 2-form.
Symplectic integrators also might possess, as a conserved quantity, a Hamiltonian which is slightly perturbed from the original one (only true for a small class of simple cases). By virtue of these advantages, the SI scheme has been widely applied to the calculations of long-term evolution of chaotic Hamiltonian systems ranging from the Kepler problem to the classical and semi-classical simulations in molecular dynamics.
Most of the usual numerical methods, like the primitive Euler scheme and the classical Runge–Kutta scheme, are not symplectic integrators.
Methods for constructing symplectic algorithms
Splitting methods for separable Hamiltonians
A widely used class of symplectic integrators is formed by the splitting methods.
Assume that the Hamiltonian is separable, meaning that it can be written in the form
This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy.
For the notational simplicity, let us introduce the symbol to denote the canonical coordinates
including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be expressed in a single expression as
where is a Poisson bracket. Furthermore, by introducing an operator , which returns a Poisson bracket of the operand with the Hamiltonian, the expression of the Hamilton's equation can be further simplified to
The formal solution of this set of equations is given as a matrix exponential:
Note the positivity of in the matrix exponential.
When the Hamiltonian has the form of equation (), the solution () is equivalent to
The SI scheme approximates the time-evolution operator in the formal solution () by a product of operators as
where and are real numbers, is an integer, which is called the order of the integrator, and where . Note that each of the operators and provides a symplectic map, so their product appearing in the right-hand side of () also constitutes a symplectic map.
Since for all , we can conclude that
By using a Taylor series,
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https://en.wikipedia.org/wiki/Bruce%20Price
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Bruce Price (December 12, 1845 – May 29, 1903) was an American architect and an innovator in the Shingle Style. The stark geometry and compact massing of his cottages in Tuxedo Park, New York, influenced Modernist architects, including Frank Lloyd Wright and Robert Venturi.
He also designed Richardsonian Romanesque institutional buildings, Beaux-Arts mansions, and Manhattan skyscrapers. In Canada, he designed Châteauesque railroad stations and grand hotels for the Canadian Pacific Railway, including Windsor Station in Montreal and Château Frontenac in Quebec City.
Life and career
Price was born in Cumberland, Maryland, the son of William and Marian Bruce Price. He studied for a short time at Princeton University. After four years of internship in the office of the Baltimore architects Niernsee & Neilson (1864–68), he began his professional work in Baltimore with Ephraim Francis Baldwin as a partner. Following a brief study trip to Europe, he opened an office in Wilkes-Barre, Pennsylvania, where he practiced from 1873 to 1876.
He settled in New York City in 1877, where he worked on a series of domestic projects. These culminated in the design and layout of the exclusive 7,000-acre planned community of Tuxedo Park (1885–86), created by Pierre Lorillard IV. The striking buildings Price designed there, with their severe geometry, compact massing and axial plans, were highly influential in the architectural profession. Eight of Price's houses – including five from Tuxedo Park – were among the one hundred buildings selected for George William Sheldon's landmark survey of American domestic architecture: Artistic Country-Seats (1886–87). The most famous of these, the Pierre Lorillard V cottage ("Cottage G"), though demolished and now known only through photographs, remains an icon of American architecture. Price's daughter wrote in 1911:
"In beginning Tuxedo, the architect's idea was to fit buildings with the surrounding woods, and the gate-lodge and keep were built of graystone with as much moss and lichen as possible. The shingled cottages were stained with the color of the woods—russets and grays and dull reds—ugly to the taste of a quarter century later, though this treatment did much to neutralize the newness of the buildings—Old World and tradition-haunted as it looks, it is new, incredibly new."
Among the Manhattan office buildings he designed were the American Surety Building, the St. James Building, the Bank of the Metropolis and the International Bank. He also collaborated with sculptor Daniel Chester French on the Richard Morris Hunt Memorial (1898) in Central Park. He designed a lecture hall and a dormitory at Yale University. His grandest residential commission was Georgian Court, the neo-Georgian estate of George Jay Gould I in Lakewood, New Jersey.
Price invented, patented, and built the parlor bay-window cars for the Pennsylvania Railroad and the Boston and Albany Railroad. This work prompted the Canadian Pacific Railways to consid
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https://en.wikipedia.org/wiki/Claude%20Mydorge
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Claude Mydorge (1585 – July 1647) was a French mathematician. His primary contributions were in geometry and physics.
Mydorge served on a scientific committee (whose members included Pierre Hérigone and Étienne Pascal) set up to determine whether Jean-Baptiste Morin's scheme for determining longitude from the Moon's motion was practical.
Works
External links
1585 births
1647 deaths
17th-century French mathematicians
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https://en.wikipedia.org/wiki/1997%20NBA%20draft
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The 1997 NBA draft took place on June 25, 1997, at Charlotte Coliseum in Charlotte, North Carolina. The Vancouver Grizzlies had the highest probability to win the NBA draft lottery, but since they were an expansion team along with the Toronto Raptors they were not allowed to select first in this draft. Although the Boston Celtics had the second-worst record in the 1996–97 season and the best odds (36 percent) of winning the lottery with two picks, the Spurs lost David Robinson and Sean Elliott to injury early in the season, finished with the third-worst record, and subsequently won the lottery. Leading up to the draft, there was no doubt that Tim Duncan would be selected at No. 1 by the Spurs as he was considered to be far and away the best prospect. After Duncan, the rest of the draft was regarded with some skepticism. The Celtics had the third and sixth picks, selecting Chauncey Billups and Ron Mercer, both of whom were traded in the next two years.
The Washington Wizards forfeited their 1997 first-round pick in connection with the signing of Juwan Howard. (Washington would have had the 17th pick.) Thus, the draft only had 28 first-round selections and 57 selections overall.
Draft selections
Notable undrafted players
These players eligible for the 1997 NBA Draft were not selected but played in the NBA.
Early entrants
College underclassmen
The following college basketball players successfully applied for early draft entrance.
Gracen Averil – G, Texas Tech (junior)
Tony Battie – F/C, Texas Tech (junior)
Chauncey Billups – G, Colorado (sophomore)
Carl Blanton – F, Sinclair CC (junior)
Mark Blount – C/F, Pittsburgh (sophomore)
C. J. Bruton – G, Indian Hills CC (sophomore)
Dan Buie – F, Washburn (junior)
James Cotton – G, Long Beach State (junior)
Tony Doyle – F, Columbia (junior)
Ian Folmar – F, Slippery Rock (junior)
Danny Fortson – F, Cincinnati (junior)
Adonal Foyle – C/F, Colgate (junior)
Darryl Hardy – F, Winston–Salem State (junior)
Antjonne Holmes – F, Central Baptist (freshman)
Troy Hudson – G, Southern Illinois (junior)
Marc Jackson – F/C, Temple
Stephen Jackson – F/G, Butler CC (freshman)
Ed Jenkins – F, Ohio State (junior)
Marcus Johnson – F, Long Beach State (junior)
Damon Jones – G, Houston (junior)
Nate Langley – G, George Mason (junior)
Keith Love – G, Rosary (junior)
Gordon Malone – F, West Virginia (junior)
Amere May – F, Shaw (junior)
Elgie McCoy – F, Kutztown (junior)
Ron Mercer – G/F, Kentucky (sophomore)
Victor Page – G, Georgetown (sophomore)
Shawn Ritzie – G, Norwalk CC (sophomore)
Paul Rogers – F/C, Gonzaga (junior)
Bryon Ruffner – F, BYU (junior)
Olivier Saint-Jean – San Jose State (junior)
Mark Sanford – F, Washington (junior)
God Shammgod – G, Providence (sophomore)
Maurice Taylor – F, Michigan (junior)
Tim Thomas – F, Villanova (freshman)
Mark Young – F, Kansas State (junior)
High school players
The following high school players successful
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https://en.wikipedia.org/wiki/Sticking%20probability
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The sticking probability is the probability that molecules are trapped on surfaces and adsorb chemically. From Langmuir's adsorption isotherm, molecules cannot adsorb on surfaces when the adsorption sites are already occupied by other molecules, so the sticking probability can be expressed as follows:
where is the initial sticking probability and is the surface coverage fraction ranging from 0 to 1.
Similarly, when molecules adsorb on surfaces dissociatively, the sticking probability is
The square is owing to the fact that a disassociation of 1 molecule into 2 parts requires 2 adsorption sites. These equations are simple and can be easily understood but cannot explain experimental results.
In 1958, P. Kisliuk presented an equation for the sticking probability that can explain experimental results. In his theory, molecules are trapped in precursor states of physisorption before chemisorption. Then the molecules meet adsorption sites that molecules can adsorb to chemically, so the molecules behave as follows.
If these sites are not occupied, molecules do the following (with probability in parentheses):
adsorb on the surface chemically ()
desorb from the surface ()
move to the next precursor state ()
and if these sites are occupied, they
desorb from the surface ()
move to the next precursor state ()
Note that an occupied site is defined as one where there is a chemically bonded adsorbate so by definition it would be . Then the sticking probability is, according to equation (6) of the reference,
When , this equation is identical in result to Langmuir's adsorption isotherm.
Notes
References
The constitution and fundamental properties of solids and liquids. part i. solids. Irving Langmuir; J. Am. Chem. Soc. 38, 2221-95 1916
Physical chemistry
Materials science
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https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz%20theorem
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In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh-Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.
Notations
Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary axis (i.e. the line Z = ic where i is the imaginary unit and c is a real number). Let us define (a polynomial of degree n) and (a nonzero polynomial of degree strictly less than n) by , respectively the real and imaginary parts of f on the imaginary line.
Furthermore, let us denote by:
p the number of roots of f in the left half-plane (taking into account multiplicities);
q the number of roots of f in the right half-plane (taking into account multiplicities);
the variation of the argument of f(iy) when y runs from −∞ to +∞;
w(x) is the number of variations of the generalized Sturm chain obtained from and by applying the Euclidean algorithm;
is the Cauchy index of the rational function r over the real line.
Statement
With the notations introduced above, the Routh–Hurwitz theorem states that:
From the first equality we can for instance conclude that when the variation of the argument of f(iy) is positive, then f(z) will have more roots to the left of the imaginary axis than to its right.
The equality p − q = w(+∞) − w(−∞) can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is p + q and the w from the right member is the number of variations of a Sturm chain (while w refers to a generalized Sturm chain in the present theorem).
Routh–Hurwitz stability criterion
We can easily determine a stability criterion using this theorem as it is trivial that f(z) is Hurwitz-stable iff p − q = n. We thus obtain conditions on the coefficients of f(z) by imposing w(+∞) = n and w(−∞) = 0.
See also
Plastic number#Geometry
References
Explaining the Routh–Hurwitz Criterion (2020)
External links
Mathworld entry
Eponymous theorems of physics
Theorems about polynomials
Theorems in complex analysis
Theorems in real analysis
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https://en.wikipedia.org/wiki/Pentation
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In mathematics, pentation (or hyper-5) is the next hyperoperation (infinite sequence of arithmetic operations) after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, means 2 to itself 2 times, or . This can then be reduced to
Etymology
The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration. It is part of his general naming scheme for hyperoperations.
Notation
There is little consensus on the notation for pentation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others.
Pentation can be written as a hyperoperation as . In this format, may be interpreted as the result of repeatedly applying the function , for repetitions, starting from the number 1. Analogously, , tetration, represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1, and the pentation represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. This will be the notation used in the rest of the article.
In Knuth's up-arrow notation, is represented as or . In this notation, represents the exponentiation function and represents tetration. The operation can be easily adapted for hexation by adding another arrow.
In Conway chained arrow notation, .
Another proposed notation is , though this is not extensible to higher hyperoperations.
Examples
The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if is defined by the Ackermann recurrence with the initial conditions and , then .
As tetration, its base operation, has not been extended to non-integer heights, pentation is currently only defined for integer values of a and b where a > 0 and b ≥ −2, and a few other integer values which may be uniquely defined. As with all hyperoperations of order 3 (exponentiation) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain:
Additionally, we can also define:
Other than the trivial cases shown above, pentation generates extremely large numbers very quickly such that there are only a few non-trivial cases that produce numbers that can be written in conventional notation, as illustrated below:
(shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note )
(a number with over 10153 digits)
(a number with more than 10102184 digits)
See also
Ackermann function
Large numb
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https://en.wikipedia.org/wiki/List%20of%20chaotic%20maps
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In mathematics, a chaotic map is a map (namely, an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because there are several different iterative procedures to generate the same fractal.
List of chaotic maps
List of fractals
Cantor set
de Rham curve
Gravity set, or Mitchell-Green gravity set
Julia set - derived from complex quadratic map
Koch snowflake - special case of de Rham curve
Lyapunov fractal
Mandelbrot set - derived from complex quadratic map
Menger sponge
Newton fractal
Nova fractal - derived from Newton fractal
Quaternionic fractal - three dimensional complex quadratic map
Sierpinski carpet
Sierpinski triangle
References
Chaotic maps
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https://en.wikipedia.org/wiki/Tent%20map
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In mathematics, the tent map with parameter μ is the real-valued function fμ defined by
the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ maps the unit interval [0, 1] into itself, thus defining a discrete-time dynamical system on it (equivalently, a recurrence relation). In particular, iterating a point x0 in [0, 1] gives rise to a sequence :
where μ is a positive real constant. Choosing for instance the parameter μ = 2, the effect of the function fμ may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval [0, 1/2] to get again the interval [0, 1]. Iterating the procedure, any point x0 of the interval assumes new subsequent positions as described above, generating a sequence xn in [0, 1].
The case of the tent map is a non-linear transformation of both the bit shift map and the r = 4 case of the logistic map.
Behaviour
The tent map with parameter μ = 2 and the logistic map with parameter r = 4 are topologically conjugate, and thus the behaviours of the two maps are in this sense identical under iteration.
Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic.
If μ is less than 1 the point x = 0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x = 0 from any initial value of x.
If μ is 1 all values of x less than or equal to 1/2 are fixed points of the system.
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get
If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ2/2 and μ/2 to themselves. This set of intervals is the Julia set of the map – that is, it is the smallest invariant subset of the real line under this map. If μ is greater than the square root of 2, these intervals merge, and the Julia set is the whole interval from μ − μ2/2 to μ/2 (see bifurcation diagram).
If μ is between 1 and 2 the interval [μ − μ2/2, μ/2] contains both periodic and non-periodic points, although all of the orbits are unstable (i.e. nearby points move away from the orbits rather than towards them). Orbits with longer lengths appear as μ increases. For example:
If μ equals 2 the system maps the interval [0, 1] onto itself. There are now periodic points with every orbit length within this interval, as well as non-periodic points. The periodic points are dense in [0, 1], so the map has become chaotic. In fact, the dynamics will be non-periodic if and only if is irrational. This can be seen by noting what the map does when is expressed in binary notation: It shifts the binary poi
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https://en.wikipedia.org/wiki/Costas%20array
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In mathematics, a Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n×n square tiling such that each row or column contains only one point, and all of the n(n − 1)/2 displacement vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto-ambiguity function, making the arrays useful in applications such as sonar and radar. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional Golomb ruler construction, and, as well as being of mathematical interest, have similar applications in experimental design and phased array radar engineering.
Costas arrays are named after John P. Costas, who first wrote about them in a 1965 technical report. Independently, Edgar Gilbert also wrote about them in the same year, publishing what is now known as the logarithmic Welch method of constructing Costas arrays.
The general enumeration of Costas arrays is an open problem in computer science and finding an algorithm that can solve it in polynomial time is an open research question.
Numerical representation
A Costas array may be represented numerically as an n×n array of numbers, where each entry is either 1, for a point, or 0, for the absence of a point. When interpreted as binary matrices, these arrays of numbers have the property that, since each row and column has the constraint that it only has one point on it, they are therefore also permutation matrices. Thus, the Costas arrays for any given n are a subset of the permutation matrices of order n.
Arrays are usually described as a series of indices specifying the column for any row. Since it is given that any column has only one point, it is possible to represent an array one-dimensionally. For instance, the following is a valid Costas array of order N = 4:
or simply
There are dots at coordinates: (1,2), (2,1), (3,3), (4,4)
Since the x-coordinate increases linearly, we can write this in shorthand as the set of all y-coordinates. The position in the set would then be the x-coordinate. Observe: {2,1,3,4} would describe the aforementioned array. This defines a permutation. This makes it easy to communicate the arrays for a given order of N.
Known arrays
Costas array counts are known for orders 1 through 29 :
Here are some known arrays:
N = 1
{1}
N = 2
{1,2} {2,1}
N = 3
{1,3,2} {2,1,3} {2,3,1} {3,1,2}
N = 4
{1,2,4,3} {1,3,4,2} {1,4,2,3} {2,1,3,4} {2,3,1,4} {2,4,3,1} {3,1,2,4} {3,2,4,1} {3,4,2,1} {4,1,3,2} {4,2,1,3} {4,3,1,2}
N = 5
{1,3,4,2,5} {1,4,2,3,5} {1,4,3,5,2} {1,4,5,3,2} {1,5,3,2,4} {1,5,4,2,3} {2,1,4,5,3} {2,1,5,3,4} {2,3,1,5,4} {2,3,5,1,4} {2,3,5,4,1} {2,4,1,5,3} {2,4,3,1,5} {2,5,1,3,4} {2,5,3,4,1} {2,5,4,1,3} {3,1,2,5,4} {3,1,4,5,2} {3,1,5,2,4} {3,2,4,5,1} {3,4,2,1,5} {3,5,1,4,2} {3,5,2,1,4} {3,5,4,1,2} {4,1,2,5,3} {4,1,3,2,5} {4,1,5,3,2} {4,2,3,5,1} {4,2,5,1,3} {4,3,1,2,5} {4,3,1,5,2} {4,3,5,1,2} {4,5,1,3,2} {4,5,2,1,3} {5,1,2,4,3} {5,1,3,4,2} {5,2,1,3,4} {5,2,3,1,4} {5,2,4,3,1}
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https://en.wikipedia.org/wiki/Stability%20radius
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In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
where denotes the nominal point, denotes the space of all possible values of the object , and the shaded area, , represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.
Abstract definition
The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful
where denotes a closed ball of radius in centered at .
History
It looks like the concept was invented in the early 1960s. In the 1980s it became popular in control theory and optimization. It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest.
Relation to Wald's maximin model
It was shown that the stability radius model is an instance of Wald's maximin model. That is,
where
The large penalty () is a device to force the player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.
Info-gap decision theory
Info-gap decision theory is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown that its robustness model, namely
is actually a stability radius model characterized by a simple stability requirement of the form where denotes the decision under consideration, denotes the parameter of interest, denotes the estimate of the true value of and denotes a ball of radius centered at .
Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the local robustness of decisions in the neighborhood of the estimate .
Sniedovich argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.
Alternate definition
There are cases where it is more convenient to define the stability radius slightly different. For example, in many applications in control theory the radius of stability is defined as the size of the smallest destabilizing perturbation in the nominal value of the parameter of interest. The picture is this:
More formally,
where denotes the distance of from .
Stability radius of functions
The stability radius of a continuous function f (in a functional space F) with respect to an open stability domain D is the distance between f and the set of unstable functions (with respect to D). We say that a function is stable with respect t
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https://en.wikipedia.org/wiki/Schwarz%20triangle
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In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group.
A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex. The value means the vertex angle is of the half-circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects.
Solution space
A fundamental domain triangle , with vertex angles , , and , can exist in different spaces depending on the value of the sum of the reciprocals of these integers:
This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to , while on a sphere they sum to an angle greater than , and on hyperbolic space they sum to less.
Graphical representation
A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.
Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden.
A Coxeter group can be used for a simpler notation, as (p q r) for cyclic graphs, and (p q 2) = [p,q] for (right triangles), and (p 2 2) = [p]×[].
A list of Schwarz triangles
Möbius triangles for the sphere
Schwarz triangles with whole numbers, also called Möbius triangles, include one 1-parameter family and three exceptional cases:
[p,2] or (p 2 2) – Dihedral symmetry,
[3,3] or (3 3 2) – Tetrahedral symmetry,
[4,3] or (4 3 2) – Octahedral symmetry,
[5,3] or (5 3 2) – Icosahedral symmetry,
Schwarz triangles for the sphere by density
The Schwarz triangles (p q r), grouped by density:
Triangles for the Euclidean plane
Density 1:
(3 3 3) – 60-60-60 (equilateral),
(4 4 2) – 45-45-90 (isosceles right),
(6 3 2) – 30-60-90,
Density 2:
(6 6 3/2) - 120-30-30 triangle
Density ∞:
(4 4/3 ∞)
(3 3/2 ∞)
(6 6/5 ∞)
Triangles for the hyperbolic plane
Density 1:
(2 3 7), (2 3 8), (2 3 9) ... (2 3 ∞)
(2 4 5), (2 4 6), (2 4 7) ... (2 4 ∞)
(2 5 5), (2 5 6), (2 5 7) ... (2 5 ∞)
(2 6 6), (2 6 7), (2 6 8) ... (2 6 ∞)
(3 3 4), (3 3 5), (3 3 6) ... (3 3 ∞)
(3 4 4), (3 4 5), (3 4 6) ... (3 4 ∞)
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https://en.wikipedia.org/wiki/Triangle%20group
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In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.
Definition
Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/l, 2π/m and 2π/n. Therefore, if the generating reflections are labeled a, b, c and the angles between them in the cyclic order are as given above, then the following relations hold:
It is a theorem that all other relations between a, b, c are consequences of these relations and that Δ(l,m,n) is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation
An abstract group with this presentation is a Coxeter group with three generators.
Classification
Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle. The sum of the angles of the triangle determines the type of the geometry by the Gauss–Bonnet theorem: it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π. Moreover, any two triangles with the given angles are congruent. Each triangle group determines a tiling, which is conventionally colored in two colors, so that any two adjacent tiles have opposite colors.
In terms of the numbers l, m, n > 1 there are the following possibilities.
The Euclidean case
The triangle group is the infinite symmetry group of a certain tessellation (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up to permutations, the triple (l, m, n) is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding triangle groups are instances of wallpaper groups.
The spherical case
The triangle group is the finite symmetry group of a tiling of a unit sphere by spherical triangles, or Möbius triangles, whose angles add up to a number greater than π. Up to permutations, the triple (l,m,n) has the form (2,3,3), (2,3,4), (2,3,5), or (2,2,n), n > 1. Spherical triangle groups can be identified with the symmetry groups of regular polyhedra in the three-dimensional Euclidean space: Δ(2,3,3) corresponds to the tetrahedron, Δ(2,3,4) to
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https://en.wikipedia.org/wiki/Lebesgue%27s%20lemma
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For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number lemma
In mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together with the operator norm of the projection.
Statement
Let be a normed vector space, a subspace of , and a linear projector on . Then for each in :
The proof is a one-line application of the triangle inequality: for any in , by writing as , it follows that
where the last inequality uses the fact that together with the definition of the operator norm .
See also
Lebesgue constant (interpolation)
References
Lemmas in analysis
Approximation theory
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https://en.wikipedia.org/wiki/Popular%20mathematics
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Popular mathematics is mathematical presentation aimed at a general audience. Sometimes this is in the form of books which require no mathematical background and in other cases it is in the form of expository articles written by professional mathematicians to reach out to others working in different areas.
Notable works of popular mathematics
Some of the most prolific popularisers of mathematics include Keith Devlin, Rintu Nath, Martin Gardner, and Ian Stewart. Titles by these three authors can be found on their respective pages.
On zero
On infinity
Rucker, Rudy (1982), Infinity and the Mind: The Science and Philosophy of the Infinite; Princeton, N.J.: Princeton University Press. .
On constants
On complex numbers
On the Riemann hypothesis
On recently solved problems
On classification of finite simple groups
On higher dimensions
Rucker, Rudy (1984), The Fourth Dimension: Toward a Geometry of Higher Reality; Houghton Mifflin Harcourt.
On introduction to mathematics for the general reader
Biographies
Magazines and journals
Popular science magazines such as New Scientist and Scientific American sometimes carry articles on mathematics.
Plus Magazine is a free online magazine run under the Millennium Mathematics Project at the University of Cambridge.
The journals listed below can be found in many university libraries.
American Mathematical Monthly is designed to be accessible to a wide audience.
The Mathematical Gazette contains letters, book reviews and expositions of attractive areas of mathematics.
Mathematics Magazine offers lively, readable, and appealing exposition on a wide range of mathematical topics.
The Mathematical Intelligencer is a mathematical journal that aims at a conversational and scholarly tone.
Notices of the AMS - Each issue contains one or two expository articles that describe current developments in mathematical research, written by professional mathematicians. The Notices also carries articles on the history of mathematics, mathematics education, and professional issues facing mathematicians, as well as reviews of books, plays, movies, and other artistic and cultural works involving mathematics.
Audio and video
Simon Singh's Fermat's Last Theorem is available in audio and there is also a Horizon television program.
3Blue1Brown, YouTube channel by Grant Sanderson.
Mathologer, YouTube channel by Burkard Polster.
Numberphile, YouTube channel by Brady Haran.
BetterExplained, YouTube channel and website by Kalid Azad.
Museums
Several museums aim at enhancing public understanding of mathematics:
In the United States:
Museum of Mathematics, New York, and its predecessor, the Goudreau Museum of Mathematics in Art and Science,
In Austria:
, Wien
In Germany:
Arithmeum, Bonn
Mathematisch-Physikalischer Salon, Dresden
Mathematikum, Gießen
, Frankfurt on Main
, Freiberg
, Oberwolfach
In Italy:
The Garden of Archimedes
References
Mathematics and culture
Mathematics literature
Recreational ma
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https://en.wikipedia.org/wiki/156%20%28number%29
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156 (one hundred [and] fifty-six) is the natural number, following 155 and preceding 157.
In mathematics
156 is an abundant number, a pronic number, a dodecagonal number, and a refactorable number.
156 is the number of graphs on 6 unlabeled nodes.
156 is a repdigit in base 5 (1111), and also in bases 25, 38, 51, 77, and 155.
156 degrees is the internal angle of a pentadecagon.
In the military
Convoy HX-156 was the 156th of the numbered series of World War II HX convoys of merchant ships from Halifax, Nova Scotia to Liverpool during World War II
The Fieseler Fi 156 Storch was a small German liaison aircraft during World War II
The
was a United States Navy T2 tanker during World War II
was a United States Navy cargo ship during World War II
was a United States Navy during World War II
was a United States Navy ship during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy fast civilian yacht during World War I
In music
156, a song by the Danish rock band Mew appearing in both their 2000 album Half the World Is Watching Me and their 2003 album Frengers.
NM 156, a 1984 song by the heavy metal band Queensrÿche from the album The Warning
156, a song by the Polish Black Metal band Blaze of Perdition from the 2010 album Towards the Blaze of Perdition
In transportation
The Alfa Romeo 156 car produced from 1997 to 2006.
The Ferrari 156 was a racecar made by Ferrari from 1961 to 1963.
The Ferrari 156/85 was a Formula One car in the 1985 Formula One season.
The Class 156 "Super Sprinter" DMU train.
The Midland Railway 156 Class, a 2-4-0 tender engine built in the United Kingdom between 1866 and 1874.
London Buses route 156.
Martin 156, known as the Russian clipper, was a large flying boat aircraft intended for transoceanic service.
In other fields
156 is also:
The year AD 156 or 156 BC
156 AH is a year in the Islamic calendar that corresponds to 772 – 773 CE
156 Xanthippe is a main belt asteroid with a dark surface
The number of hourly gongs a clock strikes in one day (78 AM gongs and 78 PM gongs)
The number of sons of Magbish in the Census of the men of Israel upon return from exile (Bible, Ezra 2:30)
United States DS-156 visa issued for U.S. Department of State Nonimmigrant Visa Application
The atomic number of an element temporarily called Unpenthexium.
The Indian Head No. 156, Saskatchewan rural municipality in the Canadian province of Saskatchewan
See also
List of highways numbered 156
United Nations Security Council Resolution 156
United States Supreme Court cases, Volume 156
Pennsylvania House of Representatives, District 156
References
External links
The Number 156
156th Street (3rd Avenue El)
Integers
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https://en.wikipedia.org/wiki/Heun%20function
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In mathematics, the local Heun function is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called a Heun function, denoted Hf, if it is also regular at z = 1, and is called a Heun polynomial, denoted Hp, if it is regular at all three finite singular points z = 0, 1, a.
Heun's equation
Heun's equation is a second-order linear ordinary differential equation (ODE) of the form
The condition is taken so that the characteristic exponents for the regular singularity at infinity are α and β (see below).
The complex number q is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β). Every second-order linear ODE on the extended complex plane with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation, can be transformed into this equation by a change of variable.
Coalescence of various regular singularities of the Heun equation into irregular singularities give rise to several confluent forms of the equation, as shown in the table below.
{| class="wikitable"
|+Forms of the Heun Equation
|-
! Form !! Singularities !! Equation
|-
| General
| 0, 1, a, ∞
|
|-
| Confluent
| 0, 1, ∞ (irregular, rank 1)
|
|-
| Doubly Confluent
| 0 (irregular, rank 1), ∞ (irregular, rank 1)
|
|-
| Biconfluent
| 0, ∞ (irregular, rank 2)
|
|-
| Triconfluent
| ∞ (irregular, rank 3)
|
|}
q-analog
The q-analog of Heun's equation has been discovered by and studied by .
Symmetries
Heun's equation has a group of symmetries of order 192, isomorphic to the Coxeter group of the Coxeter diagram D4, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer.
The symmetries fixing the local Heun function form a group of order 24 isomorphic to the symmetric group on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points. The complete list of 192 symmetries was given by using machine calculation. Several previous attempts by various authors to list these by hand contained many errors and omissions; for example, most of the 48 local solutions listed by Heun contain serious errors.
See also
Heine–Stieltjes polynomials, a generalization of Heun polynomials.
References
A. Erdélyi, F. Oberhettinger, W. Magnus and F. Tricomi Higher Transcendental functions vol. 3 (McGraw Hill, NY, 1953).
Hahn W.(1971) On linear geometric difference equations with accessory parameters.Funkcial. Ekvac., 14, 73–78
.
Ordinary differential equations
Special functions
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https://en.wikipedia.org/wiki/Lebesgue%20constant
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In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most and for the set of nodes is generally denoted by . These constants are named after Henri Lebesgue.
Definition
We fix the interpolation nodes and an interval containing all the interpolation nodes. The process of interpolation maps the function to a polynomial . This defines a mapping from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace of polynomials of degree or less.
The Lebesgue constant is defined as the operator norm of X. This definition requires us to specify a norm on C([a, b]). The uniform norm is usually the most convenient.
Properties
The Lebesgue constant bounds the interpolation error: let denote the best approximation of f among the polynomials of degree or less. In other words, minimizes among all p in Πn. Then
We will here prove this statement with the maximum norm.
by the triangle inequality. But X is a projection on Πn, so
.
This finishes the proof since . Note that this relation comes also as a special case of Lebesgue's lemma.
In other words, the interpolation polynomial is at most a factor worse than the best possible approximation. This suggests that we look for a set of interpolation nodes with a small Lebesgue constant.
The Lebesgue constant can be expressed in terms of the Lagrange basis polynomials:
In fact, we have the Lebesgue function
and the Lebesgue constant (or Lebesgue number) for the grid is its maximum value
Nevertheless, it is not easy to find an explicit expression for .
Minimal Lebesgue constants
In the case of equidistant nodes, the Lebesgue constant grows exponentially. More precisely, we have the following asymptotic estimate
On the other hand, the Lebesgue constant grows only logarithmically if Chebyshev nodes are used, since we have
We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation. However, there is an easy (linear) transformation of Chebyshev nodes that gives a better Lebesgue constant. Let denote the -th Chebyshev node. Then, define
For such nodes:
Those nodes are, however, not optimal (i.e. they do not minimize the Lebesgue constants) and the search for an optimal set of nodes (which has already been proved to be unique under some assumptions) is still an intriguing topic in mathematics today. However, this set of nodes is optimal for interpolation over the set of times differentiable functions whose -th derivatives are bounded in absolute values by a constant as shown by N. S. Hoang.
Using a computer, one can approximate the values of the minimal Lebesgue constants, here for the canonical interval :
{| class="w
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https://en.wikipedia.org/wiki/Heinz-Otto%20Kreiss
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Heinz-Otto Kreiss (14 September 1930 – 16 December 2015) was a German mathematician in the fields of numerical analysis, applied mathematics, and what was the new area of computing in the early 1960s. Born in Hamburg, Germany, he earned his Ph.D. at Kungliga Tekniska Högskolan in 1959. Over the course of his long career, Kreiss wrote a number of books in addition to the purely academic journal articles he authored across several disciplines. He was professor at Uppsala University, California Institute of Technology and University of California, Los Angeles (UCLA). He was also a member of the Royal Swedish Academy of Sciences. At the time of his death, Kreiss was a Swedish citizen, living in Stockholm. He died in Stockholm in 2015, aged 85.
Kreiss did research on the initial value problem for partial differential equations, numerical treatment of partial differential equations, difference equations, and applications to hydrodynamics and meteorology.
In 1974, he delivered a plenary lecture Initial Boundary Value Problems for Hyperbolic Partial Differential Equations at the International Congress of Mathematicians (ICM) in Vancouver. In 2002 he won the National Academy of Sciences Award in Numerical Analysis and Applied Mathematics. In 2003 he was the John von Neumann Lecturer of the Society for Industrial and Applied Mathematics (SIAM). He was elected a member of the American Academy of Arts and Sciences.
His doctoral students include Björn Engquist and Bertil Gustafsson. His daughter, Gunilla Kreiss, was a student of Engquist.
Selected publications
with Jens Lorenz: Initial-boundary value problems and the Navier-Stokes equations, Academic Press 1989, SIAM 2004
with Hedwig Ulmer Busenhart: Time-dependent partial differential equations and their numerical solution, Birkhäuser 2001
with Bertil Gustafsson, Joseph Oliger: time-dependent problems and difference methods, Wiley 1995
References
External links
Literature by and about Heinz-Otto Kreiss, Katalog der Deutschen Nationalbibliothek
20th-century German mathematicians
KTH Royal Institute of Technology alumni
California Institute of Technology faculty
University of California, Los Angeles faculty
Members of the Royal Swedish Academy of Sciences
1930 births
2015 deaths
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https://en.wikipedia.org/wiki/Q-theta%20function
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In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series.
It is given by
where one takes 0 ≤ |q| < 1. It obeys the identities
It may also be expressed as:
where is the q-Pochhammer symbol.
See also
elliptic hypergeometric series
Jacobi theta function
Ramanujan theta function
References
Q-analogs
Theta functions
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https://en.wikipedia.org/wiki/Elliptic%20gamma%20function
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In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , and can be expressed in terms of the triple gamma function. It is given by
It obeys several identities:
and
where θ is the q-theta function.
When , it essentially reduces to the infinite q-Pochhammer symbol:
Multiplication Formula
Define
Then the following formula holds with ().
References
Gamma and related functions
Q-analogs
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https://en.wikipedia.org/wiki/Lattice%20group
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In mathematics, the term lattice group is used for two distinct notions:
a lattice (group), a discrete subgroup of Rn and its generalizations
a lattice ordered group, a group that with a partial ordering that is a lattice order
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https://en.wikipedia.org/wiki/Pretopological%20space
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In general topology, a pretopological space is a generalization of the concept of topological space.
A pretopological space can be defined in terms of either filters or a preclosure operator.
The similar, but more abstract, notion of a Grothendieck pretopology is used to form a Grothendieck topology, and is covered in the article on that topic.
Let be a set. A neighborhood system for a pretopology on is a collection of filters one for each element of such that every set in contains as a member. Each element of is called a neighborhood of A pretopological space is then a set equipped with such a neighborhood system.
A net converges to a point in if is eventually in every neighborhood of
A pretopological space can also be defined as a set with a preclosure operator (Čech closure operator) The two definitions can be shown to be equivalent as follows: define the closure of a set in to be the set of all points such that some net that converges to is eventually in Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set be a neighborhood of if is not in the closure of the complement of The set of all such neighborhoods can be shown to be a neighborhood system for a pretopology.
A pretopological space is a topological space when its closure operator is idempotent.
A map between two pretopological spaces is continuous if it satisfies for all subsets
See also
References
E. Čech, Topological Spaces, John Wiley and Sons, 1966.
D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, Kluwer Academic Publishers, 1995.
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic, Springer Verlag, 1992.
External links
Recombination Spaces, Metrics, and Pretopologies B.M.R. Stadler, P.F. Stadler, M. Shpak., and G.P. Wagner. (See in particular Appendix A.)
Closed sets and closures in Pretopology M. Dalud-Vincent, M. Brissaud, and M Lamure. 2009 .
General topology
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https://en.wikipedia.org/wiki/Picard%E2%80%93Fuchs%20equation
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In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.
Definition
Let
be the j-invariant with and the modular invariants of the elliptic curve in Weierstrass form:
Note that the j-invariant is an isomorphism from the Riemann surface to the Riemann sphere ; where is the upper half-plane and is the modular group. The Picard–Fuchs equation is then
Written in Q-form, one has
Solutions
This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane. However, the ratio of two solutions of the hypergeometric equation is also known as a Schwarz triangle map.
The Picard–Fuchs equation can be cast into the form of Riemann's differential equation, and thus solutions can be directly read off in terms of Riemann P-functions. One has
At least four methods to find the j-function inverse can be given.
Dedekind defines the j-function by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the fundamental domain:
where (Sƒ)(x) is the Schwarzian derivative of ƒ with respect to x.
Generalization
In algebraic geometry, this equation has been shown to be a very special case of a general phenomenon, the Gauss–Manin connection.
References
Pedagogical
J. Harnad and J. McKay, Modular solutions to equations of generalized Halphen type, Proc. R. Soc. Lond. A 456 (2000), 261–294,
References
J. Harnad, Picard–Fuchs Equations, Hauptmoduls and Integrable Systems, Chapter 8 (Pgs. 137–152) of Integrability: The Seiberg–Witten and Witham Equation (Eds. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000)). arXiv:solv-int/9902013
For a detailed proof of the Picard-Fuchs equation:
Elliptic functions
Modular forms
Hypergeometric functions
Ordinary differential equations
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https://en.wikipedia.org/wiki/Modular%20invariant
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In mathematics, a modular invariant may be
A modular invariant of a group acting on a vector space of positive characteristic
The elliptic modular function, giving the modular invariant of an elliptic curve.
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https://en.wikipedia.org/wiki/Riemann%27s%20differential%20equation
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In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and . The equation is also known as the Papperitz equation.
The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and . That equation admits two linearly independent solutions; near a singularity , the solutions take the form , where is a local variable, and is locally holomorphic with . The real number is called the exponent of the solution at . Let α, β and γ be the exponents of one solution at 0, 1 and respectively; and let α', β' and γ' be those of the other. Then
By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying Möbius transformations will adjust the positions of the regular singular points, while other transformations (see below) can change the exponents at the regular singular points, subject to the exponents adding up to 1.
Definition
The differential equation is given by
The regular singular points are , , and . The exponents of the solutions at these regular singular points are, respectively, , , and . As before, the exponents are subject to the condition
Solutions and relationship with the hypergeometric function
The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol)
The standard hypergeometric function may be expressed as
The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is
In other words, one may write the solutions in terms of the hypergeometric function as
The full complement of Kummer's 24 solutions may be obtained in this way; see the article hypergeometric differential equation for a treatment of Kummer's solutions.
Fractional linear transformations
The P-function possesses a simple symmetry under the action of fractional linear transformations known as Möbius transformations (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group . Given arbitrary complex numbers , , , such that , define the quantities
and
then one has the simple relation
expressing the symmetry.
Exponents
If the Moebius transformation above moves the singular points but does not change the exponents,
the following transformation does not move the singular points but changes the exponents:
See also
Method of Frobenius
Monodromy
Notes
References
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
Chapter 15 Hypergeometric Functions
Section 15.6 Riemann's Differential Equation
Hypergeometric functions
Ordinary differential equations
Bernhard Riemann
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https://en.wikipedia.org/wiki/Willmore%20energy
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In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore.
Definition
Expressed symbolically, the Willmore energy of S is:
where is the mean curvature, is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic of the surface, so
which is a topological invariant and thus independent of the particular embedding in that was chosen. Thus the Willmore energy can be expressed as
An alternative, but equivalent, formula is
where and are the principal curvatures of the surface.
Properties
The Willmore energy is always greater than or equal to zero. A round sphere has zero Willmore energy.
The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the calculus of variations, and one can vary the embedding of a surface, while leaving it topologically unaltered.
Critical points
A basic problem in the calculus of variations is to find the critical points and minima of a functional.
For a given topological space, this is equivalent to finding the critical points of the function
since the Euler characteristic is constant.
One can find (local) minima for the Willmore energy by gradient descent, which in this context is called Willmore flow.
For embeddings of the sphere in 3-space, the critical points have been classified: they are all conformal transforms of minimal surfaces, the round sphere is the minimum, and all other critical values are integers greater than 4. They are called Willmore surfaces.
Willmore flow
The Willmore flow is the geometric flow corresponding to the Willmore energy;
it is an -gradient flow.
where H stands for the mean curvature of the manifold .
Flow lines satisfy the differential equation:
where is a point belonging to the surface.
This flow leads to an evolution problem in differential geometry: the surface is evolving
in time to follow variations of steepest descent of the energy. Like surface diffusion it is a fourth-order
flow, since the variation of the energy contains fourth derivatives.
Applications
Cell membranes tend to position themselves so as to minimize Willmore energy.
Willmore energy is used in constructing a class of optimal sphere eversions, the minimax eversions.
See also
Willmore conjecture
Notes
References
.
Geometric flow
Differential geometry
Surfaces
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https://en.wikipedia.org/wiki/Principal%20curvature
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In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.
Discussion
At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p, denoted k1 and k2, are the maximum and minimum values of this curvature.
Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k1 does not equal k2, a result of Euler (1760), and are called principal directions. From a modern perspective, this theorem follows from the spectral theorem because these directions are as the principal axes of a symmetric tensor—the second fundamental form. A systematic analysis of the principal curvatures and principal directions was undertaken by Gaston Darboux, using Darboux frames.
The product k1k2 of the two principal curvatures is the Gaussian curvature, K, and the average (k1 + k2)/2 is the mean curvature, H.
If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every point.
Formal definition
Let M be a surface in Euclidean space with second fundamental form . Fix a point p ∈ M, and an orthonormal basis X1, X2 of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix
If X1 and X2 are selected so that the matrix is a diagonal matrix, then they are called the principal directions. If the surface is oriented, then one often requires that the pair (X1, X2) be positively oriented with respect to the given orientation.
Without reference to a particular orthonormal basis, the principal curvatures are the eigenvalues of the shape operator, and the principal directions are its eigenvectors.
Generalizations
For hypersurfaces in higher-dimensional Euclidean spaces, the principal curvatures may be defined in a directly analogous fashion. The principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space. The principal directions are the corresponding eigenvectors.
Similarly, if M is a hypersurface in a Riemannian manifold N,
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https://en.wikipedia.org/wiki/Category%20of%20small%20categories
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In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms.
The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism.
The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories.
Free category
The category Cat has a forgetful functor U into the quiver category Quiv:
U : Cat → Quiv
This functor forgets the identity morphisms of a given category, and it forgets morphism compositions. The left adjoint of this functor is a functor F taking Quiv to the corresponding free categories:
F : Quiv → Cat
1-Categorical properties
Cat has all small limits and colimits.
Cat is a Cartesian closed category, with exponential given by the functor category .
Cat is not locally Cartesian closed.
Cat is locally finitely presentable.
See also
Nerve of a category
Universal set, the notion of a 'set of all sets'
References
External links
Small categories
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https://en.wikipedia.org/wiki/Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.
Motivating examples
Circle
After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1):
Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle:
Together, these parts cover the whole circle, and the four charts form an atlas for the circle.
The top and right charts, and respectively, overlap in their domain: their intersection lies in the quarter of the circle where both and -coordinates are positive. Both map this part into the interval , though differently. Thus a function can be constructed, which takes values from the co-domain of back to the circle using the inverse, followed by back to the interval. If a is any number in , then:
Such a function is called a transition map.
The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts
and
Here s is the sl
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https://en.wikipedia.org/wiki/Glaisher%E2%80%93Kinkelin%20constant
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In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those involving gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
Its approximate value is:
= ... .
The Glaisher–Kinkelin constant can be given by the limit:
where is the hyperfactorial. This formula displays a similarity between and which is perhaps best illustrated by noting Stirling's formula:
which shows that just as is obtained from approximation of the factorials, can also be obtained from a similar approximation to the hyperfactorials.
An equivalent definition for involving the Barnes -function, given by where is the gamma function is:
.
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
where is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
An alternative product formula, defined over the prime numbers, reads
where denotes the th prime number.
The following are some integrals that involve this constant:
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
References
(Provides a variety of relationships.)
External links
The Glaisher–Kinkelin constant to 20,000 decimal places
Mathematical constants
Number theory
Glaisher family
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https://en.wikipedia.org/wiki/Ernst%20Gottfried%20Fischer
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Ernst Gottfried Fischer (17 July 1754 – 27 January 1831) was a German chemist. He was born in Hoheneiche near Saalfeld. After studying theology and mathematics at the University of Halle, he was a teacher in Berlin before becoming Professor of Physics in 1810. He translated Claude Berthollet's publication Recherches sur les lois de l'affinitié in 1802. He proposed a system of equivalents based on sulfuric acid equal to one hundred.
Stoichiometry contribution
Jeremias Benjamin Richter's work had little impact until 1802, when it was summarized by Fischer in terms of tables, such as the one below.
According to this table, it takes 615 parts by weight of magnesia to neutralize either 1000 parts by weight of sulfuric acid or 1405 parts by weight of nitric acid. In the early literature on the subject, these weights were referred to as combining weights.
Works
References
1754 births
1831 deaths
19th-century German chemists
18th-century German chemists
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https://en.wikipedia.org/wiki/Holyhedron
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In mathematics, a holyhedron is a type of 3-dimensional geometric body: a polyhedron each of whose faces contains at least one polygon-shaped hole, and whose holes' boundaries share no point with each other or the face's boundary.
The concept was first introduced by John H. Conway; the term "holyhedron" was coined by David W. Wilson in 1997 as a pun involving polyhedra and holes. Conway also offered a prize of 10,000 USD, divided by the number of faces, for finding an example, asking:
Is there a polyhedron in Euclidean three-dimensional space that has only finitely many plane faces, each of which is a closed connected subset of the appropriate plane whose relative interior in that plane is multiply connected?
No actual holyhedron was constructed until 1999, when Jade P. Vinson presented an example of a holyhedron with a total of 78,585,627 faces; another example was subsequently given by Don Hatch, who presented a holyhedron with 492 faces in 2003, worth about 20.33 USD prize money.
References
External links
Don Hatch's 492-face holyhedron
Polyhedra
John Horton Conway
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https://en.wikipedia.org/wiki/COT
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A cot is a camp bed or infant bed.
Cot or COT may also refer to:
In arts and entertainment
Chicago Opera Theater, an opera company
In mathematics, science, and technology
Car of Tomorrow, a car design used in NASCAR racing
Cost of transport, an energy calculation
Cottage developed from the word cot, which can be seen in various forms in other languages meaning a tent / hut e.g. Goahti and Kohte
Cotangent, a trigonometric function, written as "cot"
Cyclooctatetraene, an unsaturated hydrocarbon
Finger cot, a hygienic cover for a single finger
Chain-of-thought prompting, a method of engineering language model prompts
In government and military use
Colombian Time, the time zone used in Colombia (UTC−05:00)
Comando de Operações Táticas, a Brazilian counter-terrorism force
Commitments of Traders Report, US market report
Committee on Toxicity of Chemicals in Food, Consumer Products and the Environment, in the UK
RAF Cottesmore Flying Training Unit, United Kingdom (ICAO airline designator)
People
Cot (surname)
Cot Deal (1923–2013), American baseball pitcher and coach
Other uses
Coatesville station, Amtrak station code
Cot Valley, Cornwall, England
Cottingley railway station, National Rail station code
Malbec grapes, known in the Loire Valley as Côt
Club Olympique des Transports, a football club based in Tunis, Tunisia
See also
COTS (disambiguation)
C0t analysis, a biochemical technique
Khat, a drug
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https://en.wikipedia.org/wiki/Baire%20space%20%28set%20theory%29
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In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation.
The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers.
The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.
Topology and trees
The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, here characterized as:
If any finite set of natural number coordinates I={i} is selected, and for each i a particular natural number value vi is selected, then the set of all infinite sequences of natural numbers that have value vi at position i is a basic open set. Every open set is a countable union of a collection of these.
Using more formal notation, one can define the individual cylinders as
for a fixed integer location n and integer value v. The cylinders are then the generators for the cylinder sets: the cylinder sets then consist of all intersections of a finite number of cylinders. That is, given any finite set of natural number coordinates and corresponding natural number values for each , one considers the intersection of cylinders
This intersection is called a cylinder set, and the set of all such cylinder sets provides a basis for the product topology. Every open set is a countable union of such cylinder sets.
By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:
If a sequence of natural numbers {wi : i < n} is selected, then the set of all infinite sequences of natural numbers that have value wi at position i for all i < n is a basic open set. Every open set is a countable union of a collection of these.
Thus a basic open set in the Baire space is the set of all infinite sequences of natural numbers extending a common finite initial segment τ. This leads to a representation of the Baire space as the set of all infinite paths passing through the full tree ω<ω of finite sequences of natural numbers ordered by extension. Each finite initial segment is a node of the tree of finite sequences. Each open set is determined by a (possibly infinite) union of nodes of that tree. A point in Baire space is in an open set if and only if its path goes through one of the nodes in its determining union.
The representation of the Baire space as paths through a tree also gives a characterization of closed sets. E
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