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https://en.wikipedia.org/wiki/Michael%20Artin
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Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology Mathematics Department, known for his contributions to algebraic geometry.
Life and career
Michael Artin or Artinian of Armenian origin was born in Hamburg, Germany, and brought up in Indiana. His parents were Natalia Naumovna Jasny (Natascha) and Emil Artin, preeminent algebraist of the 20th century of Armenian descent. Artin's parents left Germany in 1937, because his mother's father was Jewish. His elder sister is , who was married to mathematician John Tate until the late 1980s.
Artin did his undergraduate studies at Princeton University, receiving an A.B. in 1955; he then moved to Harvard University, where he received a Ph.D. in 1960 under the supervision of Oscar Zariski, defending a thesis about Enriques surfaces.
In the early 1960s, Artin spent time at the IHÉS in France, contributing to the SGA4 volumes of the Séminaire de géométrie algébrique, on topos theory and étale cohomology, jointly with Alexander Grothendieck.
He also collaborated with Barry Mazur to define étale homotopy theory which has become an important tool in algebraic geometry, and applied ideas from algebraic geometry (such as the Nash approximation) to the study of diffeomorphisms of compact manifolds. His work on the problem of characterising the representable functors in the category of schemes has led to the Artin approximation theorem in local algebra as well as the "Existence theorem". This work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. He also has made important contributions to the deformation theory of algebraic varieties, serving as the basis for all future work in this area of algebraic geometry. With Peter Swinnerton-Dyer, he provided a resolution of the Shafarevich-Tate conjecture for elliptic K3 surfaces and the pencil of elliptic curves over finite fields. He contributed to the theory of surface singularities which are both fundamental and seminal. The rational singularity and fundamental cycles, which are used in matroid theory, are such examples of his sheer originality and thinking. He began to turn his interest from algebraic geometry to noncommutative algebra (noncommutative ring theory), especially geometric aspects, after a talk by Shimshon Amitsur and an encounter in University of Chicago with Claudio Procesi and Lance W. Small, "which prompted [his] first foray into ring theory".
Today, he is a recognized world authority in noncommutative algebraic geometry and his impact can be felt across many related areas.
In 2002, Artin won the American Mathematical Society's annual Steele Prize for Lifetime Achievement. In 2005, he was awarded the Harvard Centennial Medal. In 2013, he won the Wolf Prize in Mathematics, and in 2015 was awarded the National Medal of Science from the President Barack Obama. He is also a member of the Natio
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https://en.wikipedia.org/wiki/Collectively%20exhaustive%20events
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In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collectively exhaustive, because they encompass the entire range of possible outcomes.
Another way to describe collectively exhaustive events is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if
where S is the sample space.
Compare this to the concept of a set of mutually exclusive events. In such a set no more than one event can occur at a given time. (In some forms of mutual exclusion only one event can ever occur.) The set of all possible die rolls is both mutually exclusive and collectively exhaustive (i.e., "MECE"). The events 1 and 6 are mutually exclusive but not collectively exhaustive. The events "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are also collectively exhaustive but not mutually exclusive. In some forms of mutual exclusion only one event can ever occur, whether collectively exhaustive or not. For example, tossing a particular biscuit for a group of several dogs cannot be repeated, no matter which dog snaps it up.
One example of an event that is both collectively exhaustive and mutually exclusive is tossing a coin. The outcome must be either heads or tails, or p (heads or tails) = 1, so the outcomes are collectively exhaustive. When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive.
Another example of events being collectively exhaustive and mutually exclusive at same time are, event "even" (2,4 or 6) and event "odd" (1,3 or 5) in a random experiment of rolling a six-sided die. These both events are mutually exclusive because even and odd outcome can never occur at same time. The union of both "even" and "odd" events give sample space of rolling the die, hence are collectively exhaustive.
History
The term "exhaustive" has been used in the literature since at least 1914. Here are a few examples:
The following appears as a footnote on page 23 of Couturat's text, The Algebra of Logic (1914):
"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse)." (italics added for emphasis)
In Stephen Kleene's discussion of cardinal numbers, in Introduction to Metamathematics (1952), he uses the term "mutual
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https://en.wikipedia.org/wiki/Nonelementary%20integral
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In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.
Examples
Examples of functions with nonelementary antiderivatives include:
(elliptic integral)
(logarithmic integral)
(error function, Gaussian integral)
and (Fresnel integral)
(sine integral, Dirichlet integral)
(exponential integral)
(in terms of the exponential integral)
(in terms of the logarithmic integral)
(incomplete gamma function); for the antiderivative can be written in terms of the exponential integral; for in terms of the error function; for any positive integer, the antiderivative elementary.
Some common non-elementary antiderivative functions are given names, defining so-called special functions, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.
Properties
Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series.
Even if it is not possible to evaluate an indefinite integral (antiderivative) in elementary terms, one can always approximate a corresponding definite integral by numerical integration. There are also cases where there is no elementary antiderivative, but specific definite integrals (often improper integrals over unbounded intervals) can be evaluated in elementary terms: most famously the Gaussian integral
The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.
See also
References
Integration of Nonelementary Functions, S.O.S MATHematics.com; accessed 7 Dec 2012.
Further reading
Williams, Dana P., NONELEMENTARY ANTIDERIVATIVES, 1 Dec 1993. Accessed January 24, 2014.
Integral calculus
Integrals
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https://en.wikipedia.org/wiki/Jean-Pierre%20Bourguignon
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Jean-Pierre Bourguignon (born 21 July 1947) is a French mathematician, working in the field of differential geometry.
Biography
Born in Lyon, he studied at École Polytechnique in Palaiseau, graduating in 1969. For his graduate studies he went to Paris Diderot University, where he obtained his PhD in 1974 under the direction of Marcel Berger.
He was president of the Société Mathématique de France from 1990 to 1992. From 1995 to 1998, he was president of the European Mathematical Society. He was director of the Institut des Hautes Études Scientifiques near Paris from 1994 to 2013. Between 1 January 2014 and 31 December 2019 he was the President of the European Research Council.
Selected publications
Articles
with H. Blaine Lawson and James Simons:
with H. Blaine Lawson:
with Jean-Pierre Ezin:
Books
with Oussama Hijazi, Jean-Louis Milhorat, Andrei Moroianu and Sergiu Moroianu:
as editor with Rolf Jeltsch, Alberto Adrego Pinto, and Marcelo Viana:
References
External links
An Interview with Jean-Pierre Bourguignon
1947 births
Living people
Scientists from Lyon
20th-century French mathematicians
21st-century French mathematicians
École Polytechnique alumni
Paris Diderot University alumni
Academic staff of École Polytechnique
Academic staff of the University of Paris
Differential geometers
Members of Academia Europaea
Presidents of the European Mathematical Society
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https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood%20circle%20method
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In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines. Hundreds of papers followed, and the method still yields results. The method is the subject of a monograph by R. C. Vaughan.
Outline
The goal is to prove asymptotic behavior of a series: to show that for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle.
The circle method is specifically how to compute these residues, by partitioning the circle into minor arcs (the bulk of the circle) and major arcs (small arcs containing the most significant singularities), and then bounding the behavior on the minor arcs. The key insight is that, in many cases of interest (such as theta functions), the singularities occur at the roots of unity, and the significance of the singularities is in the order of the Farey sequence. Thus one can investigate the most significant singularities, and, if fortunate, compute the integrals.
Setup
The circle in question was initially the unit circle in the complex plane. Assuming the problem had first been formulated in the terms that for a sequence of complex numbers for , we want some asymptotic information of the type , where we have some heuristic reason to guess the form taken by (an ansatz), we write
a power series generating function. The interesting cases are where is then of radius of convergence equal to 1, and we suppose that the problem as posed has been modified to present this situation.
Residues
From that formulation, it follows directly from the residue theorem that
for integers , where is a circle of radius and centred at 0, for any with ; in other words, is a contour integral, integrated over the circle described traversed once anticlockwise. We would like to take directly, that is, to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of may not be defined there.
Singularities on unit circle
The problem addressed by the circle method is to force the issue of taking , by a good understanding of the nature of the singularities f exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational number
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https://en.wikipedia.org/wiki/Superquadrics
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In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs.
The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation.
Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoids and the supertoroids. In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images and point clouds are covered in several computer vision literatures. Useful tools and algorithms for superquadrics visualization, sampling, and recovery are open-sourced here.
Formulas
Implicit equation
The surface of the basic superquadric is given by
where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely:
less than 1: a pointy octahedron modified to have concave faces and sharp edges.
exactly 1: a regular octahedron.
between 1 and 2: an octahedron modified to have convex faces, blunt edges and blunt corners.
exactly 2: a sphere
greater than 2: a cube modified to have rounded edges and corners.
infinite (in the limit): a cube
Each exponent can be varied independently to obtain combined shapes. For example, if r=s=2, and t=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) r = s.
If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called super-hyperboloids.
The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts A, B, C along each axis. Its general equation is
Parametric description
Parametric equations in terms of surface parameters u and v (equivalent to longitude and latitude if m equals 2) are
where the auxiliary functions are
and the sign function sgn(x) is
Spherical product
Barr introduces the spherical product which given two plane curves produces a 3D surface. If
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https://en.wikipedia.org/wiki/Jet%20%28mathematics%29
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In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.
This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces. It concludes with a description of jets between manifolds, and how these jets can be constructed intrinsically. In this more general context, it summarizes some of the applications of jets to differential geometry and the theory of differential equations.
Jets of functions between Euclidean spaces
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
One-dimensional case
Suppose that is a real-valued function having at least k + 1 derivatives in a neighborhood U of the point . Then by Taylor's theorem,
where
Then the k-jet of f at the point is defined to be the polynomial
Jets are normally regarded as abstract polynomials in the variable z, not as actual polynomial functions in that variable. In other words, z is an indeterminate variable allowing one to perform various algebraic operations among the jets. It is in fact the base-point from which jets derive their functional dependency. Thus, by varying the base-point, a jet yields a polynomial of order at most k at every point. This marks an important conceptual distinction between jets and truncated Taylor series: ordinarily a Taylor series is regarded as depending functionally on its variable, rather than its base-point. Jets, on the other hand, separate the algebraic properties of Taylor series from their functional properties. We shall deal with the reasons and applications of this separation later in the article.
Mappings from one Euclidean space to another
Suppose that is a function from one Euclidean space to another having at least (k + 1) derivatives. In this case, Taylor's theorem asserts that
The k-jet of f is then defined to be the polynomial
in , where .
Algebraic properties of jets
There are two basic algebraic structures jets can carry. The first is a product structure, although this ultimately turns out to be the least important. The second is the structure of the composition of jets.
If are a pair of real-valued functions, then we can define the product of their jets via
Here we have suppressed the indeterminate z, since it is understood that jets are formal polynomials. This product is just the product of ordinary polynomials in z, modulo . In other words, it is multiplication in the ring , where is the ideal generated by polynomials homogeneous of order ≥ k + 1.
We now move to the composition of jets. To avoid unnecessary technicalities, we consider jets
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https://en.wikipedia.org/wiki/Ptolemy%27s%20theorem
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In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.
If the vertices of the cyclic quadrilateral are A, B, C, and D in order, then the theorem states that:
This relation may be verbally expressed as follows:
If a quadrilateral is cyclic then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.
Moreover, the converse of Ptolemy's theorem is also true:
In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a cyclic quadrilateral.
Corollaries on Inscribed Polygons
Equilateral triangle
Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle.
Given An equilateral triangle inscribed on a circle and a point on the circle.
The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
Proof: Follows immediately from Ptolemy's theorem:
Square
Any square can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to then the length of the diagonal is equal to according to the Pythagorean theorem, and Ptolemy's relation obviously holds.
Rectangle
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d2, the right hand side of Ptolemy's relation is the sum a2 + b2.
Copernicus – who used Ptolemy's theorem extensively in his trigonometrical work – refers to this result as a 'Porism' or self-evident corollary:
Furthermore it is clear (manifestum est) that when the chord subtending an arc has been given, that chord too can be found which subtends the rest of the semicircle.
Pentagon
A more interesting example is the relation between the length a of the side and the (common) length b of the 5 chords in a regular pentagon. By completing the square, the relation yields the golden ratio:
Side of decagon
If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter d as one of its diagonals:
where is the golden ratio.
whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras's theorem applied t
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https://en.wikipedia.org/wiki/Schur%27s%20theorem
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In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.
Ramsey theory
In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers x, y, z with
For every positive integer c, S(c) denotes the smallest number S such that for every partition of the integers into c parts, one of the parts contains integers x, y, and z with . Schur's theorem ensures that S(c) is well-defined for every positive integer c. The numbers of the form S(c) are called Schur's number.
Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose nonempty sums belong to the same part.
Using this definition, the only known Schur numbers are S(n) 2, 5, 14, 45, and 161 () The proof that was announced in 2017 and took up 2 petabytes of space.
Combinatorics
In combinatorics, Schur's theorem tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of relatively prime numbers. In particular, if is a set of integers such that , the number of different tuples of non-negative integer numbers such that when goes to infinity is:
As a result, for every set of relatively prime numbers there exists a value of such that every larger number is representable as a linear combination of in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See Coin problem.)
Differential geometry
In differential geometry, Schur's theorem compares the distance between the endpoints of a space curve to the distance between the endpoints of a corresponding plane curve of less curvature.
Suppose is a plane curve with curvature which makes a convex curve when closed by the chord connecting its endpoints, and is a curve of the same length with curvature . Let denote the distance between the endpoints of and denote the distance between the endpoints of . If then .
Schur's theorem is usually stated for curves, but John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).
Linear algebra
In linear algebra, Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues.
Functional analysis
In functional analysis and the study of Banach spaces, Schur's theorem, due to I. Schur, often refers to Schur's property, that for certain spa
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https://en.wikipedia.org/wiki/Education%20in%20North%20Korea
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Education in North Korea is universal and state-funded schooling by the government. As of 2021, UNESCO Institute for Statistics does not report any data for North Korea's literacy rates. Some children go through one year of kindergarten, four years of primary education, six years of secondary education, and then on to university. The North Korean state claims its national literacy rate for citizens aged 15 and older is 100 percent.
In 1988, the United Nations Educational, Scientific, and Cultural Organization (UNESCO) reported that North Korea had 35,000 preprimary, 60,000 primary, 111,000 secondary, 23,000 college and university, and 4,000 other postsecondary teachers.
History
Formal education has played a central role in the social and cultural development of both traditional Korea and contemporary North Korea. During the Joseon Dynasty, the royal court established a system of schools that taught Confucian subjects in the provinces as well as in four central secondary schools in the capital. There was no state-supported system of primary education.
During the 15th century, state-supported schools declined in quality and were supplanted in importance by private academies, the seowon, centers of a Neo-Confucian revival in the 16th century. Higher education was provided by the Seonggyungwan, the Confucian national university, in Seoul. Its enrollment was limited to 200 students who had passed the lower civil-service examinations and were preparing for the highest examinations.
The late 19th and early 20th centuries saw major educational changes. The seewan were abolished by the central government. Christian missionaries established modern schools that taught Western curricula. Among them was the first school for women, Ehwa Woman's University, established by American Methodist missionaries as a primary school in Seoul in 1886. During the last years of the dynasty, as many as 3,000 private schools that taught modern subjects to both sexes were founded by missionaries and others.
After Japan annexed Korea in 1910, the colonial regime established an educational system with two goals: to give Koreans a minimal education designed to train them for roles in a modern economy and make them loyal subjects of the Japanese emperor; and to provide a higher quality education for Japanese expatriates who had settled in large numbers on the Korean Peninsula.
The Japanese invested more resources in the latter, and opportunities for Koreans were severely limited. A state university modeled on Tokyo Imperial University was established in Seoul in 1923, but the number of Koreans allowed to study there never exceeded 40 percent of its enrollment; the rest of its students were Japanese. Private universities, including those established by missionaries such as Sungsil College in Pyongyang and Chosun Christian College in Seoul, provided other opportunities for Koreans desiring higher education.
After the establishment of North Korea, an education system modeled
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https://en.wikipedia.org/wiki/Taenidia
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Taenidia (singular: taenidium) are circumferential thickenings of the cuticle inside a trachea or tracheole in an insect's respiratory system. The geometry of the Taenidiae varies across different orders of insects and even throughout the tracheae in an individual organism. Taenidia generally take the form of either hoop or spiral thickenings of the tracheal cuticle.
References
Mill, P.J., Tracheae and Tracheoles Microscopic Anatomy of Invertebrates, 11A, pp. 303–336, 1998.
Insect anatomy
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https://en.wikipedia.org/wiki/Chinese%20mathematics
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Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (base 2 and base 10), algebra, geometry, number theory and trigonometry.
Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like continued fractions are widely used and have been well-documented ever since. They deliberately find the principal nth root of positive numbers and the roots of equations. The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tiān yuán shù.
As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as the Song dynasty Chinese polymath Shen Kuo.
Early Chinese mathematics
Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching, which greatly influenced written literature during the Zhou dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers.
Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods. Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry (only Chinese geometry) and the usage of decimals.
Math was one of th
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https://en.wikipedia.org/wiki/Generalized%20canonical%20correlation
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In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.
Applications
The Helmert-Wolf blocking (HWB) method of estimating linear regression parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.
References
Afshin-Pour, B.; Hossein-Zadeh, G.A. Strother, S.C.; Soltanian-Zadeh, H. (2012), "Enhancing reproducibility of fMRI statistical maps using generalized canonical correlation analysis in NPAIRS framework", NeuroImage 60(4): 1970–1981.
Sun, Q.S., Liu, Z.D., Heng, P.A., Xia, D.S. (2005) "A Theorem on the Generalized Canonical Projective Vectors". Pattern Recognition 38 (3) 449
Kettenring, J. R. (1971) "Canonical analysis of several sets of variables". "Biometrika" 58 (3) 433
External links
FactoMineR (free exploratory multivariate data analysis software linked to R)
Covariance and correlation
Dimension reduction
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https://en.wikipedia.org/wiki/Oklahoma%20School%20of%20Science%20and%20Mathematics
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The Oklahoma School of Science and Mathematics (OSSM) is a two-year, public residential high school located in Oklahoma City, Oklahoma. Established by the Oklahoma state legislature in 1983, the school was designed to educate academically gifted high school juniors and seniors in advanced mathematics and science. OSSM opened doors to its inaugural class in 1990. It is a member of the National Consortium of Secondary STEM Schools (NCSSS).
History
Dr. Earl Mitchell is credited as the originator of the idea of starting OSSM. He was reportedly inspired by a letter about the North Carolina School of Science and Mathematics (NCSSM), written by North Carolina governor Jim Hunt. In 1982, Dr. Mitchell travelled to NCSSM to study their practices, and enlisted Speaker Dan Draper, Representative Penny Williams, and Senator Bernice Shedrick to help bring the idea to fruition.
OSSM was established by HB 1286 in 1983, during the 39th Oklahoma Legislature. The bill's principal authors included Representative Penny Williams, Senator Bernice Shedrick, and Senator Rodger Randle. The bill was signed into law by Governor George Nigh on June 23, 1983.
In 1988, Dr. Edna Manning was appointed the first president of OSSM. Manning aided in the building and development of the institution, supervising the selection of faculty and the development of the curriculum.
When OSSM's inaugural class was accepted in 1990, the school did not have its own campus yet. Students were temporarily housed in OU's Cross Center dormitory in Norman, and took daily shuttle buses to the OU Health Sciences Center (OUHSC) campus in Oklahoma City for their classes.
In 1992, OSSM moved into the newly-renovated Lincoln Elementary School, across the street from the OUHSC. During Dr. Manning's tenure as president, the school's campus grew to include a dormitory, a gymnasium, a library, and a science building.
In 2006, Chesapeake Energy gifted OSSM $500,000 to fund an endowed faculty chair in geophysics, the first endowed chair at an Oklahoma public high school.
In June 2012, Dr. Manning retired from her position as president, and was succeeded by Dr. Frank Y.H. Wang. During his tenure, Dr. Wang increased contributions to the OSSM Foundation Faculty Endowment from $4.2 million to $10 million.
On May 31, 2013, the 54th Oklahoma Legislature passed SB 1131, authored by Senator Clark Jolley, and signed into law by Governor Mary Fallin. The bill allows OSSM to accept out-of-state students for up to 10 percent of the student population and charge them tuition. It also allows the school to rent out its facilities, and create summer programs & workshops for which tuition and fees could be charged. Dr. Wang helped create the bill in an effort to generate a new revenue stream, to make up for money lost to severe cuts in state funding.
In June 2021, Dr. Wang announced he will retire at the end of the 2021-22 school year.
General
Admittance to OSSM is conducted through a highly selective application proces
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https://en.wikipedia.org/wiki/Implicit%20surface
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In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for or or .
The graph of a function is usually described by an equation and is called an explicit representation. The third essential description of a surface is the parametric one:
, where the -, - and -coordinates of surface points are represented by three functions depending on common parameters . Generally the change of representations is simple only when the explicit representation is given: (implicit), (parametric).
Examples:
The plane
The sphere
The torus
A surface of genus 2: (see diagram).
The surface of revolution (see diagram wineglass).
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example.
The implicit function theorem describes conditions under which an equation can be solved (at least implicitly) for , or . But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
If is polynomial in , and , the surface is called algebraic. Example 5 is non-algebraic.
Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
Formulas
Throughout the following considerations the implicit surface is represented by an equation
where function meets the necessary conditions of differentiability. The partial derivatives of
are .
Tangent plane and normal vector
A surface point is called regular if and only if the gradient of at is not the zero vector , meaning
.
If the surface point is not regular, it is called singular.
The equation of the tangent plane at a regular point is
and a normal vector is
Normal curvature
In order to keep the formula simple the arguments are omitted:
is the normal curvature of the surface at a regular point for the unit tangent direction . is the Hessian matrix of (matrix of the second derivatives).
The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of a parametric surface.
Applications of implicit surfaces
As in the case of implicit curves it is an easy task to generate implicit surfaces with desired shapes by applying algebraic operations (addition, multiplication) on simple primitives.
Equipotential surface of point charges
The electrical potential of a point charge at point generates at point the potential (omitting physical constants)
The equipotential surface for the potential value is the implicit surface which is a sphere with center at point .
The pot
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https://en.wikipedia.org/wiki/Omitted-variable%20bias
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In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included.
More specifically, OVB is the bias that appears in the estimates of parameters in a regression analysis, when the assumed specification is incorrect in that it omits an independent variable that is a determinant of the dependent variable and correlated with one or more of the included independent variables.
In linear regression
Intuition
Suppose the true cause-and-effect relationship is given by:
with parameters a, b, c, dependent variable y, independent variables x and z, and error term u. We wish to know the effect of x itself upon y (that is, we wish to obtain an estimate of b).
Two conditions must hold true for omitted-variable bias to exist in linear regression:
the omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient must not be zero); and
the omitted variable must be correlated with an independent variable specified in the regression (i.e., cov(z,x) must not equal zero).
Suppose we omit z from the regression, and suppose the relation between x and z is given by
with parameters d, f and error term e. Substituting the second equation into the first gives
If a regression of y is conducted upon x only, this last equation is what is estimated, and the regression coefficient on x is actually an estimate of (b + cf ), giving not simply an estimate of the desired direct effect of x upon y (which is b), but rather of its sum with the indirect effect (the effect f of x on z times the effect c of z on y). Thus by omitting the variable z from the regression, we have estimated the total derivative of y with respect to x rather than its partial derivative with respect to x. These differ if both c and f are non-zero.
The direction and extent of the bias are both contained in cf, since the effect sought is b but the regression estimates b+cf. The extent of the bias is the absolute value of cf, and the direction of bias is upward (toward a more positive or less negative value) if cf > 0 (if the direction of correlation between y and z is the same as that between x and z), and it is downward otherwise.
Detailed analysis
As an example, consider a linear model of the form
where
xi is a 1 × p row vector of values of p independent variables observed at time i or for the i th study participant;
β is a p × 1 column vector of unobservable parameters (the response coefficients of the dependent variable to each of the p independent variables in xi) to be estimated;
zi is a scalar and is the value of another independent variable that is observed at time i or for the i th study participant;
δ is a scalar and is an unobservable parameter (the response coefficient of the dependent variable to zi) to be estimated;
ui is the unobservable error term occurring at time i or for the
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https://en.wikipedia.org/wiki/Totally%20bounded%20space
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In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
In metric spaces
A metric space is totally bounded if and only if for every real number , there exists a finite collection of open balls of radius whose centers lie in M and whose union contains . Equivalently, the metric space M is totally bounded if and only if for every , there exists a finite cover such that the radius of each element of the cover is at most . This is equivalent to the existence of a finite ε-net. A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded.
Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded). The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded: every discrete ball of radius or less is a singleton, and no finite union of singletons can cover an infinite set.
Uniform (topological) spaces
A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a uniform structure. A subset of a uniform space is totally bounded if and only if, for any entourage , there exists a finite cover of by subsets of each of whose Cartesian squares is a subset of . (In other words, replaces the "size" , and a subset is of size if its Cartesian square is a subset of .)
The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact.
Examples and elementary properties
Every compact set is totally bounded, whenever the concept is defined.
Every totally bounded set is bounded.
A subset of the real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded.
The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded (in the norm topology) if and only if the space has finite dimension.
Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem.
A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.
The closure of a totally bounded subset is agai
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https://en.wikipedia.org/wiki/Aliquot
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Aliquot () may refer to:
Mathematics
Aliquot part, a proper divisor of an integer
Aliquot sum, the sum of the aliquot parts of an integer
Aliquot sequence, a sequence of integers in which each number is the aliquot sum of the previous number
Music
Aliquot stringing, in stringed instruments, the use of strings which are not struck to make a note, but which resonate sympathetically with struck notes
Aliquot stop, an organ stop that adds harmonics or overtones instead of the primary pitch
Sciences
Aliquot of a sample, in chemistry and other sciences, a precise portion of a sample or total amount of a liquid (e.g. precisely 25 mL of water taken from 250 mL)
Aliquot in pharmaceutics, a method of measuring ingredients below the sensitivity of a scale by proportional dilution with inactive known ingredients
Genome aliquoting, the problem of reconstructing an ancestral genome from the genomes of polyploid descendants
Other uses
Aliquot part, in the US Public Land Survey System, a subdivision of a section based upon an even division by distances along the edges and not by equal area
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https://en.wikipedia.org/wiki/Translation%20operator
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Translation operator can refer to these things:
Translation operator (quantum mechanics)
Shift operator, which effects a geometric translation
Translation (geometry)
Displacement operator in quantum optics
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https://en.wikipedia.org/wiki/Neighbourhood%20%28mathematics%29
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In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.
Definitions
Neighbourhood of a point
If is a topological space and is a point in then a of is a subset of that includes an open set containing ,
This is also equivalent to the point belonging to the topological interior of in
The neighbourhood need be an open subset of but when is open in then it is called an . Some authors have been known to require neighbourhoods to be open, so it is important to note conventions.
A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle.
The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
Neighbourhood of a set
If is a subset of a topological space , then a neighbourhood of is a set that includes an open set containing ,It follows that a set is a neighbourhood of if and only if it is a neighbourhood of all the points in Furthermore, is a neighbourhood of if and only if is a subset of the interior of
A neighbourhood of that is also an open subset of is called an of
The neighbourhood of a point is just a special case of this definition.
In a metric space
In a metric space a set is a neighbourhood of a point if there exists an open ball with center and radius such that
is contained in
is called uniform neighbourhood of a set if there exists a positive number such that for all elements of
is contained in
Under the same condition, for the -neighbourhood of a set is the set of all points in that are at distance less than from (or equivalently, is the union of all the open balls of radius that are centered at a point in ):
It directly follows that an -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an -neighbourhood for some value of
Examples
Given the set of real numbers with the usual Euclidean metric and a subset defined as
then is a neighbourhood for the set of natural numbers, but is a uniform neighbourhood of this set.
Topology from neighbourhoods
The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.
A neighbourhood system on is the assignment of a filter of su
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https://en.wikipedia.org/wiki/Uniform%20property
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In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space that is invariant under uniform isomorphisms.
Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphisms, every topological property of a uniform space is also a uniform property. This article is (mostly) concerned with uniform properties that are not topological properties.
Uniform properties
Separated. A uniform space X is separated if the intersection of all entourages is equal to the diagonal in X × X. This is actually just a topological property, and equivalent to the condition that the underlying topological space is Hausdorff (or simply T0 since every uniform space is completely regular).
Complete. A uniform space X is complete if every Cauchy net in X converges (i.e. has a limit point in X).
Totally bounded (or Precompact). A uniform space X is totally bounded if for each entourage E ⊂ X × X there is a finite cover {Ui} of X such that Ui × Ui is contained in E for all i. Equivalently, X is totally bounded if for each entourage E there exists a finite subset {xi} of X such that X is the union of all E[xi]. In terms of uniform covers, X is totally bounded if every uniform cover has a finite subcover.
Compact. A uniform space is compact if it is complete and totally bounded. Despite the definition given here, compactness is a topological property and so admits a purely topological description (every open cover has a finite subcover).
Uniformly connected. A uniform space X is uniformly connected if every uniformly continuous function from X to a discrete uniform space is constant.
Uniformly disconnected. A uniform space X is uniformly disconnected if it is not uniformly connected.
See also
Topological property
References
Uniform spaces
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https://en.wikipedia.org/wiki/Split%20exact%20sequence
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In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
Equivalent characterizations
A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category
is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones:
The requirement that the sequence is isomorphic means that there is an isomorphism such that the composite is the natural inclusion and such that the composite equals b. This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences.
Examples
A trivial example of a split short exact sequence is
where are R-modules, is the canonical injection and is the canonical projection.
Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis.
The exact sequence (where the first map is multiplication by 2) is not split exact.
Related notions
Pure exact sequences can be characterized as the filtered colimits of split exact sequences.
References
Sources
Abstract algebra
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https://en.wikipedia.org/wiki/Homeomorphism%20group
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In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups.
Properties and examples
There is a natural group action of the homeomorphism group of a space on that space. Let be a topological space and denote the homeomorphism group of by . The action is defined as follows:
This is a group action since for all ,
where denotes the group action, and the identity element of (which is the identity function on ) sends points to themselves. If this action is transitive, then the space is said to be homogeneous.
Topology
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology.
In the case of regular, locally compact spaces the group multiplication is then continuous.
If the space is compact and Hausdorff, the inversion is continuous as well and becomes a topological group.
If is Hausdorff, locally compact and locally connected this holds as well.
However there are locally compact separable metric spaces for which the inversion map is not continuous and therefore not a topological group.
In the category of topological spaces with homeomorphisms, group objects are exactly homeomorphism groups.
Mapping class group
In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:
The MCG can also be interpreted as the 0th homotopy group, .
This yields the short exact sequence:
In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.
See also
Mapping class group
References
Group theory
Topology
Topological groups
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https://en.wikipedia.org/wiki/Wilhelm%20Killing
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Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
Life
Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. In 1875, he married Anna Commer, who was the daughter of a music lecturer. He became a professor at the seminary college Collegium Hosianum in Braunsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town council. As a professor and administrator Killing was widely liked and respected. Finally, in 1892 he became a professor at the University of Münster.
In 1886, Killing and his wife entered the Third Order of Franciscans.
Work
In 1878 Killing wrote on space forms in terms of non-Euclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885. Recounting lectures of Weierstrass, he there introduced the hyperboloid model of hyperbolic geometry described by Weierstrass coordinates. He is also credited with formulating transformations mathematically equivalent to Lorentz transformations in n dimensions in 1885,.
Killing invented Lie algebras independently of Sophus Lie around 1880. Killing's university library did not contain the Scandinavian journal in which Lie's article appeared. (Lie later was scornful of Killing, perhaps out of competitive spirit and claimed that all that was valid had already been proven by Lie and all that was invalid was added by Killing.) In fact Killing's work was less rigorous logically than Lie's, but Killing had much grander goals in terms of classification of groups, and made a number of unproven conjectures that turned out to be true. Because Killing's goals were so high, he was excessively modest about his own achievement.
From 1888 to 1890, Killing essentially classified the complex finite-dimensional simple Lie algebras, as a requisite step of classifying Lie groups, inventing the notions of a Cartan subalgebra and the Cartan matrix. He thus arrived at the conclusion that, basically, the only simple Lie algebras were those associated to the linear, orthogonal, and symplectic groups, apart from a small number of isolated exceptions. Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper. Killing also introduced the notion of a root system. He discovered the exceptional Lie algebra g2 in 1887; his root system classification showed up all the exceptional cases, but concrete constructions came later.
As A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."
Selected works
Work on non-Euclidean geometry
Work on transformation groups
See also
Kill
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https://en.wikipedia.org/wiki/Equidistant
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A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.
In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is their perpendicular bisector. In three dimensions, the locus of points equidistant from two given points is a plane, and generalising further, in n-dimensional space the locus of points equidistant from two points in n-space is an (n−1)-space.
For a triangle the circumcentre is a point equidistant from each of the three vertices. Every non-degenerate triangle has such a point. This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices. Likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side. Every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle.
The center of a rectangle is equidistant from all four vertices, and it is equidistant from two opposite sides and also equidistant from the other two opposite sides. A point on the axis of symmetry of a kite is equidistant between two sides.
The center of a circle is equidistant from every point on the circle. Likewise the center of a sphere is equidistant from every point on the sphere.
A parabola is the set of points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix), where distance from the directrix is measured along a line perpendicular to the directrix.
In shape analysis, the topological skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries.
In Euclidean geometry, parallel lines (lines that never intersect) are equidistant in the sense that the distance of any point on one line from the nearest point on the other line is the same for all points.
In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form a hypercycle (which is a curve not a line).
See also
Equidistant set
References
Elementary geometry
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https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch%20theorem
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In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.
Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.
The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof.
Formulation
Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation:
where is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group:
Now consider a proper morphism between smooth quasi-projective schemes and a bounded complex of sheaves on
The Grothendieck–Riemann–Roch theorem relates the pushforward map
(alternating sum of higher direct images) and the pushforward
by the formula
Here is the Todd genus of (the tangent bundle of) X. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depend on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck–Riemann–Roch formula as
where is the relative tangent sheaf of f, defined as the element in . For example, when f is a smooth morphism, is simply a vector bundle, known as the tangent bundle along the fibers of f.
Using A1-homotopy theory, the Grothendieck–Riemann
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https://en.wikipedia.org/wiki/Univalent%20function
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In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.
Examples
The function is univalent in the open unit disc, as implies that . As the second factor is non-zero in the open unit disc, must be injective.
Basic properties
One can prove that if and are two open connected sets in the complex plane, and
is a univalent function such that (that is, is surjective), then the derivative of is never zero, is invertible, and its inverse is also holomorphic. More, one has by the chain rule
for all in
Comparison with real functions
For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function
given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).
See also
Biholomorphic mapping
De Branges's theorem
Koebe quarter theorem
Riemann mapping theorem
Schlicht function
Note
References
Analytic functions
is:Eintæk vörpun
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https://en.wikipedia.org/wiki/Similarity%20invariance
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In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, is invariant under similarities if where is a matrix similar to A. Examples of such functions include the trace, determinant, characteristic polynomial, and the minimal polynomial.
A more colloquial phrase that means the same thing as similarity invariance is "basis independence", since a matrix can be regarded as a linear operator, written in a certain basis, and the same operator in a new basis is related to one in the old basis by the conjugation , where is the transformation matrix to the new basis.
See also
Invariant (mathematics)
Gauge invariance
Trace diagram
Functions and mappings
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https://en.wikipedia.org/wiki/Proximity%20space
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In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize topological spaces.
The concept was described by but ignored at the time. It was rediscovered and axiomatized by V. A. Efremovič in 1934 under the name of infinitesimal space, but not published until 1951. In the interim, discovered a version of the same concept under the name of separation space.
Definition
A is a set with a relation between subsets of satisfying the following properties:
For all subsets
implies
implies
implies
implies ( or )
(For all or ) implies
Proximity without the first axiom is called (but then Axioms 2 and 4 must be stated in a two-sided fashion).
If we say is near or and are ; otherwise we say and are . We say is a or of written if and only if and are apart.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets
implies
implies
( and ) implies
implies
implies that there exists some such that
A proximity space is called if implies
A or is one that preserves nearness, that is, given if in then in Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if holds in then holds in
Properties
Given a proximity space, one can define a topology by letting be a Kuratowski closure operator. If the proximity space is separated, the resulting topology is Hausdorff. Proximity maps will be continuous between the induced topologies.
The resulting topology is always completely regular. This can be proven by imitating the usual proofs of Urysohn's lemma, using the last property of proximal neighborhoods to create the infinite increasing chain used in proving the lemma.
Given a compact Hausdorff space, there is a unique proximity whose corresponding topology is the given topology: is near if and only if their closures intersect. More generally, proximities classify the compactifications of a completely regular Hausdorff space.
A uniform space induces a proximity relation by declaring is near if and only if has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.
See also
References
External links
Closure operators
General topology
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https://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona%20theorem
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In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes. It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs. It is named after Joseph Kruskal and Gyula O. H. Katona, but has been independently discovered by several others.
Statement
Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows:
This expansion can be constructed by applying the greedy algorithm: set ni to be the maximal n such that replace N with the difference, i with i − 1, and repeat until the difference becomes zero. Define
Statement for simplicial complexes
An integral vector is the f-vector of some -dimensional simplicial complex if and only if
Statement for uniform hypergraphs
Let A be a set consisting of N distinct i-element subsets of a fixed set U ("the universe") and B be the set of all -element subsets of the sets in A. Expand N as above. Then the cardinality of B is bounded below as follows:
Lovász' simplified formulation
The following weaker but useful form is due to . Let A be a set of i-element subsets of a fixed set U ("the universe") and B be the set of all -element subsets of the sets in A. If then .
In this formulation, x need not be an integer. The value of the binomial expression is .
Ingredients of the proof
For every positive i, list all i-element subsets a1 < a2 < … ai of the set N of natural numbers in the colexicographical order. For example, for i = 3, the list begins
Given a vector with positive integer components, let Δf be the subset of the power set 2N consisting of the empty set together with the first i-element subsets of N in the list for i = 1, …, d. Then the following conditions are equivalent:
Vector f is the f-vector of a simplicial complex Δ.
Δf is a simplicial complex.
The difficult implication is 1 ⇒ 2.
History
The theorem is named after Joseph Kruskal and Gyula O. H. Katona, who published it in 1963 and 1968 respectively.
According to , it was discovered independently by , , , , and .
writes that the earliest of these references, by Schützenberger, has an incomplete proof.
See also
Sperner's theorem
References
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External links
Kruskal-Katona theorem on the polymath1 wiki
Algebraic combinatorics
Hypergraphs
Families of sets
Theorems in combinatorics
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https://en.wikipedia.org/wiki/MINQUE
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In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. Its application was originally to the problem of heteroscedasticity and the estimation of variance components in random effects models.
The theory involves three stages:
defining a general class of potential estimators as quadratic functions of the observed data, where the estimators relate to a vector of model parameters;
specifying certain constraints on the desired properties of the estimators, such as unbiasedness;
choosing the optimal estimator by minimising a "norm" which measures the size of the covariance matrix of the estimators.
References
Estimation theory
Statistical deviation and dispersion
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https://en.wikipedia.org/wiki/Newton%20da%20Costa
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Newton Carneiro Affonso da Costa (born 16 September 1929 in Curitiba, Brazil) is a Brazilian mathematician, logician, and philosopher. He studied engineering and mathematics at the Federal University of Paraná in Curitiba and the title of his 1961 Ph.D. dissertation was Topological spaces and continuous functions.
Work
Paraconsistency
Da Costa's international recognition came especially through his work on paraconsistent logic and its application to various fields such as philosophy, law, computing, and artificial intelligence. He is one of the founders of this non-classical logic. In addition, he constructed the theory of quasi-truth that constitutes a generalization of Alfred Tarski's theory of truth, and applied it to the foundations of science.
Other fields; foundations of physics
The scope of his research also includes model theory, generalized Galois theory, axiomatic foundations of quantum theory and relativity, complexity theory, and abstract logics. Da Costa has significantly contributed to the philosophy of logic, paraconsistent modal logics, ontology, and philosophy of science. He served as the President of the Brazilian Association of Logic and the Director of the Institute of Mathematics at the University of São Paulo. He received many awards and held numerous visiting scholarships at universities and centers of research in all continents.
Da Costa and physicist Francisco Antônio Dória axiomatized large portions of classical physics with the help of Patrick Suppes' predicates. They used that technique to show that for the axiomatized version of dynamical systems theory, chaotic properties of those systems are undecidable and Gödel-incomplete, that is, a sentence like X is chaotic is undecidable within that axiomatics. They later exhibited similar results for systems in other areas, such as mathematical economics.
Da Costa believes that the significant progress in the field of logic will give rise to new fundamental developments in computing and technology, especially in connection with non-classical logics and their applications.
Variable-binding term operators
Da Costa is co-discoverer of the truth-set principle and co-creator of the classical logic of variable-binding term operators—both with John Corcoran. He is also co-author with Chris Mortensen of the definitive pre-1980 history of variable-binding term operators in classical first-order logic: “Notes on the theory of variable-binding term operators”, History and Philosophy of Logic, vol.4 (1983) 63–72.
P = NP
Together with Francisco Antônio Dória, Da Costa has published two papers with conditional relative proofs of the consistency of P = NP with the usual
set-theoretic axioms ZFC. The results they obtain are similar to the results of DeMillo and Lipton (consistency of P = NP with fragments of arithmetic) and those of Sazonov and Maté (conditional proofs of the consistency of P = NP with strong systems).
Basically da Costa and Doria define a formal sentence [P = NP]
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https://en.wikipedia.org/wiki/Complex%20vector%20bundle
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In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the complexification
whose fibers are Ex ⊗R C.
Any complex vector bundle over a paracompact space admits a hermitian metric.
The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class.
A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic.
Complex structure
A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle E and itself:
such that J acts as the square root i of −1 on fibers: if is the map on fiber-level, then as a linear map. If E is a complex vector bundle, then the complex structure J can be defined by setting to be the scalar multiplication by . Conversely, if E is a real vector bundle with a complex structure J, then E can be turned into a complex vector bundle by setting: for any real numbers a, b and a real vector v in a fiber Ex,
Example: A complex structure on the tangent bundle of a real manifold M is usually called an almost complex structure. A theorem of Newlander and Nirenberg says that an almost complex structure J is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving J vanishes.
Conjugate bundle
If E is a complex vector bundle, then the conjugate bundle of E is obtained by having complex numbers acting through the complex conjugates of the numbers. Thus, the identity map of the underlying real vector bundles: is conjugate-linear, and E and its conjugate are isomorphic as real vector bundles.
The k-th Chern class of is given by
.
In particular, E and are not isomorphic in general.
If E has a hermitian metric, then the conjugate bundle is isomorphic to the dual bundle through the metric, where we wrote for the trivial complex line bundle.
If E is a real vector bundle, then the underlying real vector bundle of the complexification of E is a direct sum of two copies of E:
(since V⊗RC = V⊕iV for any real vector space V.) If a complex vector bundle E is the complexification of a real vector bundle E, then E is called a real form of E (there may be more than one real form) and E is said to be defined over the real numbers. If E has a real form, then E is isomorphic to its conjugate (since they are both sum of two copies of a real form), and consequently the odd Chern classes of E have order 2.
See also
Holomorphic vector bundle
K-theory
References
Vector bundles
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https://en.wikipedia.org/wiki/Smooth%20structure
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In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for This gives a natural equivalence relation on the set of smooth atlases.
A smooth manifold is a topological manifold together with a smooth structure on
Maximal smooth atlases
By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases.
Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.
In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas.
For example, if the manifold is compact, then one can find an atlas with only finitely many charts.
Equivalence of smooth structures
Let and be two maximal atlases on The two smooth structures associated to and are said to be equivalent if there is a diffeomorphism such that
Exotic spheres
John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.
E8 manifold
The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.
Related structures
The smoothness requirements on the transition functions can be weakened, so that we only require the transition maps to be -times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic.
See also
References
Differential topology
Structures on manifolds
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https://en.wikipedia.org/wiki/Darboux%27s%20theorem
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In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.
It is a foundational result in several fields, the chief among them being symplectic geometry. Indeed, one of its many consequences is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every -dimensional symplectic manifold can be made to look locally like the linear symplectic space with its canonical symplectic form.
There is also an analogous consequence of the theorem applied to contact geometry.
Statement
Suppose that is a differential 1-form on an -dimensional manifold, such that has constant rank . Then
if everywhere, then there is a local system of coordinates in which
if everywhere, then there is a local system of coordinates in which
Darboux's original proof used induction on and it can be equivalently presented in terms of distributions or of differential ideals.
Frobenius' theorem
Darboux's theorem for ensures the any 1-form such that can be written as in some coordinate system .
This recovers one of the formulation of Frobenius theorem in terms of differential forms: if is the differential ideal generated by , then implies the existence of a coordinate system where is actually generated by .
Darboux's theorem for symplectic manifolds
Suppose that is a symplectic 2-form on an -dimensional manifold . In a neighborhood of each point of , by the Poincaré lemma, there is a 1-form with . Moreover, satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart near in which
Taking an exterior derivative now shows
The chart is said to be a Darboux chart around . The manifold can be covered by such charts.
To state this differently, identify with by letting . If is a Darboux chart, then can be written as the pullback of the standard symplectic form on :
A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick.
Comparison with Riemannian geometry
Darboux's theorem for symplectic manifolds implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.
The difference is that Darboux's theorem states that can be made to take the standard form in an entire neighborhood around . In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.
Darboux's theorem for contact manifolds
Another partic
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https://en.wikipedia.org/wiki/Compactification
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Compactification may refer to:
Compactification (mathematics), making a topological space compact
Compactification (physics), the "curling up" of extra dimensions in string theory
See also
Compaction (disambiguation)
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https://en.wikipedia.org/wiki/Matrix%20norm
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In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field of either real or complex numbers, let be the -vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on .
This article will always write such norms with double vertical bars (like so: ). Thus, the matrix norm is a function that must satisfy the following properties:
For all scalars and matrices ,
(positive-valued)
(definite)
(absolutely homogeneous)
(sub-additive or satisfying the triangle inequality)
The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative:
Every norm on can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms.
Matrix norms induced by vector norms
Suppose a vector norm on and a vector norm on are given. Any matrix induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows:
where denotes the supremum. This norm measures how much the mapping induced by can stretch vectors.
Depending on the vector norms , used, notation other than can be used for the operator norm.
Matrix norms induced by vector p-norms
If the p-norm for vectors () is used for both spaces and then the corresponding operator norm is:
These induced norms are different from the "entry-wise" p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by
In the special cases of the induced matrix norms can be computed or estimated by
which is simply the maximum absolute column sum of the matrix;
which is simply the maximum absolute row sum of the matrix.
For example, for
we have that
In the special case of (the Euclidean norm or -norm for vectors), the induced matrix norm is the spectral norm. (The two values do not coincide in infinite dimensions — see Spectral radius for further discussion.) The spectral norm of a matrix is the largest singular value of (i.e., the square root of the largest eigenvalue of the matrix where denotes the conjugate transpose of ):
where represents the largest singular value of matrix Also,
since and similarly by singular value decomposition (SVD). There is another important inequality:
where is the Frobenius norm. Equality holds if and only if the matrix is a rank-one matrix or a zero matrix. This inequality can be derived from the fact that the trace of a matrix is equal to the sum of its eigenvalues.
When we have an equivalent definition for as It can be shown to be equivalent to the above definitions using the Cauchy–Schwarz inequality.
Matrix norms induced by vector α- and β- norms
Suppose vector norms and are used for spaces and respectiv
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https://en.wikipedia.org/wiki/Subgroup%20series
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In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups:
where is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the subgroup method.
Subgroup series are a special example of the use of filtrations in abstract algebra.
Definition
Normal series, subnormal series
A subnormal series (also normal series, normal tower, subinvariant series, or just series) of a group G is a sequence of subgroups, each a normal subgroup of the next one. In a standard notation
There is no requirement made that Ai be a normal subgroup of G, only a normal subgroup of Ai +1. The quotient groups Ai +1/Ai are called the factor groups of the series.
If in addition each Ai is normal in G, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series.
Length
A series with the additional property that Ai ≠ Ai +1 for all i is called a series without repetition; equivalently, each Ai is a proper subgroup of Ai +1. The length of a series is the number of strict inclusions Ai < Ai +1. If the series has no repetition then the length is n.
For a subnormal series, the length is the number of non-trivial factor groups. Every nontrivial group has a normal series of length 1, namely , and any nontrivial proper normal subgroup gives a normal series of length 2. For simple groups, the trivial series of length 1 is the longest subnormal series possible.
Ascending series, descending series
Series can be notated in either ascending order:
or descending order:
For a given finite series, there is no distinction between an "ascending series" or "descending series" beyond notation. For infinite series however, there is a distinction: the ascending series
has a smallest term, a second smallest term, and so forth, but no largest proper term, no second largest term, and so forth, while conversely the descending series
has a largest term, but no smallest proper term.
Further, given a recursive formula for producing a series, the terms produced are either ascending or descending, and one calls the resulting series an ascending or descending series, respectively. For instance the derived series and lower central series are descending series, while the upper central series is an ascending series.
Noetherian groups, Artinian groups
A group that satisfies the ascending chain condition (ACC) on subgroups is called a Noetherian group, and a group that satisfies the descending chain condition (DCC) is called an Artinian group (not to be confused with Artin groups), by analogy with Noetherian rings and Artinian rings. The ACC is equivalent to the maximal condition: every non-empty collection of subgroups has a maximal member, and the DCC is equivalent to the analogous minimal condition.
A group can
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https://en.wikipedia.org/wiki/Wang%20Xiaoyun
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Wang Xiaoyun (; born 1966) is a Chinese cryptographer, mathematician, and computer scientist. She is a professor in the Department of Mathematics and System Science of Shandong University and an academician of the Chinese Academy of Sciences.
Early life and education
Wang was born in Zhucheng, Shandong Province. She gained bachelor (1987), master (1990) and doctorate (1993) degrees at Shandong University, and subsequently lectured in the mathematics department from 1993. Her doctoral advisor was Pan Chengdong. Wang was appointed assistant professor in 1995, and full professor in 2001. She became the Chen Ning Yang Professor of the Center for Advanced Study, Tsinghua University in 2005.
Career and research
At the rump session of CRYPTO 2004, she and co-authors demonstrated collision attacks against MD5, SHA-0 and other related hash functions (a collision occurs when two distinct messages result in the same hash function output). They received a standing ovation for their work.
In February 2005, it was reported that Wang and co-authors Yiqun Lisa Yin and Hongbo Yu had found a method to find collisions in the SHA-1 hash function, which is used in many of today's mainstream security products. Their attack is estimated to require less than 269 operations, far fewer than the 280 operations previously thought needed to find a collision in . Their work was published at the CRYPTO '05 conference. In August 2005, an improved attack on SHA-1, discovered by Wang, Andrew Yao and Frances Yao, was announced at the CRYPTO conference rump session. The time complexity of the new attack is claimed to be 263.
Awards and honors
In 2019, she was named a Fellow of the International Association for Cryptologic Research (IACR) for "For essential contributions to the cryptanalysis and design of hash functions, and for service to the IACR." In 2019, she became the first female winner of China's Future Science Prize for her pioneering contribution in cryptography.
References
External links
Xiaoyun Wang
1966 births
Living people
Chinese cryptographers
Chinese women computer scientists
Chinese women mathematicians
Educators from Shandong
Mathematicians from Shandong
Members of the Chinese Academy of Sciences
Modern cryptographers
People from Zhucheng
Shandong University alumni
Academic staff of Shandong University
Academic staff of Tsinghua University
Women cryptographers
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https://en.wikipedia.org/wiki/Twisted%20cubic
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In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line.
Definition
The twisted cubic is most easily given parametrically as the image of the map
which assigns to the homogeneous coordinate the value
In one coordinate patch of projective space, the map is simply the moment curve
That is, it is the closure by a single point at infinity of the affine curve .
The twisted cubic is a projective variety, defined as the intersection of three quadrics. In homogeneous coordinates on P3, the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials
It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x3 for X, and so on.
More strongly, the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2.
Properties
The twisted cubic has the following properties:
It is the set-theoretic complete intersection of and , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not radical, since is in it, but is not).
Any four points on C span P3.
Given six points in P3 with no four coplanar, there is a unique twisted cubic passing through them.
The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P3 and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the tangent and secant lines of any non-planar smooth algebraic curve is three-dimensional. Further, any smooth algebraic variety with the property that every length four subscheme spans P3 has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself.
The projection of C onto a plane from a point on a tangent line of C yields a cuspidal cubic.
The projection from a point on a secant line of C yields a nodal cubic.
The projection from a point on C yields a conic section.
References
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Algebraic curves
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https://en.wikipedia.org/wiki/Hypotrochoid
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In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle.
The parametric equations for a hypotrochoid are:
where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from 0 to (where is least common multiple).
Special cases include the hypocycloid with and the ellipse with and . The eccentricity of the ellipse is
becoming 1 when (see Tusi couple).
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations
See also
Cycloid
Cyclogon
Epicycloid
Rosetta (orbit)
Apsidal precession
Spirograph
References
External links
Flash Animation of Hypocycloid
Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
Interactive hypotrochoide animation
Roulettes (curve)
de:Zykloide#Epi- und Hypozykloide
ja:トロコイド#内トロコイド
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https://en.wikipedia.org/wiki/Epitrochoid
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In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle.
The parametric equations for an epitrochoid are
The parameter is geometrically the polar angle of the center of the exterior circle. (However, is not the polar angle of the point on the epitrochoid.)
Special cases include the limaçon with and the epicycloid with .
The classic Spirograph toy traces out epitrochoid and hypotrochoid curves.
The paths of planets in the once popular geocentric system of deferents and epicycles are epitrochoids with for both the outer planets and the inner planets.
The orbit of the Moon, when centered around the Sun, approximates an epitrochoid.
The combustion chamber of the Wankel engine is an epitrochoid.
See also
Cycloid
Cyclogon
Epicycloid
Hypocycloid
Hypotrochoid
Spirograph
List of periodic functions
Rosetta (orbit)
Apsidal precession
References
External links
Epitrochoid generator
Visual Dictionary of Special Plane Curves on Xah Lee 李杀网
Interactive simulation of the geocentric graphical representation of planet paths
Plot Epitrochoid -- GeoFun
Roulettes (curve)
ja:トロコイド#外トロコイド
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https://en.wikipedia.org/wiki/Injective%20sheaf
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In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext).
There is a further group of related concepts applied to sheaves: flabby (flasque in French), fine, soft (mou in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence.
Injective sheaves
An injective sheaf is a sheaf that is an injective object of the category of abelian sheaves; in other words, homomorphisms from to can always be extended to any sheaf containing
The category of abelian sheaves has enough injective objects: this means that any sheaf is a subsheaf of an injective sheaf. This result of Grothendieck follows from the existence of a generator of the category (it can be written down explicitly, and is related to the subobject classifier). This is enough to show that right derived functors of any left exact functor exist and are unique up to canonical isomorphism.
For technical purposes, injective sheaves are usually superior to the other classes of sheaves mentioned above: they can do almost anything the other classes can do, and their theory is simpler and more general. In fact, injective sheaves are flabby (flasque), soft, and acyclic. However, there are situations where the other classes of sheaves occur naturally, and this is especially true in concrete computational situations.
The dual concept, projective sheaves, is not used much, because in a general category of sheaves there are not enough of them: not every sheaf is the quotient of a projective sheaf, and in particular projective resolutions do not always exist. This is the case, for example, when looking at the category of sheaves on projective space in the Zariski topology. This causes problems when attempting to define left derived functors of a right exact functor (such as Tor). This can sometimes be done by ad hoc means: for example, the left derived functors of Tor can be defined using a flat resolution rather than a projective one, but it takes some work to show that this is independent of the resolution. Not all categories of sheaves run into this problem; for instance, the category of sheaves on an affine scheme contains enough projectives.
Acyclic sheaves
An acyclic sheaf over X is one such that all higher sheaf cohomology groups vanish.
The cohomology groups of any sheaf can be calculated from any acyclic resolution of it (this goes by the name of De Rham-Weil theorem).
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https://en.wikipedia.org/wiki/Villarceau%20circles
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In geometry, Villarceau circles () are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the equatorial plane of the torus and another perpendicular to that plane (these are analogous to lines of latitude and longitude on the Earth). The other two are Villarceau circles. They are obtained as the intersection of the torus with a plane that passes through the center of the torus and touches it tangentially at two antipodal points. If one considers all these planes, one obtains two families of circles on the torus. Each of these families consists of disjoint circles that cover each point of the torus exactly once and thus forms a 1-dimensional foliation of the torus.
The Villarceau circles are named after the French astronomer and mathematician Yvon Villarceau (1813–1883) who wrote about them in 1848.
Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.
Example
Consider a horizontal torus in xyz space, centered at the origin and with major radius 5 and minor radius 3. That means that the torus is the locus of some vertical circles of radius three whose centers are on a circle of radius five in the horizontal xy plane. Points on this torus satisfy this equation:
Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82, the outer and inner equator. Slicing with the x = 0 plane produces two side-by-side circles, (y − 5)2 + z2 = 32 and (y + 5)2 + z2 = 32.
Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. One is centered at (0, +3, 0) and the other at (0, −3, 0); both have radius five. They can be written in parametric form as
and
The slicing plane is chosen to be tangent to the torus at two points while passing through its center. It is tangent at (16⁄5, 0, 12⁄5) and at (−16⁄5, 0, −12⁄5). The angle of slicing is uniquely determined by the dimensions of the chosen torus. Rotating any one such plane around the z-axis gives all of the Villarceau circles for that torus.
Existence and equations
A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. One characterization of a torus is that it is a surface of revolution. Without loss of generality, choose a coordinate system so that the axis of revolution is the z axis. Begin with a circle of radius r in the xz plane, centered at (R, 0, 0).
Sweeping replaces x by (x2 + y2)1/2, and clearing the square root produces a quartic equation.
The cross-section of the swept surface in the xz plane now includes a second circle.
This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse R and opp
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https://en.wikipedia.org/wiki/Joseph%20L.%20Doob
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Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, February 27, 1910, the son of a Jewish couple, Leo Doob and Mollie Doerfler Doob. The family moved to New York City before he was three years old. The parents felt that he was underachieving in grade school and placed him in the Ethical Culture School, from which he graduated in 1926. He then went on to Harvard where he received a BA in 1930, an MA in 1931, and a PhD (Boundary Values of Analytic Functions, advisor Joseph L. Walsh) in 1932. After postdoctoral research at Columbia and Princeton, he joined the department of mathematics of the University of Illinois in 1935 and served until his retirement in 1978. He was a member of the Urbana campus's Center for Advanced Study from its beginning in 1959. During the Second World War, he worked in Washington, D.C., and Guam as a civilian consultant to the Navy from 1942 to 1945; he was at the Institute for Advanced Study for the academic year 1941–1942 when Oswald Veblen approached him to work on mine warfare for the Navy.
Work
Doob's thesis was on boundary values of analytic functions. He published two papers based on this thesis, which appeared in 1932 and 1933 in the Transactions of the American Mathematical Society. Doob returned to this subject many years later when he proved a probabilistic version of Fatou's boundary limit theorem for harmonic functions.
The Great Depression of 1929 was still going strong in the thirties and Doob could not find a job. B.O. Koopman at Columbia University suggested that statistician Harold Hotelling might have a grant that would permit Doob to work with him. Hotelling did, so the Depression led Doob to probability.
In 1933 Kolmogorov provided the first axiomatic foundation for the theory of probability. Thus a subject that had originated from intuitive ideas suggested by real life experiences and studied informally, suddenly became mathematics. Probability theory became measure theory with its own problems and terminology. Doob recognized that this would make it possible to give rigorous proofs for existing probability results, and he felt that the tools of measure theory would lead to new probability results.
Doob's approach to probability was evident in his first probability paper, in which he proved theorems related to the law of large numbers, using a probabilistic interpretation of Birkhoff's ergodic theorem. Then he used these theorems to give rigorous proofs of theorems proven by Fisher and Hotelling related to Fisher's maximum likelihood estimator for estimating a parameter of a distribution.
After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing wha
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https://en.wikipedia.org/wiki/Fana
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{{Historical populations
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|1980|25050
|1990|27163
|2001|32393
|2013|40087
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Fana is a borough of the city of Bergen in Vestland county, Norway. The borough makes up the southeastern part of the municipality of Bergen. The borough was once part of the historic municipality of Fana which was incorporated into Bergen in 1972. The old municipality was much larger than the present-day borough of Fana. It also included all of the present-day boroughs of Ytrebygda and Fyllingsdalen as well as the southern part of the present-day boroughs of Årstad. As of 1 January 2012, Fana had a population of 39,216.
Toponymy
"The name is really [a] farm name, in Old Norse fani, which probably means swampland or myrlende" (or fen), according to the Store norske leksikon.
Geography
Fana is the geographically largest of the city's boroughs, with an area of . Most major industries in Fana are located near the neighborhood of Nesttun (which was the administrative centre of the old Fana municipality). The northeastern part is dominated by residential areas, being home to the majority of the borough's population, while the rest of the borough contains mostly forest, mountains, some farmland, in addition to a few settlements. The mountain Livarden lies along the northeastern boundary of the borough.
Villages and neighborhoods
The villages and neighborhoods in the borough include: Fanahammeren, Nattland, Nesttun, Paradis, Nordvåg, Skjold, and Krokvåg.
Landmarks
Gamlehaugen is located by the lake Nordåsvannet in northern Fana, south of the present-day Fjøsanger residential area. The mansion is the residence of the Royal Family in Bergen, and is surrounded by a park. It was commissioned by Christian Michelsen, a shipping magnate and later Prime Minister of Norway, in 1899, and he lived there until his death in 1925. While the park is open to the public at almost all times, the building is only open for a few hours a day in the summer and receives about 2000 visitors a year.
Hop is the location of Troldhaugen, a museum and home of the composer Edvard Grieg 1885-1907.
A replica of the Fantoft Stave Church has been located in Fana since 1997. The original was built in 1150 and it burned down in 1992, 109 years after it was moved to Bergen in 1883. Fana Church is a more recently built church that is used for most church functions in the area.
Culture
Fashion
The Fanakofte is an old cardigan (sweater) pattern.
Food
The "Fanaost" gouda cheese won World Cheese Awards in 2018.
Sport
Sports teams include IL Gneist, IL Bjarg and Fana IL and the athletics club FIK BFG Fana.
Choirs
Choirs include Sola Fide, Fana Mannskor, Korall.
Transport
The European route E39 highway passes through the borough of Fana. From the border with Årstad borough (north of Fjøsanger) to the neighborhood of Hop, the E39 highway is the 4-lane dual carriageway called Fritz C. Riebers veg. At Hop, the E39 highway branches of
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https://en.wikipedia.org/wiki/Rafael%20Bombelli
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Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.
He was the one who finally managed to address the problem with imaginary numbers. In his 1572 book, L'Algebra, Bombelli solved equations using the method of del Ferro/Tartaglia. He introduced the rhetoric that preceded the representative symbols +i and -i and described how they both worked.
Life
Rafael Bombelli was baptised on 20 January 1526 in Bologna, Papal States. He was born to Antonio Mazzoli, a wool merchant, and Diamante Scudieri, a tailor's daughter. The Mazzoli family was once quite powerful in Bologna. When Pope Julius II came to power, in 1506, he exiled the ruling family, the Bentivoglios. The Bentivoglio family attempted to retake Bologna in 1508, but failed. Rafael's grandfather participated in the coup attempt, and was captured and executed. Later, Antonio was able to return to Bologna, having changed his surname to Bombelli to escape the reputation of the Mazzoli family. Rafael was the oldest of six children. Rafael received no college education, but was instead taught by an engineer-architect by the name of Pier Francesco Clementi.
Bombelli felt that none of the works on algebra by the leading mathematicians of his day provided a careful and thorough exposition of the subject. Instead of another convoluted treatise that only mathematicians could comprehend, Rafael decided to write a book on algebra that could be understood by anyone. His text would be self-contained and easily read by those without higher education.
Bombelli died in 1572 in Rome.
Bombelli's Algebra
In the book that was published in 1572, entitled Algebra, Bombelli gave a comprehensive account of the algebra known at the time. He was the first European to write down the way of performing computations with negative numbers. The following is an excerpt from the text:
"Plus times plus makes plus
Minus times minus makes plus
Plus times minus makes minus
Minus times plus makes minus
Plus 8 times plus 8 makes plus 64
Minus 5 times minus 6 makes plus 30
Minus 4 times plus 5 makes minus 20
Plus 5 times minus 4 makes minus 20"
As was intended, Bombelli used simple language as can be seen above so that anybody could understand it. But at the same time, he was thorough.
Notation
Bombelli introduced, for the first time in a printed text (in Book II of his Algebra), a form of index notation in which the equation
appeared as
1U3 a. 6U1 p. 40.
in which he wrote the U3 as a raised bowl-shape (like the curved part of the capital letter U) with the number 3 above it. Full symbolic notation was developed shortly thereafter by the French mathematician François_Viète.
Complex numbers
Perhaps more importantly than his work with algebra, however, the book also includes Bombelli's monumental contributions to complex number theory. Befor
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https://en.wikipedia.org/wiki/Fractional-order%20control
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Fractional-order control (FOC) is a field of control theory that uses the fractional-order integrator as part of the control system design toolkit. The use of fractional calculus (FC) can improve and generalize well-established control methods and strategies.
The fundamental advantage of FOC is that the fractional-order integrator weights history using a function that decays with a power-law tail. The effect is that the effects of all time are computed for each iteration of the control algorithm. This creates a 'distribution of time constants,' the upshot of which is there is no particular time constant, or resonance frequency, for the system.
In fact, the fractional integral operator is different from any integer-order rational transfer function , in the sense that it is a non-local operator that possesses an infinite memory and takes into account the whole history of its input signal.
Fractional-order control shows promise in many controlled environments that suffer from the classical problems of overshoot and resonance, as well as time diffuse applications such as thermal dissipation and chemical mixing. Fractional-order control has also been demonstrated to be capable of suppressing chaotic behaviors in mathematical models of, for example, muscular blood vessels.
Initiated from the 80's by the Pr. Oustaloup's group, the CRONE approach is one of the most developed control-system design methodologies that uses fractional-order operator properties.
See also
Differintegral
Fractional calculus
Fractional-order system
External links
Dr. YangQuan Chen's latest homepage for the applied fractional calculus (AFC)
Dr. YangQuan Chen's page about fractional calculus on Google Sites
References
Control theory
Cybernetics
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https://en.wikipedia.org/wiki/Semiregular%20polyhedron
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In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.
Definitions
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). These polyhedra include:
The thirteen Archimedean solids.
The elongated square gyrobicupola, also called a pseudo-rhombicuboctahedron, a Johnson solid, has identical vertex figures 3.4.4.4, but is not vertex-transitive including a twist has been argued for inclusion as a 14th Archimedean solid by Branko Grünbaum.
An infinite series of convex prisms.
An infinite series of convex antiprisms (their semiregular nature was first observed by Kepler).
These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, in order as they occur around a vertex. For example: represents the icosidodecahedron, which alternates two triangles and two pentagons around each vertex. In contrast: is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive.
Since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures uniform, with only a quite restricted subset classified as semiregular.
Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
Three sets of star polyhedra which meet Gosset's definition, analogous to the three convex sets listed above.
The duals of the above semiregular solids, arguing that since the dual polyhedra share the same symmetries as the originals, they too should be regarded as semiregular. These duals include the Catalan solids, the convex dipyramids, and the convex antidipyramids or trapezohedra, and their nonconvex analogues.
A further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing.
Gosset's definition of semiregular includes figures of higher symmetry: the regular and quasiregular polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra are then said to include the regular, quasiregular, and semiregular ones. This naming system works well, and reconciles many (but by no means all) of the confusions.
In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or Archimedean, and then assuming (or even stating) a different set in subsequent discussions. Assuming that one's stated definition applies only to convex polyhedra is probably the most common failing. Coxeter, Cromwell, and Cundy & Rollett are all guil
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https://en.wikipedia.org/wiki/Cuisenaire%20rods
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Cuisenaire rods are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors. In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by Georges Cuisenaire (1891–1975), a Belgian primary school teacher, who called the rods réglettes.
According to Gattegno, "Georges Cuisenaire showed in the early 1950s that students who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods."
History
The educationalists Maria Montessori and Friedrich Fröbel had used rods to represent numbers, but it was Georges Cuisenaire who introduced the rods that were to be used across the world from the 1950s onwards. In 1952 he published Les nombres en couleurs, Numbers in Color, which outlined their use. Cuisenaire, a violin player, taught music as well as arithmetic in the primary school in Thuin. He wondered why children found it easy and enjoyable to pick up a tune and yet found mathematics neither easy nor enjoyable. These comparisons with music and its representation led Cuisenaire to experiment in 1931 with a set of ten rods sawn out of wood, with lengths from 1 cm to 10 cm. He painted each length of rod a different colour and began to use these in his teaching of arithmetic. The invention remained almost unknown outside the village of Thuin for about 23 years until, in April 1953, British mathematician and mathematics education specialist Caleb Gattegno was invited to see students using the rods in Thuin. At this point he had already founded the International Commission for the Study and Improvement of Mathematics Education (CIEAEM) and the Association of Teachers of Mathematics, but this marked a turning point in his understanding:
Then Cuisenaire took us to a table in one corner of the room where pupils were standing round a pile of colored sticks and doing sums which seemed to me to be unusually hard for children of that age. At this sight, all other impressions of the surrounding vanished, to be replaced by a growing excitement. After listening to Cuisenaire asking his first and second grade pupils questions and hearing their answers immediately and with complete self-assurance and accuracy, the excitement then turned into irrepressible enthusiasm and a sense of illumination.
Gattegno named the rods "Cuisenaire rods" and began trialing and popularizing them. Seeing that the rods allowed students "to expand on their latent mathematical abilities in a creative and enjoyable fashion", Gattegno's pedagogy shifted radically as he began to stand back and allow students to take a leading role:
Cuisenaire's gift of the rods led me to teach by non-interference making it necessary to watch and listen for the signs
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https://en.wikipedia.org/wiki/Index%20calculus%20algorithm
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In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.
Dedicated to the discrete logarithm in where is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm with respect to the discrete logarithms of small primes.
Description
Roughly speaking, the discrete log problem asks us to find an x such that , where g, h, and the modulus n are given.
The algorithm (described in detail below) applies to the group where q is prime. It requires a factor base as input. This factor base is usually chosen to be the number −1 and the first r primes starting with 2. From the point of view of efficiency, we want this factor base to be small, but in order to solve the discrete log for a large group we require the factor base to be (relatively) large. In practical implementations of the algorithm, those conflicting objectives are compromised one way or another.
The algorithm is performed in three stages. The first two stages depend only on the generator g and prime modulus q, and find the discrete logarithms of a factor base of r small primes. The third stage finds the discrete log of the desired number h in terms of the discrete logs of the factor base.
The first stage consists of searching for a set of r linearly independent relations between the factor base and power of the generator g. Each relation contributes one equation to a system of linear equations in r unknowns, namely the discrete logarithms of the r primes in the factor base. This stage is embarrassingly parallel and easy to divide among many computers.
The second stage solves the system of linear equations to compute the discrete logs of the factor base. A system of hundreds of thousands or millions of equations is a significant computation requiring large amounts of memory, and it is not embarrassingly parallel, so a supercomputer is typically used. This was considered a minor step compared to the others for smaller discrete log computations. However, larger discrete logarithm records were made possible only by shifting the work away from the linear algebra and onto the sieve (i.e., increasing the number of equations while reducing the number of variables).
The third stage searches for a power s of the generator g which, when multiplied by the argument h, may be factored in terms of the factor base gsh = (−1)f0 2f1 3f2···prfr.
Finally, in an operation too simple to really be called a fourth stage, the results of the second and third stages can be rearranged by simple algebraic manipulation to work out the desired discrete logarithm x = f0logg(−1) + f1logg2 + f2logg3 + ··· + frloggpr − s.
The first and third stages are both embarrassingly parallel, and in fact the third sta
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https://en.wikipedia.org/wiki/Linear%20complex%20structure
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In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.
Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure.
Definition and properties
A complex structure on a real vector space V is a real linear transformation
such that
Here means composed with itself and is the identity map on . That is, the effect of applying twice is the same as multiplication by . This is reminiscent of multiplication by the imaginary unit, . A complex structure allows one to endow with the structure of a complex vector space. Complex scalar multiplication can be defined by
for all real numbers and all vectors in . One can check that this does, in fact, give the structure of a complex vector space which we denote .
Going in the other direction, if one starts with a complex vector space then one can define a complex structure on the underlying real space by defining for all .
More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers , thought of as an associative algebra over the real numbers. This algebra is realized concretely as
which corresponds to . Then a representation of is a real vector space , together with an action of on (a map ). Concretely, this is just an action of , as this generates the algebra, and the operator representing (the image of in ) is exactly .
If has complex dimension then must have real dimension . That is, a finite-dimensional space admits a complex structure only if it is even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. One can define on pairs of basis vectors by and and then extend by linearity to all of . If is a basis for the complex vector space then is a basis for the underlying real space .
A real linear transformation is a complex linear transformation of the corresponding complex space if and only if commutes with , i.e. if and only if
Likewise, a real subspace of is a complex subspace of if and only if preserves , i.e. if and only if
Examples
Elementary example
The collection of 2x2 real matrices M(2,R) over the real field is 4-dimensional. Any matrix
with a2 + bc = –1
has square equal to the negative of the identity matrix. A complex structure may be formed in M(2,R): with identity matrix I, elemen
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https://en.wikipedia.org/wiki/136%20%28number%29
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136 (one hundred [and] thirty-six) is the natural number following 135 and preceding 137.
In mathematics
136 is itself a factor of the Eddington number. With a total of 8 divisors, 8 among them, 136 is a refactorable number. It is a composite number.
136 is a centered triangular number and a centered nonagonal number.
The sum of the ninth row of Lozanić's triangle is 136.
136 is a self-descriptive number in base 4, and a repdigit in base 16. In base 10, the sum of the cubes of its digits is . The sum of the cubes of the digits of 244 is .
136 is a triangular number, because it's the sum of the first 16 positive integers.
In the military
Force 136 branch of the British organization, the Special Operations Executive (SOE), in the South-East Asian Theatre of World War II
USNS Mission Soledad (T-AO-136) was a United States Navy Mission Buenaventura-class fleet oiler during World War II
USS Admirable (AM-136) was a United States Navy Admirable class minesweeper
USS Ara (AK-136) was a United States Navy during World War II
was a United States Navy during World War II
USS Botetourt (APA-136) was a United States Navy during World War II and the Korean War
was a United States Navy tanker during World War II
was a United States Navy during World War II
was a United States Navy heavy cruiser during World War II
USS Frederick C. Davis (DE-136) was a United States Navy during World War II
was a United States Navy General G. O. Squier-class transport ship during World War II
Electronic Attack Squadron 136 (VAQ-136) also known as "The Gauntlets" is a United States Navy attack squadron at Naval Air Station Atsugi, Japan
Strike Fighter Squadron 136 (VFA-136) is a United States Navy strike fighter squadron based at Naval Air Station Oceana, Virginia
In transportation
London Buses route 136 is a Transport for London contracted bus route in London
In TV and radio
136 kHz band is the lowest frequency band amateur radio operators are allowed to transmit
In other fields
The year AD 136 or 136 BC
136 AH is a year in the Islamic calendar that corresponds to 753 – 754 CE
136 Austria is a main-belt asteroid discovered in 1874
WR 136 is a Wolf–Rayet red supergiant star
136P/Mueller, or Mueller 3, is a periodic comet in our Solar System
Section 136 of the Mental Health Act 1983 (UK law) details removing a mentally ill person from a public place to a place of safety. It details police powers and the rights of someone in this position.
Sonnet 136 by William Shakespeare
See also
List of highways numbered 136
United Nations Security Council Resolution 136
External links
136 cats (video)
References
Integers
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https://en.wikipedia.org/wiki/173%20%28number%29
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173 (one hundred [and] seventy-three) is the natural number following 172 and preceding 174.
In mathematics
173 is:
an odd number.
a deficient number.
an odious number.
a balanced prime.
an Eisenstein prime with no imaginary part.
a Sophie Germain prime.
an inconsummate number.
the sum of 2 squares: 22 + 132.
the sum of three consecutive prime numbers: 53 + 59 + 61.
Palindromic number in bases 3 (201023) and 9 (2129).
In astronomy
173 Ino is a large dark main belt asteroid
173P/Mueller is a periodic comet in the Solar System
Arp 173 (VV 296, KPG 439) is a pair of galaxies in the constellation Boötes
In the military
173rd Air Refueling Squadron unit of the Nebraska Air National Guard
173rd Airborne Brigade Combat Team of the United States Army based in Vicenza
173rd Battalion unit of the Canadian Expeditionary Force during the World War I
173rd Special Operations Aviation Squadron of the Australian Army
K-173 Chelyabinsk Russian
was a U.S. Navy Phoenix-class auxiliary ship following World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War II
was a U.S. Navy yacht during World War I
was a U.S. Navy ship during World War II
was a U.S. Navy submarine chaser during World War II
was a U.S. Navy Porpoise-class submarine during World War II
was a U.S. Navy following World War II
Vought V-173 (Flying Pancake) was a U.S. Navy experimental test aircraft during World War II
In transportation
The Georgia Railroad, the world longest railroad in 1845, ran for from Augusta to Marthasville (Atlanta, Georgia)
United Airlines Flight 173 en route from Denver to Portland crashed on December 28, 1978
The Velocity 173 was a kit aircraft produced by Velocity Aircraft in the early 1990s.
In popular culture
The book 173 Hours in Captivity (2000)
SCP-173, a fictional statue
In other fields
173 is also:
The year AD 173 or 173 BC
173 AH is a year in the Islamic calendar that corresponds to 789 – 790 CE
The atomic number of an element temporarily called unsepttrium
Topic of discussion during the podcast "Skeptics with a K" episode 180
See also
List of highways numbered 173
United Nations Security Council Resolution 173
United States Supreme Court cases, Volume 173
External links
Number Facts and Trivia: 173
Prime curiosities: 173
Number Gossip: 173
References
Integers
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https://en.wikipedia.org/wiki/Martin%27s%20axiom
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In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, , behave roughly like . The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.
Statement
For any cardinal 𝛋, consider the following statement:
MA(𝛋) For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense subsets of P such that |D| ≤ 𝛋, there is a filter F on P such that F ∩ d is non-empty for every d in D.
In this case (for application of ccc), an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of trees.
MA(ℵ0) is simply true — the Rasiowa–Sikorski lemma. MA(2ℵ0) is false: [0, 1] is a separable compact Hausdorff space, and so (P, the poset of open subsets under inclusion, is) ccc. But now consider the following two size-2ℵ0= families of dense sets in P: no x∈[0, 1] is isolated, and so each x defines the dense subset {S : x∉S}. And each r∈(0, 1], defines the dense subset {S : diam(S)<r}. The two families combined are also of size , and a filter meeting both must simultaneously avoid all points of [0, 1] while containing sets of arbitrarily small diameter. But a filter F containing sets of arbitrarily small diameter must contain a point in ⋂F by compactness. (See also .)
Martin's axiom is then that MA(κ) holds "as long as possible":
Martin's axiom (MA) For every 𝛋 < , MA(𝛋) holds.
Equivalent forms of MA(𝛋)
The following statements are equivalent to MA(𝛋):
If X is a compact Hausdorff topological space that satisfies the ccc then X is not the union of 𝛋 or fewer nowhere dense subsets.
If P is a non-empty upwards ccc poset and Y is a family of cofinal subsets of P with |Y| ≤ 𝛋 then there is an upwards-directed set A such that A meets every element of Y.
Let A be a non-zero ccc Boolean algebra and F a family of subsets of A with |F| ≤ 𝛋. Then there is a boolean homomorphism φ: A → Z/2Z such that for every X in F either there is an a in X with φ(a) = 1 or there is an upper bound b for X with φ(b) = 0.
Consequences
Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences:
The union of 𝛋 or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null. In particular, the union of 𝛋 or fewer subsets of R of Lebesgue measure 0 also has Lebesgue measure 0.
A compact Hausdorff space X with |X| < 2𝛋
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https://en.wikipedia.org/wiki/Samuel%20Horsley
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Samuel Horsley (15 September 1733 – 4 October 1806) was a British churchman, bishop of Rochester from 1793. He was also well versed in physics and mathematics, on which he wrote a number of papers and thus was elected a Fellow of the Royal Society in 1767; and secretary in 1773, but, in consequence of a difference with the president (Sir Joseph Banks) he withdrew in 1784.
Life
He was the son of Rev John Horsley of Newington Butts and his first wife Anne Hamilton, daughter of Rev Prof William Hamilton of Edinburgh and Mary Robertson.
Entering Trinity Hall, Cambridge in 1751, he became LL.B. in 1758 without graduating in arts. In the following year he succeeded his father in the living of Newington Butts in Surrey. In 1768 he attended the son and heir of the 3rd Earl of Aylesford to Oxford as private tutor; and, after receiving through the earl and Bishop of London various minor preferments, which by dispensations he combined with his first living, he was installed in 1781 as archdeacon of St Albans.
Horsley now entered his controversy with Joseph Priestley, who denied that the early Christians held the doctrine of the Trinity. In this fierce debate, Horsley's aim was to lessen the influence which Priestley's name gave to his views, by pointing to (what he claimed were) inaccuracies in his scholarship. Horsley was rewarded by Lord Chancellor Thurlow with a prebendal stall at Gloucester; and in 1788 Thurlow procured his promotion to the see of St David's.
As a bishop, Horsley was active both in his diocese, and in parliament. The effective support which he afforded the government was acknowledged by his successive translations to Rochester in 1793, and to St Asaph in 1802. With the see of Rochester he held the deanery of Westminster.
Family
He married firstly Mary Botham (died 1777), daughter of John Botham, Rector of Albury, Hertfordshire, and secondly Sarah Wright, who died in 1805. Sarah had been a servant of his first wife, but her elegant manners impressed Queen Charlotte when she was presented at Court. By Mary, he had one surviving son, the Rev Heneage Horsley, and a daughter who died young.
He died at Brighton in 1806, and was buried in St Mary's Church, Newington Butts. He died heavily in debt, due largely it was said to his generous and charitable nature.
His granddaughter Harriet Horsley married Robert Jebb QC and had numerous distinguished descendants, including Richard Claverhouse Jebb.
Works
Besides the controversial Tracts, which appeared in 1783–1785, 1786, and were republished in 1789 and 1812, Horsley's more important works are:
Apollonii Pergaei inclinationum libri duo (1770)
Remarks on the Observations ... for determining the acceleration of the Pendulum in Lat. 7o 51''' (1774)Isaaci Newtoni Opera quae extant Omnia, with a commentary (5 vols 4to, 1779–1785)On the Incarnation. A Sermon, Preached in the Parish Church of St. Mary Newington, in Surrey, Dec. 25, 1785 A Sermon Preached in the Cathedral Church of St. Paul (
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https://en.wikipedia.org/wiki/Science%20Foundation%20Ireland
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Science Foundation Ireland (SFI; ) is the statutory body in Ireland with responsibility for funding oriented basic and applied research in the areas of science, technology, engineering and mathematics (STEM) with a strategic focus. The agency was established in 2003 under the Industrial Development (Science Foundation Ireland) Act 2003 and is run by a board appointed by the Minister for Further and Higher Education, Research, Innovation and Science. SFI is an agency of the Department of Further and Higher Education, Research, Innovation and Science.
Organisation
Remit
Science Foundation Ireland (SFI) is the national foundation for investment in scientific and engineering research. SFI invests in academic researchers and research teams who are most likely to generate new knowledge, leading edge technologies and competitive enterprises in the fields of science, technology, engineering and maths (STEM).
The foundation also promotes and supports the study of, education in, and engagement with STEM and promotes an awareness and understanding of the value of STEM to society and, in particular, to the growth of the economy.
SFI makes grants based upon the merit review of distinguished scientists.
SFI also facilitates co-operative efforts among education, government, and industry that support its fields of emphasis and promotes Ireland's ensuing achievements around the world.
When applying to SFI, applicants will be asked to justify the alignment of their research with Call- or Programme-specific themes and/or they will be required to describe the alignment of their research or activities with SFI's legal remit, as outlined below. Eligible research areas, or themes, may vary according to the scope and objectives of an individual programme and are described in the relevant call documentation. Details of individual programmes are included in the SFI Annual Plan.
Structure
Chairpersons of SFI
Directors General of SFI
Graham Love acted as interim Director General of SFI from the departure of Frank Gannon until the appointment of Mark Ferguson.
Agenda 2020
Agenda 2020 is a strategic plan to position Ireland as a global knowledge leader, a society with scientific and engineering research at its core, driving economic, social and cultural development. The plan was devised to build on the early investment in Ireland's scientific and enterprise communities since Science Foundation Ireland (SFI) was established and to set out goals for further development and growth by 2020.
It has four primary objectives:
To be the best science funding agency in the world at creating impact from excellent research and demonstrating clear value for money invested.
To be the exemplar in building partnerships that fund excellent science and drive it out into the market and society.
To have the most engaged and scientifically informed public
To represent the ideal modern public service organisation, staffed in a lean and flexible manner, with efficient and effective ma
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https://en.wikipedia.org/wiki/CAMS
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CAMS or cams may refer to:
Organizations
Chinese Academy of Medical Sciences
California Academy of Mathematics and Science, a high school in Carson, California, US
Calexico Mission School, a Seventh-day Adventist Church school, California, US
Center for Advanced Media Studies, Johns Hopkins University
Chantiers Aéro-Maritimes de la Seine, a French aircraft manufacturer of the 1920s and 1930s
Coalition Against Militarism in Our Schools in the United States
Copernicus Atmosphere Monitoring Service (CAMS)
Motorsport Australia, formerly the Confederation of Australian Motor Sport, the national sporting organisation vested with the authority to conduct motor sport in Australia by the FIA
Cameras for All-Sky Meteor Surveillance, a project from the SETI Institute that tracks meteor showers globally
Other uses
Camshafts, which can be found on Internal combustion engines
Spring-loaded camming device or cams, a piece of rock climbing or mountaineering protection equipment
Cell adhesion molecules
See also
Cam (disambiguation)
Child and Adolescent Mental Health Services (CAMHS)
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https://en.wikipedia.org/wiki/Linkage%20%28mechanical%29
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A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain.
Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph.
The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the degrees of freedom (DOF) of the joint.
Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement. The ratio of the output force to the input force is known as the mechanical advantage of the linkage, while the ratio of the input speed to the output speed is known as the speed ratio. The speed ratio and mechanical advantage are defined so they yield the same number in an ideal linkage.
A kinematic chain, in which one link is fixed or stationary, is called a mechanism, and a linkage designed to be stationary is called a structure.
History
Archimedes applied geometry to the study of the lever. Into the 1500s the work of Archimedes and Hero of Alexandria were the primary sources of machine theory. It was Leonardo da Vinci who brought an inventive energy to machines and mechanism.
In the mid-1700s the steam engine was of growing importance, and James Watt realized that efficiency could be increased by using different cylinders for expansion and condensation of the steam. This drove his search for a linkage that could transform rotation of a crank into a linear slide, and resulted in his discovery of what is called Watt's linkage. This led to the study of linkages that could generate straight lines, even if only approximately; and inspired the mathematician J. J. Sylvester, who lectured on the Peaucellier linkage, which generates an exact straight line from a rotating crank.
The work of Sylvester inspired A. B. Kempe, who showed that linkages for addition and multiplication could be assembled into a system that traced a given algebraic curve. Kempe's design procedure has inspired research at the intersection of geometry and computer science.
In the late 1800s F. Reuleaux, A. B. W. Kennedy, and L. Burmester formalized the analysis and synthesis of linkage systems using descriptive geometry, and P. L. Chebyshev introduced analytical techniques for the study and invention of linkages.
In the mid-1900s F. Freudenstein and G. N. Sandor used the newly developed digital computer to solve the
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https://en.wikipedia.org/wiki/Brianchon%27s%20theorem
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In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864).
Formal statement
Let be a hexagon formed by six tangent lines of a conic section. Then lines (extended diagonals each connecting opposite vertices) intersect at a single point , the Brianchon point.
Connection to Pascal's theorem
The polar reciprocal and projective dual of this theorem give Pascal's theorem.
Degenerations
As for Pascal's theorem there exist degenerations for Brianchon's theorem, too: Let coincide two neighbored tangents. Their point of intersection becomes a point of the conic. In the diagram three pairs of neighbored tangents coincide. This procedure results in a statement on inellipses of triangles. From a projective point of view the two triangles and lie perspectively with center . That means there exists a central collineation, which maps the one onto the other triangle. But only in special cases this collineation is an affine scaling. For example for a Steiner inellipse, where the Brianchon point is the centroid.
In the affine plane
Brianchon's theorem is true in both the affine plane and the real projective plane. However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane. Consider, for example, five tangent lines to a parabola. These may be considered sides of a hexagon whose sixth side is the line at infinity, but there is no line at infinity in the affine plane. In two instances, a line from a (non-existent) vertex to the opposite vertex would be a line parallel to one of the five tangent lines. Brianchon's theorem stated only for the affine plane would therefore have to be stated differently in such a situation.
The projective dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane.
Proof
Brianchon's theorem can be proved by the idea of radical axis or reciprocation.
To prove it take an arbitrary length (MN) and carry it on the tangents starting from the contact points: PL = RJ = QH = MN etc. Draw circles a, b, c tangent to opposite sides of the hexagon at the created points (H,W), (J,V) and (L,Y) respectively. One sees easily that the concurring lines coincide with the radical axes ab, bc, ca resepectively, of the three circles taken in pairs. Thus O coincides with the radical center of these three circles.
The theorem takes particular forms in the case of circumscriptible pentagons e.g. when R and Q tend to coincide with F, a case where AFE is transformed to the tangent at F. Then, taking a further similar identification of points T,C and U, we obtain a corresponding theorem for quadrangles.
See also
Seven circles theorem
Pascal's theorem
References
Conic sections
Theorems in projective geometry
Euclidean plane geometry
Theorems about
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https://en.wikipedia.org/wiki/Positive%20and%20negative%20predictive%20values
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The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV describe the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic. The PPV and NPV are not intrinsic to the test (as true positive rate and true negative rate are); they depend also on the prevalence. Both PPV and NPV can be derived using Bayes' theorem.
Although sometimes used synonymously, a positive predictive value generally refers to what is established by control groups, while a post-test probability refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the positive predictive value, the two are numerically equal.
In information retrieval, the PPV statistic is often called the precision.
Definition
Positive predictive value (PPV)
The positive predictive value (PPV), or precision, is defined as
where a "true positive" is the event that the test makes a positive prediction, and the subject has a positive result under the gold standard, and a "false positive" is the event that the test makes a positive prediction, and the subject has a negative result under the gold standard. The ideal value of the PPV, with a perfect test, is 1 (100%), and the worst possible value would be zero.
The PPV can also be computed from sensitivity, specificity, and the prevalence of the condition:
cf. Bayes' theorem
The complement of the PPV is the false discovery rate (FDR):
Negative predictive value (NPV)
The negative predictive value is defined as:
where a "true negative" is the event that the test makes a negative prediction, and the subject has a negative result under the gold standard, and a "false negative" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard. With a perfect test, one which returns no false negatives, the value of the NPV is 1 (100%), and with a test which returns no true negatives the NPV value is zero.
The NPV can also be computed from sensitivity, specificity, and prevalence:
The complement of the NPV is the (FOR):
Although sometimes used synonymously, a negative predictive value generally refers to what is established by control groups, while a negative post-test probability rather refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the negative predictive value, then the two are numerically equal.
Relationship
The following diagram illustrates how the positive predictive value, negative predictive value, sensitivity, and specificity are related.
Note that the positive and negat
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https://en.wikipedia.org/wiki/Pushforward%20%28differential%29
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In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of at to the tangent space of at , . Hence it can be used to push tangent vectors on forward to tangent vectors on . The differential of a map is also called, by various authors, the derivative or total derivative of .
Motivation
Let be a smooth map from an open subset of to an open subset of . For any point in , the Jacobian of at (with respect to the standard coordinates) is the matrix representation of the total derivative of at , which is a linear map
between their tangent spaces. Note the tangent spaces are isomorphic to and , respectively. The pushforward generalizes this construction to the case that is a smooth function between any smooth manifolds and .
The differential of a smooth map
Let be a smooth map of smooth manifolds. Given the differential of at is a linear map
from the tangent space of at to the tangent space of at The image of a tangent vector under is sometimes called the pushforward of by The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).
If tangent vectors are defined as equivalence classes of the curves for which then the differential is given by
Here, is a curve in with and is tangent vector to the curve at In other words, the pushforward of the tangent vector to the curve at is the tangent vector to the curve at
Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by
for an arbitrary function and an arbitrary derivation at point (a derivation is defined as a linear map that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of is in and therefore itself is a derivation, .
After choosing two charts around and around is locally determined by a smooth map between open sets of and , and
in the Einstein summation notation, where the partial derivatives are evaluated at the point in corresponding to in the given chart.
Extending by linearity gives the following matrix
Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from to . In general, the differential need not be invertible. However, if is a local diffeomorphism, then is invertible, and the inverse gives the pullback of
The differential is frequently expressed using a variety of othe
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https://en.wikipedia.org/wiki/Hipodil
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Hipodil ( ) was a Bulgarian rock band, founded in the late 1980s in Sofia by four classmates from the local Mathematics High School.
Hipodil's popularity was based in large on their aggressive, sarcastic, sometimes vulgar and explicit but yet humorous lyrics. Because of that Hipodil were known as a "scandalous and rebellious" band. Their main goal was to entertain the listener and themselves; most of the lyrics concerned topics such as alcohol, sex, women, the status quo, etc.; they frequently ridiculed famous Bulgarian and non-Bulgarian people and even politicians (and sometimes songs). Nevertheless, the band had some pretty serious tracks with complex and socially oriented lyrics. All the lyrics were in Bulgarian and were written by vocalist Svetoslav Vitkov (Svetlyo). Most of the music was composed by guitarist Petar Todorov (Pesho). In a great number of songs, guest musicians took part, adding mainly to the brass section.
History
The four classmates from the Sofia's Mathematics High School Nikola Kavaldjiev, Miroslav Tellalov, Nikolay Savov and Petar Todorov formed in 1988 a punk band which played only their own songs in Bulgarian. The first public performance of the band was at Sofia's Summer Theatre, an open-air stage in the largest city park, where the band performed the song "Zidaromazachi" (Wallplasterers), a parody of the ruling communist regime, which got them into minor trouble with the authorities.
In 1992, after a couple of line-up changes and recruiting the new vocalist Svetoslav Vitkov, Hipodil recorded the songs "Bira s vodka" ("Beer With Vodka"), "Bira" ("Beer"), "Chift ochi" ("A Pair of Eyes", "Himna" ("The Anthem"), "Jenata" ("The Woman") and "Klitoren orgasm" ("Clitoral Orgasm"). Next year, the band released their first studio album called Alkoholen delirium (Alcoholic Delirium). The band immediately launched a self-bankrolled national tour, which turned quite successful. At one of the tour concerts in Varna, a mass alcohol-fuelled disorder erupted and all band members and some of the audience were arrested. This event, along with the explicit lyrics of most of their songs gave Hipodil their "scandal aura".
The Varna incident inspired a song - "Hipodili" - which Hipodil put in their next album, Nekuf ujas, nekuf at (Some Kind of Horror, Some Kind of Hell), recorded and released in 1994. This song became a Hipodil anthem and a concert favorite. The album release was followed by a national tour with some 20 dates across the country.
In their fourth album Nadurveni vuglishta (Horny Charcoal), released in 1998, the band showed obvious growth in terms of music and production and experimented with other styles, mostly ska. This album was the most commercially successful album of the band and the first to be released on a CD. It was also the most controversial as some of the lyrics provoked Bulgarian Ministry of Culture to mull legal actions against the band for obscenity. The ministry later dropped its plans but the dispute
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https://en.wikipedia.org/wiki/Guido%20Castelnuovo
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Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also significant.
Life
Early life
Castelnuovo was born in Venice. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy. His mother Emma Levi was a relative of Cesare Lombroso and David Levi. His wife Elbina Marianna Enriques was the sister of mathematician Federigo Enriques and zoologist Paolo Enriques.
After attending a grammar school at in Venice, he went to the University of Padua, from where he graduated in 1886. At the University of Padua he was taught by Giuseppe Veronese. He also achieved minor fame due to winning the university salsa dancing competition. After his graduation, he sent one of his papers to Corrado Segre, whose replies he found remarkably helpful. It marked the beginning of a long period of collaboration.
Career
Castelnuovo spent one year in Rome to research advanced geometry. After that, he was appointed as an assistant of Enrico D'Ovidio at the University of Turin, where he was strongly influenced by Corrado Segre. Here he worked with Alexander von Brill and Max Noether. In 1891 he moved back to Rome to work at the chair of Analytic and Projective Geometry. Here he was a colleague of Luigi Cremona, his former teacher, and took over his job when he later died in 1903. He also founded the University of Rome's School of Statistics and Actuarial Sciences (1927). He influenced a younger generation of Italian mathematicians and statisticians, including Corrado Gini and Francesco Paolo Cantelli.
Retirement and World War II
Castelnuovo retired from teaching in 1935. It was a period of great political difficulty in Italy. In 1922 Benito Mussolini had risen to power and in 1938 a large number of anti-semitic laws were declared, which excluded him, like all other Jews, from public work. With the rise of Nazism, he was forced into hiding. However, during World War II, he organised and taught secret courses for Jewish students — the latter were not allowed to attend university either.
Final years and death
After the liberation of Rome, Castelnuovo was appointed as a special commissioner of the Consiglio Nazionale delle Ricerche in June 1944. He was given the task of repairing the damage done to Italian scientific institutions by the twenty years of Mussolini's rule. He became president of the Accademia dei Lincei until his death and was elected a member of the Académie des Sciences in Paris. On 5 December 1949, he became a life senator of the Italian Republic.
Castelnuovo died at the age of 86 on 27 April 1952 in Rome. He is buried in the Verano cemetery, in Rome, together with his wife, Elbina Enriques Castelnuovo and his mathematician daughter, Emma Castelnuovo.
Work
In Turin Castelnuovo was strongly influenced by Corrado Segre. In this period he publ
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https://en.wikipedia.org/wiki/ASMS
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ASMS may stand for:
American Society for Mass Spectrometry, a professional society, as well as the society's annual meeting
Arkansas School for Mathematics, Sciences, and the Arts
Association of Salaried Medical Specialists, a New Zealand trade union.
Australian Science and Mathematics School on the campus of Flinders University.
Annie Sullivan Middle School Franklin, Massachusetts
Advanced Surface Missile System – US naval combat system, predecessor of Aegis.
Alternative School for Math and Science in Corning, NY
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https://en.wikipedia.org/wiki/Ronald%20Coifman
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Ronald Raphael Coifman is the Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata.
Coifman is a member of the American Academy of Arts and Sciences, the Connecticut Academy of Science and Engineering, and the National Academy of Sciences. He is a recipient of the 1996 DARPA Sustained Excellence Award, the 1996 Connecticut Science Medal, the 1999 Pioneer Award of the International Society for Industrial and Applied Science, and the 1999 National Medal of Science.
In 2013, he co-founded ThetaRay, a cyber security and big data analytics company.
In 2018, he received the Rolf Schock Prize for Mathematics. In 2024 he will be awarded the George David Birkhoff Prize.
References
External links
Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It, Wired
Members of the United States National Academy of Sciences
Living people
20th-century American mathematicians
21st-century American mathematicians
Israeli Jews
Israeli mathematicians
Yale University faculty
National Medal of Science laureates
1941 births
Israeli emigrants to the United States
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https://en.wikipedia.org/wiki/Canadian%20Global%20Almanac
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The Canadian Global Almanac is a Canadian reference book containing a large collection of facts and statistics. It grew out of the American World Almanac and Book of Facts when in 1986 an all-Canadian version was published, edited by John Filion and published by Susan Yates. John Robert Columbo later became its editor. While it was being published, a new edition was released each year in November. The almanac has not been published since 2005.
Almanacs
Canadian non-fiction books
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https://en.wikipedia.org/wiki/Arthur%20Wieferich
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Arthur Josef Alwin Wieferich (April 27, 1884 – September 15, 1954) was a German mathematician and teacher, remembered for his work on number theory, as exemplified by a type of prime numbers named after him.
He was born in Münster, attended the University of Münster (1903–1909) and then worked as a school teacher and tutor until his retirement in 1949. He married in 1916 and had no children.
Wieferich abandoned his studies after his graduation and did not publish any paper after 1909. His mathematical reputation is founded on five papers he published while a student at Münster:
.
.
.
.
.
The first three papers are related to Waring's problem. His fourth paper led to the term Wieferich prime, which are p such that p^2 divides 2^(p-1) - 1.
See also
Wieferich pair
Wieferich's theorem
Wieferich prime
External links
Arthur Josef Alwin Wieferich, obituary at numbertheory.org
References
1884 births
1954 deaths
20th-century German mathematicians
Number theorists
People from Münster
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https://en.wikipedia.org/wiki/Principles%20and%20Standards%20for%20School%20Mathematics
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Principles and Standards for School Mathematics (PSSM) are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada. It is the primary model for standards-based mathematics.
The NCTM employed a consensus process that involved classroom teachers, mathematicians, and educational researchers. The resulting document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe NCTM's recommended framework for mathematics programs, and ten general strands or standards that cut across the school mathematics curriculum. These strands are divided into mathematics content (Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability) and processes (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation). Specific expectations for student learning are described for ranges of grades (preschool to 2, 3 to 5, 6 to 8, and 9 to 12).
Origins
The Principles and Standards for School Mathematics was developed by the NCTM. The NCTM's stated intent was to improve mathematics education. The contents were based on surveys of existing curriculum materials, curricula and policies from many countries, educational research publications, and government agencies such as the U.S. National Science Foundation. The original draft was widely reviewed at the end of 1998 and revised in response to hundreds of suggestions from teachers.
The PSSM is intended to be "a single resource that can be used to improve mathematics curricula, teaching, and assessment." The latest update was published in 2000. The PSSM is available as a book, and in hypertext format on the NCTM web site.
The PSSM replaces three prior publications by NCTM:
Curriculum and Evaluation Standards for School Mathematics (1989), which was the first such publication by an independent professional organization instead of a government agency and outlined what students should learn and how to measure their learning.
Professional Standards for Teaching Mathematics (1991), which added information about best practices for teaching mathematics.
Assessment Standards for School Mathematics (1995), which focused on the use of accurate assessment methods.
Six principles
Equity: The NCTM standards for equity, as outlined in the PSSM, encourage equal access to mathematics for all students, "especially students who are poor, not native speakers of English, disabled, female, or members of minority groups." The PSSM makes explicit the goal that all students should learn higher level mathematics, particularly underserved groups such as minorities and women. This principle encourages provision of extra help to students who are struggling and advocates high expectations and excellent teaching for all students.
Curri
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https://en.wikipedia.org/wiki/National%20Council%20of%20Teachers%20of%20Mathematics
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Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds annual national and regional conferences for teachers and publishes five journals.
Journals
NCTM publishes five official journals. All are available in print and online versions.
Teaching Children Mathematics supports improvement of pre-K–6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics idea, activities, and pedagogical strategies, and or sharing and interpreting research.
Mathematics Teaching in the Middle School supports the improvement of grade 5–9 mathematics education by serving as a resource for practicing and prospective teachers, as well as supervisors and teacher educators. It is a forum for the exchange of mathematics idea, activities, and pedagogical strategies, and or sharing and interpreting research.
Mathematics Teacher is devoted to improving mathematics instruction for grades 8–14 and supporting teacher education programs. It provides a forum for sharing activities and pedagogical strategies, deepening understanding of mathematical ideas, and linking mathematical education research to practice.
Mathematics Teacher Educator, published jointly with the Association of Mathematics Teacher Educators, contributes to building a professional knowledge base for mathematics teacher educators that stems from, develops, and strengthens practitioner knowledge. The journal provides a means for practitioner knowledge related to the preparation and support of teachers of mathematics to be not only public, shared, and stored, but also verified and improved over time (Hiebert, Gallimore, and Stigler 2002).
NCTM does not conduct research in mathematics education, but it does publish the Journal for Research in Mathematics Education (JRME). JRME is devoted to the interests of teachers of mathematics and mathematics education at all levels—preschool through adult. JRME is a forum for disciplined inquiry into the teaching and learning of mathematics. The editors encourage the submission of a variety of manuscripts: reports of research, including experiments, case studies, surveys, philosophical studies, and historical studies; articles about research, including literature reviews and theoretical analyses; brief reports of research; critiques of articles and books; and brief commentaries on issues pertaining to research.
NCTM Standards
NCTM has published a series of math Standards outlining a vision for school mathematics in the USA and Canada. In 1989, NCTM developed the Curriculum and Evaluation Standards for School Mathematics, followed by the Professional Standards for Teaching Mathematics (1991) and the Assessment Standards for School Mathematics (1995). Education officials lauded
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https://en.wikipedia.org/wiki/Heesch%27s%20problem
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In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it with no overlaps and no gaps. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 (the union of a square, equilateral triangle, and 30-60-90 right triangle) and proposed the more general problem.
For example, a square may be surrounded by infinitely many layers of congruent squares in the square tiling, while a circle cannot be surrounded by even a single layer of congruent circles without leaving some gaps. The Heesch number of the square is infinite and the Heesch number of the circle is zero. In more complicated examples, such as the one shown in the illustration, a polygonal tile can be surrounded by several layers, but not by infinitely many; the maximum number of layers is the tile's Heesch number.
Formal definitions
A tessellation of the plane is a partition of the plane into smaller regions called tiles. The zeroth corona of a tile is defined as the tile itself, and for k > 0 the kth corona is the set of tiles sharing a boundary point with the (k − 1)th corona. The Heesch number of a figure S is the maximum value k such that there exists a tiling of the plane, and tile t within that tiling, for which that all tiles in the zeroth through kth coronas of t are congruent to S. In some work on this problem this definition is modified to additionally require that the union of the zeroth through kth coronas of t is a simply connected region.
If there is no upper bound on the number of layers by which a tile may be surrounded, its Heesch number is said to be infinite. In this case, an argument based on Kőnig's lemma can be used to show that there exists a tessellation of the whole plane by congruent copies of the tile.
Example
Consider the non-convex polygon P shown in the figure to the right, which is formed from a regular hexagon by adding projections on two of its sides and matching indentations on three sides. The figure shows a tessellation consisting of 61 copies of P, one large infinite region, and four small diamond-shaped polygons within the fourth layer. The first through fourth coronas of the central polygon consist entirely of congruent copies of P, so its Heesch number is at least four. One cannot rearrange the copies of the polygon in this figure to avoid creating the small diamond-shaped polygons, because the 61 copies of P have too many indentations relative to the number of projections that could fill them. By formalizing this argument, one can prove that the Heesch number of P is exactly four. According to the modified definition that requires that coronas be simply connected, the Heesch number is three. This example was discovered by Robert Ammann.
Known results
It is unknown whether all positive integers can be Heesch numbers. The first examples of polygons with Heesch num
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https://en.wikipedia.org/wiki/Contorsion%20tensor
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The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) 11-dimensional supergravity. That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role.
The elimination of torsion in a connection is referred to as the absorption of torsion, and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures.
Definition in metric geometry
In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol and the unique torsion-free Levi-Civita connection for the same metric.
The contorsion tensor is defined in terms of the torsion tensor as (up to a sign, see below)
where the indices are being raised and lowered with respect to the metric:
.
The reason for the non-obvious sum in the definition of the contorsion tensor is due to the sum-sum difference that enforces metric compatibility. The contorsion tensor is antisymmetric in the first two indices, whilst the torsion tensor itself is antisymmetric in its last two indices; this is shown below.
The full metric compatible affine connection can be written as:
Where the torsion-free Levi-Civita connection:
Definition in affine geometry
In affine geometry, one does not have a metric nor a metric connection, and so one is not free to raise and lower indices on demand. One can still achieve a similar effect by making use of the solder form, allowing the bundle to be related to what is happening on its base space. This is an explicitly geometric viewpoint, with tensors now being geometric objects in the vertical and horizontal bundles of a fiber bundle, instead of being indexed algebraic objects defined only on the base space. In this case, one may construct a contorsion tensor, living as a one-form on the tangent bundle.
Recall that the torsion of a connection can be expressed as
where is the solder form (tautological one-form). The subscript serves only as a reminder that this torsion tensor was obtained from the connection.
By analogy to the lowering of the index on torsion tensor on the section above, one can perform a similar operation with the solder form, and construct a tensor
Here is the scalar product. This tensor can be expressed as
The quantity is the contorsion form and is exactly what is needed to add to an arbitrary connection to get the torsion-free Levi-Civita connection. That is, given an Ehresmann connection , there is another connection that is torsion-free.
The vanishing of the torsion is th
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https://en.wikipedia.org/wiki/Martin%20Nowak
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Martin Andreas Nowak (born April 7, 1965) is an Austrian-born professor of mathematics and biology at Harvard University. He is one of the leading researchers in the field of mathematical biology. He made contributions to the theory of evolution, cooperation, virus dynamics, and cancer dynamics. Nowak held professorships at Oxford University and at the Institute for Advanced Study, Princeton, before being recruited by Harvard in 2003. He was the director of Harvard's program for evolutionary dynamics from 2003 until 2020. He is a professor in the Department of Mathematics and in the Department of Organismic and Evolutionary Biology.
Nowak has authored more than 500 academic papers and has been cited more than 140,000 times. In addition, Nowak has authored four books, who have received critical praise. Nowak's best known work outside of academia is his 2011 book SuperCooperators: Altruism, Evolution and Why We Need Each Other to Succeed. Another work, Evolution, Games, and God, explores the interplay between theology and evolutionary theology. Nowak, a Roman Catholic, frequently lectures about religion and was co-director with Sarah Coakley of the Evolution and Theology of Cooperation project at Harvard University.
Early life and education
Nowak was born April 7, 1965 in Vienna, Austria. He studied at Albertus Magnus Gymnasium and the University of Vienna, earning a doctorate in biochemistry and mathematics in 1989. He worked with Peter Schuster on quasi-species theory and with Karl Sigmund on evolution of cooperation. Nowak received the highest Austrian honors (Sub auspiciis Praesidentis) when awarded his degree.
Career
From 1989 to 1998, Nowak worked at the University of Oxford with Robert May as an Erwin Schrödinger postdoctoral Scholar and Wellcome Trust Senior Research Fellow. From 1997 to 1998, Nowak was a professor of mathematical biology. After 1998, he conducted research at the Institute for Advanced Study at Princeton and established a program in theoretical biology.
In 2003, Nowak was recruited to Harvard University as Professor of Mathematics and Biology. When Harvard received a large donation for the founding of the Program for Evolutionary Dynamics (PED), he was appointed its director. Scientific American reported that Nowak's team received US$6.5 million initially, with nothing released to him after 2007, a couple of hundred thousand dollars remained unspent.
Nowak has authored books and scientific papers on topics in evolutionary game theory, cancer, viruses, infectious disease, the evolution of language, and the evolution of cooperation.
His first book, Virus Dynamics (written with Robert May) was published by Oxford University Press in 2001. Nowak is a corresponding member of the Austrian Academy of Sciences. He won the Weldon Memorial Prize, the Albert Wander Prize, the Akira Okubo Prize, the David Starr Jordan Prize and the Henry Dale Prize. Nowak's 2006 book Evolutionary Dynamics: Exploring the Equations of Life earne
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https://en.wikipedia.org/wiki/Disjoint%20union%20%28topology%29
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In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.
The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
Definition
Let {Xi : i ∈ I} be a family of topological spaces indexed by I. Let
be the disjoint union of the underlying sets. For each i in I, let
be the canonical injection (defined by ). The disjoint union topology on X is defined as the finest topology on X for which all the canonical injections are continuous (i.e.: it is the final topology on X induced by the canonical injections).
Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage is open in Xi for each i ∈ I. Yet another formulation is that a subset V of X is open relative to X iff its intersection with Xi is open relative to Xi for each i.
Properties
The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and fi : Xi → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute:
This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff fi = f o φi is continuous for all i in I.
In addition to being continuous, the canonical injections φi : Xi → X are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X.
Examples
If each Xi is homeomorphic to a fixed space A, then the disjoint union X is homeomorphic to the product space A × I where I has the discrete topology.
Preservation of topological properties
Every disjoint union of discrete spaces is discrete
Separation
Every disjoint union of T0 spaces is T0
Every disjoint union of T1 spaces is T1
Every disjoint union of Hausdorff spaces is Hausdorff
Connectedness
The disjoint union of two or more nonempty topological spaces is disconnected
See also
product topology, the dual construction
subspace topology and its dual quotient topology
topological union, a generalization to the case where the pieces are not disjoint
General topology
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https://en.wikipedia.org/wiki/De%20Bruijn%20sequence
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In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a contiguous subsequence). Such a sequence is denoted by and has length , which is also the number of distinct strings of length n on A. Each of these distinct strings, when taken as a substring of , must start at a different position, because substrings starting at the same position are not distinct. Therefore, must have at least symbols. And since has exactly symbols, de Bruijn sequences are optimally short with respect to the property of containing every string of length n at least once.
The number of distinct de Bruijn sequences is
The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn, who wrote about them in 1946. As he later wrote, the existence of de Bruijn sequences for each order together with the above properties were first proved, for the case of alphabets with two elements, by . The generalization to larger alphabets is due to . Automata for recognizing these sequences are denoted as de Bruijn automata.
In most applications, A = {0,1}.
History
The earliest known example of a de Bruijn sequence comes from Sanskrit prosody where, since the work of Pingala, each possible three-syllable pattern of long and short syllables is given a name, such as 'y' for short–long–long and 'm' for long–long–long. To remember these names, the mnemonic yamātārājabhānasalagām is used, in which each three-syllable pattern occurs starting at its name: 'yamātā' has a short–long–long pattern, 'mātārā' has a long–long–long pattern, and so on, until 'salagām' which has a short–short–long pattern. This mnemonic, equivalent to a de Bruijn sequence on binary 3-tuples, is of unknown antiquity, but is at least as old as Charles Philip Brown's 1869 book on Sanskrit prosody that mentions it and considers it "an ancient line, written by Pāṇini".
In 1894, A. de Rivière raised the question in an issue of the French problem journal L'Intermédiaire des Mathématiciens, of the existence of a circular arrangement of zeroes and ones of size that contains all binary sequences of length . The problem was solved (in the affirmative), along with the count of distinct solutions, by Camille Flye Sainte-Marie in the same year. This was largely forgotten, and proved the existence of such cycles for general alphabet size in place of 2, with an algorithm for constructing them. Finally, when in 1944 Kees Posthumus conjectured the count for binary sequences, de Bruijn proved the conjecture in 1946, through which the problem became well-known.
Karl Popper independently describes these objects in his The Logic of Scientific Discovery (1934), calling them "shortest random-like sequences".
Examples
Taking A = {0, 1}, there are two distinct B(2, 3): 00010111 and 11101000, one being the reverse or negation of the other.
Two of the 16 possible B(2, 4) in the
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https://en.wikipedia.org/wiki/Estimation%20theory
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Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.
In estimation theory, two approaches are generally considered:
The probabilistic approach (described in this article) assumes that the measured data is random with probability distribution dependent on the parameters of interest
The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.
Examples
For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters. Alternatively, it is desired to estimate the probability of a voter voting for a particular candidate, based on some demographic features, such as age.
Or, for example, in radar the aim is to find the range of objects (airplanes, boats, etc.) by analyzing the two-way transit timing of received echoes of transmitted pulses. Since the reflected pulses are unavoidably embedded in electrical noise, their measured values are randomly distributed, so that the transit time must be estimated.
As another example, in electrical communication theory, the measurements which contain information regarding the parameters of interest are often associated with a noisy signal.
Basics
For a given model, several statistical "ingredients" are needed so the estimator can be implemented. The first is a statistical sample – a set of data points taken from a random vector (RV) of size N. Put into a vector,
Secondly, there are M parameters
whose values are to be estimated. Third, the continuous probability density function (pdf) or its discrete counterpart, the probability mass function (pmf), of the underlying distribution that generated the data must be stated conditional on the values of the parameters:
It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the Bayesian probability
After the model is formed, the goal is to estimate the parameters, with the estimates commonly denoted , where the "hat" indicates the estimate.
One common estimator is the minimum mean squared error (MMSE) estimator, which utilizes the error between the estimated parameters and the actual value of the parameters
as the basis for optimality. This error term is then squared and the expected value of this squared value is minimized for the MMSE estimator.
Estimators
Commonly used estimators (estimation methods) and topics related to them include:
Maximum likelihood estimators
Bayes estimators
Method of moments estimators
Cramér–Rao b
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https://en.wikipedia.org/wiki/Pi%20Mu%20Epsilon
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Pi Mu Epsilon ( or ) is the U.S. honorary national mathematics society.
The society was founded at Syracuse University on , by Professor Edward Drake Roe, Jr, and currently has chapters at 371 institutions across the US.
Goals
Pi Mu Epsilon is dedicated to the promotion of mathematics and recognition of students who successfully pursue mathematical understanding. To promote mathematics, the National Pi Mu Epsilon Council co-sponsors an annual conference in conjunction with the Mathematical Association of America.
The society also publishes a semi-annual journal, the Pi Mu Epsilon Journal, which both presents research papers particularly focusing on student authored papers, as well as a problem section.
The Richard V. Andree Awards are given by the organization to undergraduates whose articles in the Journal have been judged as containing the best content for the year. Andree served as the editor of the journal, as well as President and Secretary-Treasurer of the organization.
Membership
A person meeting any one of the following four sets of qualifications may be elected to membership by a chapter. This election shall be irrespective of sex, religion, race, or national origin:
Undergraduate students who have completed at least the equivalent of two semesters of calculus and two additional courses in mathematics, at or above the calculus level, all of which lead to the fulfillment of the requirements for a major in the mathematical sciences. In addition, such students must have maintained a grade point average equivalent to that of at least 3.0 on a 4-point scale, both for all courses that lead to fulfillment of requirements for a major in the mathematical sciences, and also for all courses that lead to fulfillment of requirements for an undergraduate degree.
Graduate students whose mathematical work is at least equivalent to that required of qualified undergraduates, and who have maintained at least a B average in mathematics during their last school year prior to their election.
Members of the faculty in mathematics or related subjects.
Any person who has some special distinction in mathematics (e.g. major math publication of importance, Putnam competition winners).
See also
Kappa Mu Epsilon, (mathematics)
Mu Alpha Theta, (mathematics, high school)
Mu Sigma Rho, (statistics)
Professional fraternities and sororities
References
Honor societies
Mathematical societies
Student organizations established in 1914
Student societies in the United States
1914 establishments in New York (state)
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https://en.wikipedia.org/wiki/Approach%20space
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In topology, a branch of mathematics, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989, in a series of papers on approach theory between 1988 and 1995.
Definition
Given a metric space (X, d), or more generally, an extended pseudoquasimetric (which will be abbreviated ∞pq-metric here), one can define an induced map d: X × P(X) → [0,∞] by d(x, A) = inf{d(x, a) : a ∈ A}. With this example in mind, a distance on X is defined to be a map X × P(X) → [0,∞] satisfying for all x in X and A, B ⊆ X,
d(x, {x}) = 0,
d(x, Ø) = ∞,
d(x, A∪B) = min(d(x, A), d(x, B)),
For all 0 ≤ ε ≤ ∞, d(x, A) ≤ d(x, A(ε)) + ε,
where we define A(ε) = {x : d(x, A) ≤ ε}.
(The "empty infimum is positive infinity" convention is like the nullary intersection is everything convention.)
An approach space is defined to be a pair (X, d) where d is a distance function on X. Every approach space has a topology, given by treating A → A(0) as a Kuratowski closure operator.
The appropriate maps between approach spaces are the contractions. A map f: (X, d) → (Y, e) is a contraction if e(f(x), f[A]) ≤ d(x, A) for all x ∈ X and A ⊆ X.
Examples
Every ∞pq-metric space (X, d) can be distanced to (X, d), as described at the beginning of the definition.
Given a set X, the discrete distance is given by d(x, A) = 0 if x ∈ A and d(x, A) = ∞ if x ∉ A. The induced topology is the discrete topology.
Given a set X, the indiscrete distance is given by d(x, A) = 0 if A is non-empty, and d(x, A) = ∞ if A is empty. The induced topology is the indiscrete topology.
Given a topological space X, a topological distance is given by d(x, A) = 0 if x ∈ A, and d(x, A) = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.
Let P = [0, ∞] be the extended non-negative reals. Let d+(x, A) = max(x − sup A, 0) for x ∈ P and A ⊆ P. Given any approach space (X, d), the maps (for each A ⊆ X) d(., A) : (X, d) → (P, d+) are contractions.
On P, let e(x, A) = inf{|x − a| : a ∈ A} for x < ∞, let e(∞, A) = 0 if A is unbounded, and let e(∞, A) = ∞ if A is bounded. Then (P, e) is an approach space. Topologically, P is the one-point compactification of [0, ∞). Note that e extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.
Let βN be the Stone–Čech compactification of the integers. A point U ∈ βN is an ultrafilter on N. A subset A ⊆ βN induces a filter F(A) = ∩ {U : U ∈ A}. Let b(U, A) = sup{ inf{ |n − j| : n ∈ X, j ∈ E } : X ∈ U, E ∈ F(A) }. Then (βN, b) is an approach space that extends the ordinary Euclidean distance on N. In contrast, βN is not metrizable.
Equivalent definitions
Lowen has offered at least seven equivalent formulations. Two of them are below.
Let XPQ(X) denote the set of xpq-metrics on X. A subfamily G of XPQ(X) is called a gauge if
0 ∈ G, where 0 is the zero metr
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https://en.wikipedia.org/wiki/Derived%20set%20%28mathematics%29
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In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
Definition
The derived set of a subset of a topological space denoted by is the set of all points that are limit points of that is, points such that every neighbourhood of contains a point of other than itself.
Examples
If is endowed with its usual Euclidean topology then the derived set of the half-open interval is the closed interval
Consider with the topology (open sets) consisting of the empty set and any subset of that contains 1. The derived set of is
Properties
If and are subsets of the topological space then the derived set has the following properties:
implies
implies
A subset of a topological space is closed precisely when that is, when contains all its limit points. For any subset the set is closed and is the closure of (that is, the set ).
The derived set of a subset of a space need not be closed in general. For example, if with the trivial topology, the set has derived set which is not closed in But the derived set of a closed set is always closed.
In addition, if is a T1 space, the derived set of every subset of is closed in
Two subsets and are separated precisely when they are disjoint and each is disjoint from the other's derived set
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset.
A space is a T1 space if every subset consisting of a single point is closed. In a T1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T1 space). It follows that in T1 spaces, the derived set of any finite set is empty and furthermore,
for any subset and any point of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points. It can also be shown that in a T1 space, for any subset
A set with is called dense-in-itself and can contain no isolated points. A set with is called a perfect set. Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.
The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.
Topology in terms of derived sets
Because homeomorphisms can be described
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https://en.wikipedia.org/wiki/Elementary%20arithmetic
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Elementary arithmetic is a branch of mathematics involving basic numerical operations, namely addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first critical branch of mathematics to be taught in schools.
Digits
Symbols called digits are used to represent the value of numbers in a numeral system. The most commonly used digits are the Arabic numerals (0 to 9). The Hindu-Arabic numeral system is the most commonly used numeral system, being a positional notation system used to represent numbers using these digits.
Successor function and size
In elementary arithmetic, the successor of a natural number (including zero) is the result of adding one to that number, whereas the predecessor of a natural number (excluding zero) is the result obtained by subtracting one from that number. For example, the successor of zero is one and the predecessor of eleven is ten ( and ). Every natural number has a successor, and all natural numbers (except zero) have a predecessor.
If one number is greater than () another number, then the latter is less than () the first one. For example, three is less than eight (), and eight is greater than three ().
Counting
Counting involves assigning a natural number to each object in a set, starting with one for the first object and increasing by one for each subsequent object. The number of objects in the set is the count which is equal to the highest natural number assigned to an object in the set. This count is also known as the cardinality of the set.
Counting can also be the process of tallying using tally marks, drawing a mark for each object in a set.
In more advanced mathematics, the process of counting can be thought of as constructing a one-to-one correspondence (or bijection), between the elements of a set and the set , where is a natural number, and the size of the set is .
Addition
Addition is a mathematical operation that combines two or more numbers, called addends or summands, to produce the final number, called the sum. The addition of two numbers is expressed using the plus sign "+" and is performed according to the following rules:
The sum of two numbers is equal to the number obtained by adding their individual values.
The order in which the addends are added does not affect the sum. This property is known as the commutative property of addition.
The sum of two numbers is unique, meaning that there is only one correct answer for the sum of any given pair of numbers.
Addition has an inverse operation, called subtraction, which can be used to find the difference between two numbers.
Addition is used in a variety of contexts, including comparing quantities, joining quantities, and measuring. When the sum of a pair of digits results in a two-digit number, the "tens" digit is referred to as the "carry digit" in the addition algorithm. In eleme
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https://en.wikipedia.org/wiki/Sigma-ideal
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In mathematics, particularly measure theory, a -ideal, or sigma ideal, of a σ-algebra (, read "sigma") is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory.
Let be a measurable space (meaning is a -algebra of subsets of ). A subset of is a -ideal if the following properties are satisfied:
;
When and then implies ;
If then
Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of -ideal is dual to that of a countably complete (-) filter.
If a measure is given on the set of -negligible sets ( such that ) is a -ideal.
The notion can be generalized to preorders with a bottom element as follows: is a -ideal of just when
(i')
(ii') implies and
(iii') given a sequence there exists some such that for each
Thus contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being upwards directed.
A -ideal of a set is a -ideal of the power set of That is, when no -algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the -ideal generated by the collection of closed subsets with empty interior.
See also
References
Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
Measure theory
Families of sets
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https://en.wikipedia.org/wiki/Todd%20class
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In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle.
The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem.
History
It is named for J. A. Todd, who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the Todd-Eger class. The general definition in higher dimensions is due to Friedrich Hirzebruch.
Definition
To define the Todd class where is a complex vector bundle on a topological space , it is usually possible to limit the definition to the case of a Whitney sum of line bundles, by means of a general device of characteristic class theory, the use of Chern roots (aka, the splitting principle). For the definition, let
be the formal power series with the property that the coefficient of in is 1, where denotes the -th Bernoulli number. Consider the coefficient of in the product
for any . This is symmetric in the s and homogeneous of weight : so can be expressed as a polynomial in the elementary symmetric functions of the s. Then defines the Todd polynomials: they form a multiplicative sequence with as characteristic power series.
If has the as its Chern roots, then the Todd class
which is to be computed in the cohomology ring of (or in its completion if one wants to consider infinite-dimensional manifolds).
The Todd class can be given explicitly as a formal power series in the Chern classes as follows:
where the cohomology classes are the Chern classes of , and lie in the cohomology group . If is finite-dimensional then most terms vanish and is a polynomial in the Chern classes.
Properties of the Todd class
The Todd class is multiplicative:
Let be the fundamental class of the hyperplane section.
From multiplicativity and the Euler exact sequence for the tangent bundle of
one obtains
Computations of the Todd class
For any algebraic curve the Todd class is just . Since is projective, it can be embedded into some and we can find using the normal sequenceand properties of chern classes. For example, if we have a degree plane curve in , we find the total chern class iswhere is the hyperplane class in restricted to .
Hirzebruch-Riemann-Roch formula
For any coherent sheaf F on a smooth
compact complex manifold M, one has
where is its holomorphic Euler characteristic,
and its Chern characte
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https://en.wikipedia.org/wiki/Half-integer
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In mathematics, a half-integer is a number of the form
where is a whole number. For example,
are all half-integers. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers such as 1 (being half the integer 2). A name such as "integer-plus-half" may be more accurate, but even though not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient.
Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two).
Notation and algebraic structure
The set of all half-integers is often denoted
The integers and half-integers together form a group under the addition operation, which may be denoted
However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. The smallest ring containing them is , the ring of dyadic rationals.
Properties
The sum of half-integers is a half-integer if and only if is odd. This includes since the empty sum 0 is not half-integer.
The negative of a half-integer is a half-integer.
The cardinality of the set of half-integers is equal to that of the integers. This is due to the existence of a bijection from the integers to the half-integers: , where is an integer
Uses
Sphere packing
The densest lattice packing of unit spheres in four dimensions (called the D4 lattice) places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to the Hurwitz integers: quaternions whose real coefficients are either all integers or all half-integers.
Physics
In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.
The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.
Sphere volume
Although the factorial function is defined only for integer arguments, it can be extended to fractional arguments using the gamma function. The gamma function for half-integers is an important part of the formula for the volume of an -dimensional ball of radius ,
The values of the gamma function on half-integers are integer multiples of the square root of pi:
where denotes the double factorial.
References
Rational numbers
Elementary number theory
Parity (mathematics)
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https://en.wikipedia.org/wiki/Unitarian%20trick
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In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some group G is in a qualitative way controlled by that of some other compact group K. An important example is that in which G is the complex general linear group, and K the unitary group acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for those of G, at least in finite dimensions.
The relationship between G and K that drives this connection is traditionally expressed in the terms that the Lie algebra of K is a real form of that of G. In the theory of algebraic groups, the relationship can also be put that K is a dense subset of G, for the Zariski topology.
The trick works for reductive Lie groups, of which an important case are semisimple Lie groups.
Weyl's theorem
The complete reducibility of finite-dimensional linear representations of compact groups, or connected semisimple Lie groups and complex semisimple Lie algebras goes sometimes under the name of Weyl's theorem. A related result, that the universal cover of a compact semisimple Lie group is also compact, also goes by the same name.
History
Adolf Hurwitz had shown how integration over a compact Lie group could be used to construct invariants, in the cases of unitary groups and compact orthogonal groups. Issai Schur in 1924 showed that this technique can be applied to show complete reducibility of representations for such groups via the construction of an invariant inner product. Weyl extended Schur's method to complex semisimple Lie algebras by showing they had a compact real form.
Notes
References
V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups (1999), p. 49.
Wulf Rossmann, Lie groups: an introduction through linear groups (2006), p. 225.
Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants (2009), p. 171.
Representation theory of Lie groups
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https://en.wikipedia.org/wiki/Veronese%20surface
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In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety.
The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a Steiner surface.
Definition
The Veronese surface is the image of the mapping
given by
where denotes homogeneous coordinates. The map is known as the Veronese embedding.
Motivation
The Veronese surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation:
The pairing between coefficients and variables is linear in coefficients and quadratic in the variables; the Veronese map makes it linear in the coefficients and linear in the monomials. Thus for a fixed point the condition that a conic contains the point is a linear equation in the coefficients, which formalizes the statement that "passing through a point imposes a linear condition on conics".
Veronese map
The Veronese map or Veronese variety generalizes this idea to mappings of general degree d in n+1 variables. That is, the Veronese map of degree d is the map
with m given by the multiset coefficient, or more familiarly the binomial coefficient, as:
The map sends to all possible monomials of total degree d (of which there are ); we have since there are variables to choose from; and we subtract since the projective space has coordinates. The second equality shows that for fixed source dimension n, the target dimension is a polynomial in d of degree n and leading coefficient
For low degree, is the trivial constant map to and is the identity map on so d is generally taken to be 2 or more.
One may define the Veronese map in a coordinate-free way, as
where V is any vector space of finite dimension, and are its symmetric powers of degree d. This is homogeneous of degree d under scalar multiplication on V, and therefore passes to a mapping on the underlying projective spaces.
If the vector space V is defined over a field K which does not have characteristic zero, then the definition must be altered to be understood as a mapping to the dual space of polynomials on V. This is because for fields with finite characteristic p, the pth powers of elements of V are not rational normal curves, but are of course a line. (See, for example additive polynomial for a treatment of polynomials over a field of finite characteristic).
Rational normal curve
For the Veronese variety is known as the rational normal curve, of which the lower-degree examples are familiar.
For the Veronese map is simply the identity map on the projective line.
For the Veronese variety is
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https://en.wikipedia.org/wiki/Society%20of%20Mathematicians%2C%20Physicists%20and%20Astronomers%20of%20Slovenia
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The Society of Mathematicians, Physicists and Astronomers of Slovenia (Slovene: Društvo matematikov, fizikov in astronomov Slovenije, DMFA) is the main Slovene society in the field of mathematics, physics and astronomy.
The Society is occupied with pedagogical activity and with the popularization of mathematics, recreational mathematics, physics, astronomy and with organizing competitions at all levels of education.
It takes care of publicistic and editorial activity, where we should mention its gazette Obzornik za matematiko in fiziko (A Review for Mathematics and Physics), a magazine for secondary schools Presek (A Section), literary collection Sigma and other literary editions.
The current president of the Society is Dragan Mihailovic (since 2017) and the vice-president is Nada Razpet.
The DMFA collaborates with the European Mathematical Society (EMS), the European Physical Society (EPS) and many other related societies around the world.
Honourable members
The Society grants an honourable membership to a person or persons, which have contributed significantly to advance of mathematical and natural sciences in Slovenia, and to development of the Society.
External links
DMFA Slovenije
Info at EPS
Mathematical societies
Non-profit organizations based in Slovenia
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https://en.wikipedia.org/wiki/Mathematical%20chemistry
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Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena. Mathematical chemistry has also sometimes been called computer chemistry, but should not be confused with computational chemistry.
Major areas of research in mathematical chemistry include chemical graph theory, which deals with topology such as the mathematical study of isomerism and the development of topological descriptors or indices which find application in quantitative structure-property relationships; and chemical aspects of group theory, which finds applications in stereochemistry and quantum chemistry. Another important area is molecular knot theory and circuit topology that describe the topology of folded linear molecules such as proteins and Nucleic Acids.
The history of the approach may be traced back to the 19th century. Georg Helm published a treatise titled "The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena" in 1894. Some of the more contemporary periodical publications specializing in the field are MATCH Communications in Mathematical and in Computer Chemistry, first published in 1975, and the Journal of Mathematical Chemistry, first published in 1987. In 1986 a series of annual conferences MATH/CHEM/COMP taking place in Dubrovnik was initiated by the late Ante Graovac.
The basic models for mathematical chemistry are molecular graph and topological index.
In 2005 the International Academy of Mathematical Chemistry (IAMC) was founded in Dubrovnik (Croatia) by Milan Randić. The Academy has 82 members (2009) from all over the world, including six scientists awarded with a Nobel Prize.
See also
Bibliography
Molecular Descriptors for Chemoinformatics, by R. Todeschini and V. Consonni, Wiley-VCH, Weinheim, 2009.
Mathematical Chemistry Series, by D. Bonchev, D. H. Rouvray (Eds.), Gordon and Breach Science Publisher, Amsterdam, 2000.
Chemical Graph Theory, by N. Trinajstic, CRC Press, Boca Raton, 1992.
Mathematical Concepts in Organic Chemistry, by I. Gutman, O. E. Polansky, Springer-Verlag, Berlin, 1986.
Chemical Applications of Topology and Graph Theory, ed. by R. B. King, Elsevier, 1983.
Topological approach to the chemistry of conjugated molecules, by A. Graovac, I. Gutman, and N. Trinajstic, Lecture Notes in Chemistry, no.4, Springer-Verlag, Berlin, 1977.
Notes
References
N. Trinajstić, I. Gutman, Mathematical Chemistry, Croatica Chemica Acta, 75(2002), pp. 329–356.
A. T. Balaban, Reflections about Mathematical Chemistry, Foundations of Chemistry, 7(2005), pp. 289–306.
G. Restrepo, J. L. Villaveces, Mathematical Thinking in Chemistry, HYLE, 18(2012), pp. 3–22.
Advances in Mathematical Chemistry and Applications. Volume 2. Basak S. C., Restrepo G., Villaveces J. L. (Bentham Science eBooks, 2015)
External links
Journal of Mathematical Chemistry
MATCH Communications in Mathematical and in Comput
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https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek%20set%20theory%20with%20urelements
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The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set theory. It is considerably weaker than the (relatively) familiar system ZFU. The purpose of allowing urelements is to allow large or high-complexity objects (such as the set of all reals) to be included in the theory's transitive models without disrupting the usual well-ordering and recursion-theoretic properties of the constructible universe; KP is so weak that this is hard to do by traditional means.
Preliminaries
The usual way of stating the axioms presumes a two sorted first order language with a single binary relation symbol .
Letters of the sort designate urelements, of which there may be none, whereas letters of the sort designate sets. The letters may denote both sets and urelements.
The letters for sets may appear on both sides of , while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: , .
The statement of the axioms also requires reference to a certain collection of formulae called -formulae. The collection consists of those formulae that can be built using the constants, , , , , and bounded quantification. That is quantification of the form or where is given set.
Axioms
The axioms of KPU are the universal closures of the following formulae:
Extensionality:
Foundation: This is an axiom schema where for every formula we have .
Pairing:
Union:
Δ0-Separation: This is again an axiom schema, where for every -formula we have the following .
Δ0-SCollection: This is also an axiom schema, for every -formula we have .
Set Existence:
Additional assumptions
Technically these are axioms that describe the partition of objects into sets and urelements.
Applications
KPU can be applied to the model theory of infinitary languages. Models of KPU considered as sets inside a maximal universe that are transitive as such are called admissible sets.
See also
Axiomatic set theory
Admissible set
Admissible ordinal
Kripke–Platek set theory
References
.
.
External links
Logic of Abstract Existence
Systems of set theory
Urelements
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https://en.wikipedia.org/wiki/KPU
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KPU is an abbreviation that can mean:
Kenya People's Union, a historic political party in Kenya
Korea Polytechnic University, South Korea
Kripke–Platek set theory with urelements, an axiom system for set theory
Kwantlen Polytechnic University, a public university located in Surrey, British Columbia, Canada.
Kyoto Prefectural University, Japan
General Elections Commission (Indonesia), Komisi Pemilihan Umum
Communist Party of Ukraine, Komunistychna Partiya Ukrayiny
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https://en.wikipedia.org/wiki/Von%20Mises
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The Mises family or von Mises is the name of an Austrian noble family. Members of the family excelled especially in mathematics and economy.
Notable members
Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises
Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von Mises
Richard von Mises, an Austrian-American scientist and mathematician, younger brother of Ludwig von Mises
Von Mises distribution, named after Richard von Mises
Von Mises yield criterion, named after Richard von Mises
Dr. Mises, pseudonym of Gustav Fechner, a German philosopher, physicist and experimental psychologist.
Surnames of Jewish origin
Austrian noble families
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https://en.wikipedia.org/wiki/Hilbert%27s%20fourth%20problem
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In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added.
If one assumes the continuity axiom in addition, then, in the case of the Euclidean plane, we come to the problem posed by Jean Gaston Darboux: "To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines."
There are several interpretations of the original statement of David Hilbert. Nevertheless, a solution was sought, with the German mathematician Georg Hamel being the first to contribute to the solution of Hilbert's fourth problem.
A recognized solution was given by Soviet mathematician Aleksei Pogorelov in 1973.<ref name="Pogorelov1973">А. В. Погорелов, Полное решение IV проблемы Гильберта, ДАН СССР № 208, т.1 (1973), 46–49. English translation: {{cite journal
| last1=Pogorelov | first1=A. V.
| title=A complete solution of "Hilbert's fourth problem| journal=Doklady Akademii Nauk SSSR
| volume=208
| issue=1
| date=1973
| pages=48–52}}</ref> In 1976, Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert's fourth problem.
Original statement
Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry
...a geometry in which all the axioms of ordinary euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.
Due to the idea that a 'straight line' is defined as the shortest path between two points, he mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows:
The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable.
Flat metrics
Desargues's theorem:If two triangles l
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https://en.wikipedia.org/wiki/Cardinality%20of%20the%20continuum
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In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or .
The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is
This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.
Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with This is also true for several other infinite sets, such as any n-dimensional Euclidean space (see space filling curve). That is,
The smallest infinite cardinal number is (aleph-null). The second smallest is (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between and means that . The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).
Properties
Uncountability
Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite. That is, is strictly greater than the cardinality of the natural numbers, :
In practice, this means that there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. For more information on this topic, see Cantor's first uncountability proof and Cantor's diagonal argument.
Cardinal equalities
A variation of Cantor's diagonal argument can be used to prove Cantor's theorem, which states that the cardinality of any set is strictly less than that of its power set. That is, (and so that the power set of the natural numbers is uncountable). In fact, one can show that the cardinality of is equal to as follows:
Define a map from the reals to the power set of the rationals, , by sending each real number to the set of all rationals less than or equal to (with the reals viewed as Dedekind cuts, this is nothing other than the inclusion map in the set of sets of rationals). Because the rationals are dense in , this map is injective, and because the rationals are countable, we have that .
Let be the set of infinite sequences with values in set . This set has cardinality (the natural bijection between the set of binary sequences and is given b
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https://en.wikipedia.org/wiki/Arthur%20Lyon%20Bowley
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Sir Arthur Lyon Bowley, FBA (6 November 1869 – 21 January 1957) was an English statistician and economist who worked on economic statistics and pioneered the use of sampling techniques in social surveys.
Early life
Bowley's father, James William Lyon Bowley, was a minister in the Church of England. He died at the age of 40 when Arthur was one, leaving Arthur's mother as mother or stepmother to seven children. Arthur was educated at Christ's Hospital, and won a scholarship to Trinity College, Cambridge to study mathematics. He graduated as Tenth Wrangler.
At Cambridge Bowley had a short course of study with the economist Alfred Marshall who had also been a Cambridge wrangler. Under Marshall's influence Bowley became an economic statistician. His Account of England's Foreign Trade won the Cobden Essay Prize and was published as a book. Marshall watched over Bowley's career, recommending him for jobs and offering him advice. Most notoriously Marshall told him the Elements of Statistics contained "too much mathematics."
Academic career
After leaving Cambridge Bowley taught mathematics at St John's School in Leatherhead from 1893 to 1899. Meanwhile, he was publishing in economic statistics; his first article for the journal of the Royal Statistical Society) appeared in 1895. In that year the London School of Economics opened. Bowley was appointed as a part-time lecturer and he would be connected with the School until he retired in 1936. He can be considered one of the School's intellectual fathers. However, he continued to teach elsewhere; for more than a decade he taught at University College, Reading (now the University of Reading). He was the Newmarch lecturer at University College London (1897–98 and 1927–28). At the LSE he became Reader in 1908, and Professor in 1915. In 1919, he was appointed to a newly established Chair of Statistics, probably the first of its kind in Britain. In Bowley's time, however, the LSE statistics group was very small: Margaret Hogg arrived in 1919 and left for the United States in 1925, E. C. Rhodes arrived in 1924 and R. G. D. Allen in 1928. Bowley's students included Ronald George, Lewis Connor and Winifred Mackenzie, first recipient of the Frances Wood memorial prize. As a post-graduate student Josiah Stamp worked "nominally" under Bowley's supervision.
Bowley produced a stream of studies of British economic statistics, beginning in the 1890s with work on trade and on wages and income. His 1900 publication Wages in the United Kingdom in the Nineteenth Century was created using the unpaid assistance of Edith Marvin when she was a researcher at the London School of Economics. Proceeding to studies of national income in the 1920s and –30s. Especially noteworthy was his collaboration with Josiah Stamp on a comparison of the UK national income in 1911 and 1924. (Official national income statistics date only from the Second World War.) From around 1910 Bowley worked on social statistics as well. In aim, the work was
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https://en.wikipedia.org/wiki/Mimesis%20%28mathematics%29
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In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of numerical analysis is to approximate the continuum, so instead of solving a partial differential equation one aims to solve a discrete version of the continuum problem. Properties of the continuum problem commonly imitated by numerical methods are conservation laws, solution symmetries, and fundamental identities and theorems of vector and tensor calculus like the divergence theorem.
Both finite difference or finite element method can be mimetic; it depends on the properties that the method has.
For example, a mixed finite element method applied to Darcy flows strictly conserves the mass of the flowing fluid.
The term geometric integration denotes the same philosophy.
References
Numerical differential equations
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https://en.wikipedia.org/wiki/Paul%20Seymour%20%28mathematician%29
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Paul D. Seymour (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website.
Seymour is currently the Albert Baldwin Dod Professor of Mathematics at Princeton University. He won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2003; and (sometimes with others) won the Fulkerson Prize in 1979, 1994, 2006 and 2009, and the Pólya Prize in 1983 and 2004. He received an honorary doctorate from the University of Waterloo in 2008, one from the Technical University of Denmark in 2013, and one from the École normale supérieure de Lyon in 2022. He was an invited speaker in the 1986 International Congress of Mathematicians and a plenary speaker in the 1994 International Congress of Mathematicians. He became a Fellow of the Royal Society in 2022.
Early life
Seymour was born in Plymouth, Devon, England. He was a day student at Plymouth College, and then studied at Exeter College, Oxford, gaining a BA degree in 1971, and D.Phil in 1975.
Career
From 1974 to 1976 he was a college research fellow at University College of Swansea, and then returned to Oxford for 1976–1980 as a Junior Research Fellow at Merton College, Oxford, with the year 1978–79 at University of Waterloo. He became an associate and then a full professor at Ohio State University, Columbus, Ohio, between 1980 and 1983, where he began research with Neil Robertson,
a fruitful collaboration that continued for many years. From 1983 until 1996, he was at Bellcore (Bell Communications Research), Morristown, New Jersey (now Telcordia Technologies). He was also an adjunct professor at Rutgers University from 1984 to 1987 and at the University of Waterloo from 1988 to 1993. He became professor at Princeton University in 1996. He is Editor-in-Chief (jointly with Carsten Thomassen) for the Journal of Graph Theory, and an editor for Combinatorica and the Journal of Combinatorial Theory, Series B.
Personal life
He married Shelley MacDonald of Ottawa in 1979, and they have two children, Amy and Emily. The couple separated amicably in 2007. His brother Leonard W. Seymour is Professor of gene therapy at Oxford University.
Major contributions
Combinatorics in Oxford in the 1970s was dominated by matroid theory, due to the influence of Dominic Welsh and Aubrey William Ingleton. Much of Seymour's early work, up to about 1980, was on matroid theory, and included three important matroid results: his D.Phil. thesis on matroids with the max-flow min-cut property (for which he won his first Fulkerson prize); a characterisation by excluded minors of the matroids representable over the three-element f
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https://en.wikipedia.org/wiki/Series%20expansion
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In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.
Types of series expansions
There are several kinds of series expansions, listed below.
Taylor series
A Taylor series is a power series based on a function's derivatives at a single point. More specifically, if a function is infinitely differentiable around a point , then the Taylor series of f around this point is given by
under the convention . The Maclaurin series of f is its Taylor series about .
Laurent series
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form and converges in an annulus. In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Dirichlet series
A general Dirichlet series is a series of the form One important special case of this is the ordinary Dirichlet series Used in number theory.
Fourier series
A Fourier series is an expansion of periodic functions as a sum of many sine and cosine functions. More specifically, the Fourier series of a function of period is given by the expressionwhere the coefficients are given by the formulae
Other series
In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.
Newtonian series
Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.
Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.
The Stirling seriesis an approximation of the log-gamma function.
Examples
The following is the Taylor series of :
The Dirichlet series of the Riemann zeta function is
References
Algebra
Polynomials
Mathematical analysis
Mathematical series
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https://en.wikipedia.org/wiki/Hyperbolic%20partial%20differential%20equation
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In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is
The equation has the property that, if and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time .
The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed. They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.
Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration. There is a well-developed theory for linear differential operators, due to Lars Gårding, in the context of microlocal analysis. Nonlinear differential equations are hyperbolic if their linearizations are hyperbolic in the sense of Gårding. There is a somewhat different theory for first order systems of equations coming from systems of conservation laws.
Definition
A partial differential equation is hyperbolic at a point provided that the Cauchy problem is uniquely solvable in a neighborhood of for any initial data given on a non-characteristic hypersurface passing through . Here the prescribed initial data consist of all (transverse) derivatives of the function on the surface up to one less than the order of the differential equation.
Examples
By a linear change of variables, any equation of the form
with
can be transformed to the wave equation, apart from lower order terms which are inessential for the qualitative understanding of the equation. This definition is analogous to the definition of a planar hyperbola.
The one-dimensional wave equation:
is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.
Hyperbolic system of partial differential equa
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https://en.wikipedia.org/wiki/Variadic%20function
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In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages.
The term variadic is a neologism, dating back to 1936–1937. The term was not widely used until the 1970s.
Overview
There are many mathematical and logical operations that come across naturally as variadic functions. For instance, the summing of numbers or the concatenation of strings or other sequences are operations that can be thought of as applicable to any number of operands (even though formally in these cases the associative property is applied).
Another operation that has been implemented as a variadic function in many languages is output formatting. The C function and the Common Lisp function are two such examples. Both take one argument that specifies the formatting of the output, and any number of arguments that provide the values to be formatted.
Variadic functions can expose type-safety problems in some languages. For instance, C's , if used incautiously, can give rise to a class of security holes known as format string attacks. The attack is possible because the language support for variadic functions is not type-safe: it permits the function to attempt to pop more arguments off the stack than were placed there, corrupting the stack and leading to unexpected behavior. As a consequence of this, the CERT Coordination Center considers variadic functions in C to be a high-severity security risk.
In functional languages variadics can be considered complementary to the apply function, which takes a function and a list/sequence/array as arguments, and calls the function with the arguments supplied in that list, thus passing a variable number of arguments to the function. In the functional language Haskell, variadic functions can be implemented by returning a value of a type class ; if instances of are a final return value and a function , this allows for any number of additional arguments .
A related subject in term rewriting research is called hedges, or hedge variables. Unlike variadics, which are functions with arguments, hedges are sequences of arguments themselves. They also can have constraints ('take no more than 4 arguments', for example) to the point where they are not variable-length (such as 'take exactly 4 arguments') - thus calling them variadics can be misleading. However they are referring to the same phenomenon, and sometimes the phrasing is mixed, resulting in names such as variadic variable (synonymous to hedge). Note the double meaning of the word variable and the difference between arguments and variables in functional programming and term rewriting. For example, a term (function) can have three variables, one of them a hedge, thus allowing the term to take three or more arguments (or two or more if the hedge is allowed to be empty).
Examples
In C
To portably implement variadic func
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https://en.wikipedia.org/wiki/Incidence%20geometry
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In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.
Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects. In graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics. In the examples selected for this article we use only those with a natural geometric flavor.
A special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of (straight) lines they determine. Some results of this situation can extend to more general settings since only incidence properties are considered.
Incidence structures
An incidence structure consists of a set whose elements are called points, a disjoint set whose elements are called lines and an incidence relation between them, that is, a subset of whose elements are called flags. If is a flag, we say that is incident with or that is incident with (the terminology is symmetric), and write . Intuitively, a point and line are in this relation if and only if the point is on the line. Given a point and a line which do not form a flag, that is, the point is not on the line, the pair is called an anti-flag.
Distance in an incidence structure
There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance b
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https://en.wikipedia.org/wiki/Fundamenta%20Mathematicae
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Fundamenta Mathematicae is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems.
The first specialized journal in the field of mathematics, originally it covered only topology, set theory, and foundations of mathematics. It is published by the Mathematics Institute of the Polish Academy of Sciences.
History
The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland. Janiszewski posited that, to achieve its goal, the journal should not compel Polish mathematicians to submit articles written exclusively in Polish, and should be devoted only to a specialized topic in mathematics; Fundamenta Mathematicae thus became the first specialized journal in the field of mathematics.
Despite Janiszewski having, in a 1918 article, given the initial impetus for the creation of the journal, he did not live long enough to see the first issue published, in Warsaw, as he died on 3 January 1920. Wacław Sierpiński and Stefan Mazurkiewicz took over as editors-in-chief. Soon after its launch, the founding editors were joined by Kazimierz Kuratowski and, later, by Karol Borsuk.
Abstracting and indexing
The journal is abstracted and indexed in the Science Citation Index Expanded, Scopus, and Zentralblatt MATH. According to the Journal Citation Reports, the journal has a 2016 impact factor of 0.609.
Notes
References
External links
Online archive: 1920-2000 at Polish Digital Mathematical Library
Fundamenta Mathematicae (1920-2016) at European Digital Mathematics Library
Mathematics journals
Mass media in Poland
Publications established in 1920
Polish Academy of Sciences academic journals
English-language journals
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