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https://en.wikipedia.org/wiki/Fisher%20consistency | In statistics, Fisher consistency, named after Ronald Fisher, is a desirable property of an estimator asserting that if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained.
Definition
Suppose we have a statistical sample X1, ..., Xn where each Xi follows a cumulative distribution Fθ which depends on an unknown parameter θ. If an estimator of θ based on the sample can be represented as a functional of the empirical distribution function F̂n:
the estimator is said to be Fisher consistent if:
As long as the Xi are exchangeable, an estimator T defined in terms of the Xi can be converted into an estimator T′ that can be defined in terms of F̂n by averaging T over all permutations of the data. The resulting estimator will have the same expected value as T and its variance will be no larger than that of T.
If the strong law of large numbers can be applied, the empirical distribution functions F̂n converge pointwise to Fθ, allowing us to express Fisher consistency as a limit — the estimator is Fisher consistent if
Finite population example
Suppose our sample is obtained from a finite population Z1, ..., Zm. We can represent our sample of size n in terms of the proportion of the sample ni / n taking on each value in the population. Writing our estimator of θ as T(n1 / n, ..., nm / n), the population analogue of the estimator is T(p1, ..., pm), where pi = P(X = Zi). Thus we have Fisher consistency if T(p1, ..., pm) = θ.
Suppose the parameter of interest is the expected value μ and the estimator is the sample mean, which can be written
where I is the indicator function. The population analogue of this expression is
so we have Fisher consistency.
Role in maximum likelihood estimation
Maximising the likelihood function L gives an estimate that is Fisher consistent for a parameter b if
where b0 represents the true value of b.
Relationship to asymptotic consistency and unbiasedness
The term consistency in statistics usually refers to an estimator that is asymptotically consistent. Fisher consistency and asymptotic consistency are distinct concepts, although both aim to define a desirable property of an estimator. While many estimators are consistent in both senses, neither definition encompasses the other. For example, suppose we take an estimator Tn that is both Fisher consistent and asymptotically consistent, and then form Tn + En, where En is a deterministic sequence of nonzero numbers converging to zero. This estimator is asymptotically consistent, but not Fisher consistent for any n.
The sample mean is a Fisher consistent and unbiased estimate of the population mean, but not all Fisher consistent estimates are unbiased. Suppose we observe a sample from a uniform distribution on (0,θ) and we wish to estimate θ. The sample maximum is Fisher consistent, but downwardly biased. Conversely, the sample variance is an unbiased estimate of the p |
https://en.wikipedia.org/wiki/Tony%20Brooker | Ralph Anthony Brooker (22 September 1925 – 20 November 2019), was a British computer scientist known for developing the Mark 1 Autocode.
He was educated at Emanuel School and graduated in Mathematics from Imperial College in 1945 and returned there in 1947 as assistant lecturer. His first computer project was the construction of a fast multiplier unit from electro-mechanical relays. This was taken over by Sid Michaelson and K. D. Tocher and incorporated into ICCE, the Imperial College Computing Engine based on the same technology. By then (1949)Brooker had moved to the University of Cambridge Computer Laboratory to work for Maurice Wilkes on software development for EDSAC.
In October 1951 Brooker joined the Computing Machine Laboratory at Manchester University, where he took over from Alan Turing the task of writing programming manuals and running a user service on the Ferranti Mark 1 computer. It was his experience with the rather tedious Manchester machine-coding conventions that led him to devise what was probably the world's first publicly available High-Level Language. This was the Mark 1 Autocode available from March 1954 and therefore about two years ahead of the first Fortran compiler.
Throughout the 1950s Brooker led a group at Manchester working on the theoretical underpinnings of compilers. This culminated in the compiler-compiler, a seminal idea first presented at a British Computer Society Conference in July 1960 by Brooker and Derrick Morris. This was subsequently implemented on the Ferranti ATLAS and used for high-level language development. The ATLAS was regarded as the world's most powerful computer when it was brought into service in December 1962.
In the mid-1960s Brooker helped to inaugurate the UK's first Computer Science degree course at Manchester. He moved to Essex University in 1967 to take up the university's founding Chair of Computer Science. The first Essex Computer Science graduates obtained their degrees in the summer of 1970. He retired in 1988 and died on 20 November 2019 in Hexham.
Further reading
References
External links
https://web.archive.org/web/20080816195357/http://hopl.murdoch.edu.au/showperson.prx?PeopleID=115
http://www.computer50.org/mark1/gac1.html#brooker
https://web.archive.org/web/20041031031946/http://www.computer50.org/mark1/gethomas/manchester_autocodes.html
Listen to an oral history interview with Tony Brooker - a life story interview recorded for An Oral History of British Science at the British Library
1925 births
2019 deaths
British computer scientists
Alumni of Imperial College London
People from Fulham |
https://en.wikipedia.org/wiki/High-dimensional%20statistics | In statistical theory, the field of high-dimensional statistics studies data whose dimension is larger than typically considered in classical multivariate analysis. The area arose owing to the emergence of many modern data sets in which the dimension of the data vectors may be comparable to, or even larger than, the sample size, so that justification for the use of traditional techniques, often based on asymptotic arguments with the dimension held fixed as the sample size increased, was lacking.
Examples
Parameter estimation in linear models
The most basic statistical model for the relationship between a covariate vector and a response variable is the linear model
where is an unknown parameter vector, and is random noise with mean zero and variance . Given independent responses , with corresponding covariates , from this model, we can form the response vector , and design matrix . When and the design matrix has full column rank (i.e. its columns are linearly independent), the ordinary least squares estimator of is
When , it is known that . Thus, is an unbiased estimator of , and the Gauss-Markov theorem tells us that it is the Best Linear Unbiased Estimator.
However, overfitting is a concern when is of comparable magnitude to : the matrix in the definition of may become ill-conditioned, with a small minimum eigenvalue. In such circumstances will be large (since the trace of a matrix is the sum of its eigenvalues). Even worse, when , the matrix is singular. (See Section 1.2 and Exercise 1.2 in .)
It is important to note that the deterioration in estimation performance in high dimensions observed in the previous paragraph is not limited to the ordinary least squares estimator. In fact, statistical inference in high dimensions is intrinsically hard, a phenomenon known as the curse of dimensionality, and it can be shown that no estimator can do better in a worst-case sense without additional information (see Example 15.10). Nevertheless, the situation in high-dimensional statistics may not be hopeless when the data possess some low-dimensional structure. One common assumption for high-dimensional linear regression is that the vector of regression coefficients is sparse, in the sense that most coordinates of are zero. Many statistical procedures, including the Lasso, have been proposed to fit high-dimensional linear models under such sparsity assumptions.
Covariance matrix estimation
Another example of a high-dimensional statistical phenomenon can be found in the problem of covariance matrix estimation. Suppose that we observe , which are i.i.d. draws from some zero mean distribution with an unknown covariance matrix . A natural unbiased estimator of is the sample covariance matrix
In the low-dimensional setting where increases and is held fixed, is a consistent estimator of in any matrix norm. When grows with , on the other hand, this consistency result may fail to hold. As an illustration, suppose that each an |
https://en.wikipedia.org/wiki/William%20McCune | William Walker McCune (December 17, 1953 – May 2, 2011) was an American computer scientist and logician working in the fields of automated reasoning, algebra, logic, and formal methods. He was best known for the development of the Otter, Prover9, and Mace4 automated reasoning systems, and the automated proof of the Robbins conjecture using the EQP theorem prover.
In 2000, McCune received the Herbrand Award for Distinguished Contributions to Automated Reasoning. In 2013, Automated Reasoning and Mathematics - Essays in Memory of William W. McCune was published in his honour.
References
External links
Prover9 software
William McCune home page
Updated version of Prover9 software
2011 deaths
1953 births
American computer scientists |
https://en.wikipedia.org/wiki/Tammes%20problem | In geometry, the Tammes problem is a problem in packing a given number of points on the surface of a sphere such that the minimum distance between points is maximized. It is named after the Dutch botanist Pieter Merkus Lambertus Tammes (the nephew of pioneering botanist Jantina Tammes) who posed the problem in his 1930 doctoral dissertation on the distribution of pores on pollen grains. Mathematicians independent of Tammes began studying circle packing on the sphere in the early 1940s; it was not until twenty years later that the problem became associated with his name.
It can be viewed as a particular special case of the generalized Thomson problem of minimizing the total Coulomb force of electrons in a spherical arrangement. Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24. There are conjectured solutions for many other cases, including those in higher dimensions.
See also
Spherical code
Kissing number problem
Cylinder sphere packings
References
Bibliography
Journal articles
Books
External links
How to Stay Away from Each Other in a Spherical Universe (PDF).
Packing and Covering of Congruent Spherical Caps on a Sphere.
Science of Spherical Arrangements (PPT).
General discussion of packing points on surfaces, with focus on tori (PDF).
Circle packing
Spherical geometry
Palynology |
https://en.wikipedia.org/wiki/Five%20circles%20theorem | In geometry, the five circles theorem states that, given five circles centered on a common sixth circle and intersecting each other chainwise on the same circle, the lines joining their second intersection points forms a pentagram whose points lie on the circles themselves.
See also
Clifford's circle theorems
Miquel's theorem
Six circles theorem
Seven circles theorem
References
External links
Theorems about circles |
https://en.wikipedia.org/wiki/Six%20circles%20theorem | In geometry, the six circles theorem relates to a chain of six circles together with a triangle, such that each circle is tangent to two sides of the triangle and also to the preceding circle in the chain. The chain closes, in the sense that the sixth circle is always tangent to the first circle. It is assumed in this construction that all circles lie within the triangle, and all points of tangency lie on the sides of the triangle. If the problem is generalized to allow circles that may not be within the triangle, and points of tangency on the lines extending the sides of the triangle, then the sequence of circles eventually reaches a periodic sequence of six circles, but may take arbitrarily many steps to reach this periodicity.
The name may also refer to Miquel's six circles theorem, the result that if five circles have four triple points of intersection then the remaining four points of intersection lie on a sixth circle.
References
External links
Theorems about circles |
https://en.wikipedia.org/wiki/Seven%20circles%20theorem | In geometry, the seven circles theorem is a theorem about a certain arrangement of seven circles in the Euclidean plane. Specifically, given a chain of six circles all tangent to a seventh circle and each tangent to its two neighbors, the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same point. Though elementary in nature, this theorem was not discovered until 1974 (by Evelyn, Money-Coutts, and Tyrrell).
See also
Brianchon's theorem
References
External links
Interactive Applet by Michael Borcherds showing The Seven Circles Theorem made using GeoGebra.
the Seven Circles Theorem by Stanley Rabinowitz, with a proof based on Ceva's theorems.
Seven Circles Theorem at Cut-the-knot.
Theorems about circles |
https://en.wikipedia.org/wiki/Miquel%27s%20theorem | Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal Journal de mathématiques pures et appliquées.
Formally, let ABC be a triangle, with arbitrary points A´, B´ and C´ on sides BC, AC, and AB respectively (or their extensions). Draw three circumcircles (Miquel's circles) to triangles AB´C´, A´BC´, and A´B´C. Miquel's theorem states that these circles intersect in a single point M, called the Miquel point. In addition, the three angles MA´B, MB´C and MC´A (green in the diagram) are all equal, as are the three supplementary angles MA´C, MB´A and MC´B.
The theorem (and its corollary) follow from the properties of cyclic quadrilaterals. Let the circumcircles of A'B'C and AB'C' meet at Then hence BA'MC' is cyclic as desired.
Pivot theorem
If in the statement of Miquel's theorem the points A´, B´ and C´ form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in . (In the diagram these points are labeled P, Q and R.)
If A´, B´ and C´ are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then A´, B´ and C´ are on a line.
Trilinear coordinates of the Miquel point
If the fractional distances of A´, B´ and C´ along sides BC (a), CA (b) and AB (c) are da, db and dc, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by:
where da = 1 - da, etc.
In the case da = db = dc = ½ the Miquel point is the circumcentre .
A converse of Miquel's theorem
The theorem can be reversed to say: for three circles intersecting at M, a line can be drawn from any point A on one circle, through its intersection C´ with another to give B (at the second intersection). B is then similarly connected, via intersection at A´ of the second and third circles, giving point C. Points C, A and the remaining point of intersection, B´, will then be collinear, and triangle ABC will always pass through the circle intersections A´, B´ and C´.
Similar inscribed triangle
If the inscribed triangle XYZ is similar to the reference triangle ABC, then the point M of concurrence of the three circles is fixed for all such XYZ.
Miquel and Steiner's quadrilateral theorem
The circumcircles of all four triangles of a complete quadrilateral meet at a point M. In the diagram above these are ∆ABF, ∆CDF, ∆ADE and ∆BCE.
This result was announced, in two lines, by Jakob Steiner in the 1827/1828 issue of Gergonne's Annales de Mathématiques, but a detailed proof was given by Miquel.
Miquel's pentagon theorem
Let ABCDE be a convex pentagon. Extend all sides until they meet in five points F,G,H,I,K and draw the circumcircles of the five triangles CFD, DGE, EHA, AIB and BKC. Then |
https://en.wikipedia.org/wiki/Clifford%27s%20circle%20theorems | In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.
Statement
The first theorem considers any four circles passing through a common point M and otherwise in general position, meaning that there are six additional points where exactly two of the circles cross and that no three of these crossing points are collinear. Every set of three of these four circles has among them three crossing points, and (by the assumption of non-collinearity) there exists a circle passing through these three crossing points. The conclusion is that, like the first set of four circles, the second set of four circles defined in this way all pass through a single point P (in general not the same point as M).
The second theorem considers five circles in general position passing through a single point M. Each subset of four circles defines a new point P according to the first theorem. Then these five points all lie on a single circle C.
The third theorem considers six circles in general position that pass through a single point M. Each subset of five circles defines a new circle by the second theorem. Then these six new circles C all pass through a single point.
The sequence of theorems can be continued indefinitely.
See also
Cox's chain
Five circles theorem
Miquel's six circles theorem
References
W. K. Clifford (1882). Mathematical Papers, pages 51,2 via Internet Archive
H. S. M. Coxeter (1965). Introduction to Geometry, page 262, John Wiley & Sons
Further reading
H. Martini & M. Spirova (2008) "Clifford’s chain of theorems in strictly convex Minkowski planes", Publicationes Mathematicae Debrecen 72: 371–83
External links
Theorems about circles |
https://en.wikipedia.org/wiki/Robert%20Edouard%20Moritz | Robert Edouard Moritz (2 Jun 1868 – 28 Dec 1940) was a German-American mathematician.
He published about 75 books and papers. For over 30 years he was head of the mathematics department at the University of Washington.
Biography
Moritz was born in Schleswig-Holstein to Karl R. and Maria Stahlhut Moritz, and emigrated to the United States at the age of twelve where the family settled on a farm in Nebraska. From 1885 to 1892 he attended Hastings College in Hastings, Nebraska, and then studied another year at the University of Chicago. After two summer quarters in the next years he received his MA in mathematics in 1896.
In 1893 he started his academic career as assistant professor in mathematics and physics back at Hastings College. In 1898 he moved to the University of Nebraska, where he was appointed instructor in mathematics, and obtained his PhD in 1901. The next year he studied in Europe under Heinrich Martin Weber and Theodor Reye, obtaining a second PhD at the University of Strassburg in 1902. Back in the United States he worked two years at the University of Nebraska, before moving to the University of Washington where he was appointed professor and head of the Department of Mathematics and Astronomy.
He died in Seattle in the state of Washington in 1940 at the age of 72.
Publications by Moritz, a selection
Plane and Spherical Trigonometry. John Wiley and Sons, 1911, 1913.
On Mathematics and Mathematicians, Macmillan 1914, 1942, 1958; also published as Memorabilia Mathematica; Or, The Philomath's Quotation-book, Macmillan 1914.
A Short Course in College Mathematics, Macmillan, 1919, 1920.
References
External links
19th-century German mathematicians
19th-century American mathematicians
1868 births
1940 deaths
20th-century American mathematicians
Emigrants from the German Empire to the United States |
https://en.wikipedia.org/wiki/Zoll%20surface | In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-dimensional family of geometrically distinct deformations that are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature.
Zoll, a student of David Hilbert, discovered the first non-trivial examples.
See also
Funk transform: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
References
External links
Tannery's pear, an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight.
Surfaces |
https://en.wikipedia.org/wiki/Riemann%20hypothesis | In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:
Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers where t is a real number and i is the imaginary unit.
Riemann zeta function
The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series
Leonhard Euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product
where the infinite product extends over all prime numbers p.
The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. Because the zeta function is meromorphic, all choices of how to perform this analytic continuation will lead to the same result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation
within the region of convergence for both series. However, the zeta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part. Thus, the zeta function can be redefined as , extending it from to a larger domain: , except for the points where is zero. These are the points where can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see ), giving a finite value for all values of s with positive real part except fo |
https://en.wikipedia.org/wiki/John%20W.%20Dawson%20Jr. | John W. Dawson Jr. (born February 4, 1944) is an American academic who is an emeritus professor of mathematics at Penn State York.
Early life and education
Born in Wichita, Kansas, Dawson attended the Massachusetts Institute of Technology as a National Merit Scholar before earning a doctorate in mathematical logic from the University of Michigan in 1972.
Career
An internationally recognized authority on the life and work of Kurt Gödel, Dawson is the author of numerous articles on axiomatic set theory and the history of modern logic. From 1982 to 1984, he catalogued Gödel's papers at the Institute for Advanced Study. He also served as a co-editor of Gödel's Collected Works. He retired as co-editor-in-chief of the journal History and Philosophy of Logic.
Books by Dawson
John W. Dawson Jr, 1997. Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA,
John W. Dawson Jr, 2015. Why Prove it Again? Alternative Proofs in Mathematical Practice, Springer, Chaim,
Additional publications
Dawson, John W. Jr. The published work of Kurt Gödel: an annotated bibliography. Notre Dame J. Formal Logic 24 (1983), no. 2, 255–284.
References
External links
Dr. John Dawson, personal webpage, Pennsylvania State University at York
1944 births
Living people
People from Wichita, Kansas
20th-century American mathematicians
21st-century American mathematicians
Mathematical logicians
American philosophy academics
American historians of philosophy
Set theorists
Massachusetts Institute of Technology alumni
University of Michigan alumni
Institute for Advanced Study visiting scholars
American book editors |
https://en.wikipedia.org/wiki/Tangent%20circles | In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the use of materials.
Two given circles
Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radii
Steiner chains
Pappus chains
Three given circles: Apollonius' problem
Apollonius' problem is to construct circles that are tangent to three given circles.
Apollonian gasket
If a circle is iteratively inscribed into the interstitial curved triangles between three mutually tangent circles, an Apollonian gasket results, one of the earliest fractals described in print.
Malfatti's problem
Malfatti's problem is to carve three cylinders from a triangular block of marble, using as much of the marble as possible. In 1803, Gian Francesco Malfatti conjectured that the solution would be obtained by inscribing three mutually tangent circles into the triangle (a problem that had previously been considered by Japanese mathematician Ajima Naonobu); these circles are now known as the Malfatti circles, although the conjecture has been proven to be false.
Six circles theorem
A chain of six circles can be drawn such that each circle is tangent to two sides of a given triangle and also to the preceding circle in the chain. The chain closes; the sixth circle is always tangent to the first circle.
Generalizations
Problems involving tangent circles are often generalized to spheres. For example, the Fermat problem of finding sphere(s) tangent to four given spheres is a generalization of Apollonius' problem, whereas Soddy's hexlet is a generalization of a Steiner chain.
See also
Tangent lines to circles
Circle packing theorem, the result that every planar graph may be realized by a system of tangent circles
Hexafoil, the shape formed by a ring of six tangent circles
Feuerbach's theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles
Descartes' theorem
Ford circle
Bankoff circle
Archimedes' twin circles
Archimedean circle
Schoch circles
Woo circles
Arbelos
Ring lemma
References
External links
Circles |
https://en.wikipedia.org/wiki/Timeline%20of%20category%20theory%20and%20related%20mathematics | This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as:
Categories of abstract algebraic structures including representation theory and universal algebra;
Homological algebra;
Homotopical algebra;
Topology using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology;
Categorical logic and set theory in the categorical context such as algebraic set theory;
Foundations of mathematics building on categories, for instance topos theory;
Abstract geometry, including algebraic geometry, categorical noncommutative geometry, etc.
Quantization related to category theory, in particular categorical quantization;
Categorical physics relevant for mathematics.
In this article, and in category theory in general, ∞ = ω.
Timeline to 1945: before the definitions
1945–1970
1971–1980
1981–1990
1991–2000
2001–present
See also
EGA
FGA
SGA
Notes
References
nLab, just as a higher-dimensional Wikipedia, started in late 2008; see nLab
Zhaohua Luo; Categorical geometry homepage
John Baez, Aaron Lauda; A prehistory of n-categorical physics
Ross Street; An Australian conspectus of higher categories
Elaine Landry, Jean-Pierre Marquis; Categories in context: historical, foundational, and philosophical
Jim Stasheff; A survey of cohomological physics
John Bell; The development of categorical logic
Jean Dieudonné; The historical development of algebraic geometry
Charles Weibel; History of homological algebra
Peter Johnstone; The point of pointless topology
George Whitehead; Fifty years of homotopy theory
Haynes Miller; The origin of sheaf theory
Category theory
Higher category theory
Category theory and related mathematics |
https://en.wikipedia.org/wiki/Farid%20Abedi | Farid Abedi (born August 28, 1977 in Firouzabad) is an Iranian footballer. He currently plays for Bargh Shiraz in Azadegan League.
Club Career Statistics
Last Update 5 June 2010
Assist Goals
References
Iran Pro League Stats
1977 births
Living people
Iranian men's footballers
Rah Ahan Tehran F.C. players
Bargh Shiraz F.C. players
Persian Gulf Pro League players
Azadegan League players
People from Firuzabad, Fars
Men's association football forwards
Sportspeople from Fars province |
https://en.wikipedia.org/wiki/Krull%20ring | In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.
In this article, a ring is commutative and has unity.
Formal definition
Let be an integral domain and let be the set of all prime ideals of of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then is a Krull ring if
is a discrete valuation ring for all ,
is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ).
Any nonzero element of is contained in only a finite number of height 1 prime ideals.
It is also possible to characterize Krull rings by mean of valuations only:
An integral domain is a Krull ring if there exists a family
of discrete valuations on the field of fractions of such that:
for any and all , except possibly a finite number of them, ;
for any , belongs to if and only if for all .
The valuations are called essential valuations of .
The link between the two definitions is as follows: for every , one can associate a unique normalized valuation of whose valuation ring is . Then the set satisfies the conditions of the equivalent definition. Conversely, if the set is as above, and the have been normalized, then may be bigger than , but it must contain . In other words, is the minimal set of normalized valuations satisfying the equivalent definition.
There are other ways to introduce and define Krull rings. The theory of Krull rings can be exposed in synergy with the theory of divisorial ideals. One of the best references on the subject is Lecture on Unique Factorization Domains by P. Samuel.
Properties
With the notations above, let denote the normalized valuation corresponding to the valuation ring , denote the set of units of , and its quotient field.
An element belongs to if, and only if, for every . Indeed, in this case, for every , hence ; by the intersection property, . Conversely, if and are in , then , hence , since both numbers must be .
An element is uniquely determined, up to a unit of , by the values , . Indeed, if for every , then , hence by the above property (q.e.d). This shows that the application is well defined, and since for only finitely many , it is an embedding of into the free Abelian group generated by the elements of . Thus, using the multiplicative notation "" for the later group, there holds, for every , , where the are the elements of containing , and .
The valuations are pairwise independent. As a consequence, there holds the so-called weak approximation theorem, an homologue of the Chinese remainder theorem: if are distinct elements of , belong to (resp. ), and are natural numbers, then there exist (resp. ) such that for every .
Two elements |
https://en.wikipedia.org/wiki/Senior%20Wrangler | The Senior Wrangler is the top mathematics undergraduate at the University of Cambridge in England, a position which has been described as "the greatest intellectual achievement attainable in Britain".
Specifically, it is the person who achieves the highest overall mark among the Wranglers – the students at Cambridge who gain first-class degrees in mathematics. The Cambridge undergraduate mathematics course, or Mathematical Tripos, is famously difficult.
Many Senior Wranglers have become world-leading figures in mathematics, physics, and other fields. They include George Airy, Jacob Bronowski, Christopher Budd, Kevin Buzzard, Arthur Cayley, Donald Coxeter, Arthur Eddington, Ben Green, John Herschel, James Inman, J. E. Littlewood, Lee Hsien Loong, Jayant Narlikar, Morris Pell, John Polkinghorne, Frank Ramsey, Lord Rayleigh (John Strutt), Sir George Stokes, Isaac Todhunter, Sir Gilbert Walker, and James H. Wilkinson.
Senior Wranglers were once fêted with torchlit processions and took pride of place in the University's graduation ceremony. Years in Cambridge were often remembered by who had been Senior Wrangler that year.
The annual ceremony in which the Senior Wrangler becomes known was first held in the 18th century. Standing on the balcony of the University's Senate House, the examiner reads out the class results for mathematics, and printed copies of the results are then thrown to the audience below. The examiner no longer announces the students' exact rankings, but they still identify the Senior Wrangler, nowadays by tipping their academic hat when reading out the person's name.
Others who finished in the top 12
Those who have achieved second place, known as Second Wranglers, include Alfred Marshall, James Clerk Maxwell, J. J. Thomson, Lord Kelvin, William Clifford, and William Whewell.
Those who have finished between third and 12th include Archibald Hill, Karl Pearson and William Henry Bragg (third), George Green, G. H. Hardy, and Alfred North Whitehead (fourth), Adam Sedgwick (fifth), John Venn (sixth), Bertrand Russell, Nevil Maskelyne and Sir James Timmins Chance (seventh), Thomas Malthus (ninth), and John Maynard Keynes and William Henry Fox Talbot (12th).
History
Between 1748 and 1909, the University publicly announced the ranking, which was then reported in newspapers such as The Times. The examination was considered to be by far the most important in Britain and the Empire. The prestige of being a high Wrangler was great; the respect accorded to the Senior Wrangler was immense. Andrew Warwick, author of Masters of Theory, describes the term 'Senior Wrangler' as "synonymous with academic supremacy".
Since 1910, successful students in the examinations have been told their rankings privately, and not all Senior Wranglers have become publicly known as such. In recent years, the custom of discretion regarding ranking has progressively vanished, and all Senior Wranglers since 2010 have announced their identity publicly.
The youngest |
https://en.wikipedia.org/wiki/Locally%20finite%20poset | In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements.
Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval [x, y] of P a real number ƒ(x, y). These functions form an associative algebra with a product defined by
There is also a definition of incidence coalgebra.
In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime.
References
Stanley, Richard P. Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.
Order theory |
https://en.wikipedia.org/wiki/Lions%E2%80%93Lax%E2%80%93Milgram%20theorem | In mathematics, the Lions–Lax–Milgram theorem (or simply Lions's theorem) is a result in functional analysis with applications in the study of partial differential equations. It is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear function can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Jacques-Louis Lions, Peter Lax and Arthur Milgram.
Statement of the theorem
Let H be a Hilbert space and V a normed space. Let B : H × V → R be a continuous, bilinear function. Then the following are equivalent:
(coercivity) for some constant c > 0,
(existence of a "weak inverse") for each continuous linear functional f ∈ V∗, there is an element h ∈ H such that
Related results
The Lions–Lax–Milgram theorem can be applied by using the following result, the hypotheses of which are quite common and easy to verify in practical applications:
Suppose that V is continuously embedded in H and that B is V-elliptic, i.e.
for some c > 0 and all v ∈ V,
for some α > 0 and all v ∈ V,
Then the above coercivity condition (and hence the existence result) holds.
Importance and applications
Lions's generalization is an important one since it allows one to tackle boundary value problems beyond the Hilbert space setting of the original Lax–Milgram theory. To illustrate the power of Lions's theorem, consider the heat equation in n spatial dimensions (x) and one time dimension (t):
where Δ denotes the Laplace operator. Two questions arise immediately: on what domain in spacetime is the heat equation to be solved, and what boundary conditions are to be imposed? The first question — the shape of the domain — is the one in which the power of the Lions–Lax–Milgram theorem can be seen. In simple settings, it suffices to consider cylindrical domains: i.e., one fixes a spatial region of interest, Ω, and a maximal time, T ∈(0, +∞], and proceeds to solve the heat equation on the "cylinder"
One can then proceed to solve the heat equation using classical Lax–Milgram theory (and/or Galerkin approximations) on each "time slice" {t} × Ω. This is all very well if one only wishes to solve the heat equation on a domain that does not change its shape as a function of time. However, there are many applications for which this is not true: for example, if one wishes to solve the heat equation on the polar ice cap, one must take account of the changing shape of the volume of ice as it evaporates and/or icebergs break away. In other words, one must at least be able to handle domains G in spacetime that do not look the same along each "time slice". (There is also the added complication of domains whose shape changes according to the solution u of the problem itself.) Such domains and boundary conditions are beyond the reach of classical Lax–Milgram theory, but can be attacked using Lions's theorem.
See also
Babuška–Lax–Milgram theorem
Refe |
https://en.wikipedia.org/wiki/Contact%20type | In mathematics, more precisely in symplectic geometry, a hypersurface of a symplectic manifold is said to be of contact type if there is 1-form such that and is a contact manifold, where is the natural inclusion. The terminology was first coined by Alan Weinstein.
See also
Weinstein conjecture
References
Symplectic geometry |
https://en.wikipedia.org/wiki/Common%20chord | Common chord may refer to:
Common chord (geometry), the secant line that joins the intersection points of two curves
Common chord (music), a chord shared by two musical keys
The Common Chord, a 1947 short story collection by Frank O'Connor
Common Chord, a 1993 album by David Grisman |
https://en.wikipedia.org/wiki/Quaternionic%20vector%20space | In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions.
The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:
for quaternions q and q1, q2, ... qn.
Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.
See also
Vector space
General linear group
Special linear group
SL(n,H)
Symplectic group
References
Quaternions
Linear algebra |
https://en.wikipedia.org/wiki/Intersecting%20chords%20theorem | In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle.
It states that the products of the lengths of the line segments on each chord are equal.
It is Proposition 35 of Book 3 of Euclid's Elements.
More precisely, for two chords and intersecting in a point the following equation holds:
The converse is true as well, that is if for two line segments and intersecting in the equation above holds true, then their four endpoints lie on a common circle. Or in other words, if the diagonals of a quadrilateral intersect in and fulfill the equation above, then it is a cyclic quadrilateral.
The value of the two products in the chord theorem depends only on the distance of the intersection point from the circle's center and is called the absolute value of the power of ; more precisely, it can be stated that:
where is the radius of the circle, and is the distance between the center of the circle and the intersection point . This property follows directly from applying the chord theorem to a third chord going through and the circle's center (see drawing).
The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles and :
This means the triangles and are similar and therefore
Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chords theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.
References
Paul Glaister: Intersecting Chords Theorem: 30 Years on. Mathematics in School, Vol. 36, No. 1 (Jan., 2007), p. 22 (JSTOR)
Bruce Shawyer: Explorations in Geometry. World scientific, 2010, , p. 14
Hans Schupp: Elementargeometrie. Schöningh, Paderborn 1977, , p. 149 (German).
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)
External links
Intersecting Chords Theorem at cut-the-knot.org
Intersecting Chords Theorem at proofwiki.org
Two interactive illustrations: and
Theorems about circles |
https://en.wikipedia.org/wiki/Intersecting%20secants%20theorem | In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
For two lines and that intersect each other at and for which all lie on the same circle, the following equation holds:
The theorem follows directly from the fact that the triangles and are similar. They share and as they are inscribed angles over . The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above:
Next to the intersecting chords theorem and the tangent-secant theorem, the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.
References
S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, , pp. 175-176
Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, , p. 161
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, , pp. 415-417 (German)
External links
Secant Secant Theorem at proofwiki.org
Power of a Point Theorem auf cut-the-knot.org
Theorems about circles |
https://en.wikipedia.org/wiki/Geometric%20calculus | In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including vector calculus, differential geometry, and differential forms.
Differentiation
With a geometric algebra given, let and be vectors and let be a multivector-valued function of a vector. The directional derivative of along at is defined as
provided that the limit exists for all , where the limit is taken for scalar . This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued.
Next, choose a set of basis vectors and consider the operators, denoted , that perform directional derivatives in the directions of :
Then, using the Einstein summation notation, consider the operator:
which means
where the geometric product is applied after the directional derivative. More verbosely:
This operator is independent of the choice of frame, and can thus be used to define what in geometric calculus is called the vector derivative:
This is similar to the usual definition of the gradient, but it, too, extends to functions that are not necessarily scalar-valued.
The directional derivative is linear regarding its direction, that is:
From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction can be written , so that:
For this reason, is often noted .
The standard order of operations for the vector derivative is that it acts only on the function closest to its immediate right. Given two functions and , then for example we have
Product rule
Although the partial derivative exhibits a product rule, the vector derivative only partially inherits this property. Consider two functions and :
Since the geometric product is not commutative with in general, we need a new notation to proceed. A solution is to adopt the overdot notation, in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define
then the product rule for the vector derivative is
Interior and exterior derivative
Let be an -grade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives,
In particular, if is grade 1 (vector-valued function), then we can write
and identify the divergence and curl as
Unlike the vector derivative, neither the interior derivative operator nor the exterior derivative operator is invertible.
Multivector derivative
The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the multivector derivative.
Let be a multivector-valued function of a multivector. The directional derivative of with respect to in the direction , where and are multivectors, is defined |
https://en.wikipedia.org/wiki/Calibrated%20geometry | In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
φ is closed: dφ = 0, where d is the exterior derivative
for any x ∈ M and any oriented p-dimensional subspace ξ of TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g.
Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(φ) for x in M.
The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.
Calibrated submanifolds
A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).
A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then
where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem (as φ is closed), and the inequality holds because φ is a calibration.
Examples
On a Kähler manifold, suitably normalized powers of the Kähler form are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality.
On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds are special Lagrangian submanifolds.
On a G2-manifold, both the 3-form and the Hodge dual 4-form define calibrations. The corresponding calibrated submanifolds are called associative and coassociative submanifolds.
On a Spin(7)-manifold, the defining 4-form, known as the Cayley form, is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.
References
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Differential geometry
Riemannian geometry
Structures on manifolds |
https://en.wikipedia.org/wiki/Riemannian%20circle | In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the intrinsic Riemannian metric of a compact one-dimensional manifold of total length 2, or the extrinsic metric obtained by restriction of the intrinsic metric to the two-dimensional surface of the sphere, rather than the extrinsic metric obtained by restriction of the Euclidean metric to the unit circle of the two-dimensional Cartesian plane. The distance between a pair of points on the surface of the sphere is defined to be the length of the shorter of the two arcs into which the circle is partitioned by the two points.
It is named after German mathematician Bernhard Riemann.
Properties
The diameter of the Riemannian circle is , in contrast with the usual value of 2 for the Euclidean diameter of the unit circle.
The inclusion of the Riemannian circle as the equator (or any great circle) of the 2-sphere of constant Gaussian curvature +1, is an isometric imbedding in the sense of metric spaces (there is no isometric imbedding of the Riemannian circle in Hilbert space in this sense).
Gromov's filling conjecture
A long-standing open problem, posed by Mikhail Gromov, concerns the calculation of the filling area of the Riemannian circle. The filling area is conjectured to be 2, a value attained by the hemisphere of constant Gaussian curvature +1.
References
Gromov, M.: "Filling Riemannian manifolds", Journal of Differential Geometry 18 (1983), 1–147.
Riemannian geometry
Circles
Metric geometry
Bernhard Riemann |
https://en.wikipedia.org/wiki/Gail%20Carpenter | Gail Alexandra Carpenter (born 1948) is an American cognitive scientist, neuroscientist and mathematician. She is now a "Professor Emerita of Mathematics and Statistics, Boston University." She had also been a Professor of Cognitive and Neural Systems at Boston University, and the director of the Department of Cognitive and Neural Systems (CNS) Technology Lab at Boston University.
Early life
Gail Carpenter is the only daughter of Chadwick Hunter "Chad" Carpenter (1920-1996) and Ruth M. (née Stevenson) Carpenter (1920-2010). She has four brothers.
Carpenter attended the International School of Geneva (1961-1966) then went to the University of Colorado in Boulder earning a B.A. in 1970 (summa cum laude, mathematics). She then earned a Ph.D. in mathematics at the University of Wisconsin–Madison. Carpenter then taught at MIT and Northeastern University before moving to Boston University.
Carpenter married Stephen Grossberg on June 16, 1979, in Boston University Castle in Boston, Massachusetts.
Adaptive resonance theory
Carpenter's "neural modeling" efforts were clearly seen in her 1974 mathematics PhD thesis on Traveling wave solutions of nerve impulse equations at the University of Wisconsin Department of Mathematics while working with Charles C. Conley. Later she re-defined and expanded her theories in various papers during the mid to late 1970s. She defined the "generalized Hodgkin-Huxley models, used dynamical systems techniques to analyze their solutions, and characterized the qualitative properties of the burst suppression patterns that a typical neuron may propagate: while investigating normal and abnormal signal patterns in nerve cells.
Adaptive resonance theory (ART) is a theory developed by Stephen Grossberg and Gail Carpenter on aspects of how the brain processes information. It describes a number of neural network models which use supervised and unsupervised learning methods, and address problems such as pattern recognition and prediction.
The primary intuition behind the ART model is that object identification and recognition generally occur as a result of the interaction of 'top-down' observer expectations with 'bottom-up' sensory information. The model postulates that 'top-down' expectations take the form of a memory template or prototype that is then compared with the actual features of an object as detected by the senses. This comparison gives rise to a measure of category belongingness. As long as this difference between sensation and expectation does not exceed a set threshold called the 'vigilance parameter', the sensed object will be considered a member of the expected class. The system thus offers a solution to the 'plasticity/stability' problem, i.e. the problem of acquiring new knowledge without disrupting existing knowledge that is also called incremental learning.
Academic acknowledgements
Per Boston University, where Carpenter is a "Professor Emerita of Mathematics and Statistics," she is acknowledged as having been |
https://en.wikipedia.org/wiki/David%20Preiss | David Preiss FRS (born January 21, 1947) is a Czech and British mathematician, specializing in mathematical analysis.
He is a professor of mathematics at the University of Warwick
Preiss is a recipient of the Ostrowski Prize (2011)
and the winner of the 2008 London Mathematical Society Pólya Prize for his 1987 result on Geometry of Measures, where he solved the remaining problem in the geometric theoretic structure of sets and measures in Euclidean space.
He was an invited speaker at the ICM 1990 in Kyoto.
He is a Fellow of the Royal Society (2004) and a Foreign Fellow of the Learned Society of the Czech Republic (2003).
He is associate editor of the mathematical journal Real Analysis Exchange.
Publications
References
External links
David Preiss webpage at University of Warwick
1947 births
Living people
English mathematicians
Academics of the University of Warwick
Fellows of the Royal Society
Measure theorists
Functional analysts |
https://en.wikipedia.org/wiki/Timeline%20of%20probability%20and%20statistics | The following is a timeline of probability and statistics.
Before 1600
8th century – Al-Khalil, an Arab mathematician studying cryptology, wrote the Book of Cryptographic Messages. The work has been lost, but based on the reports of later authors, it contained the first use of permutations and combinations to list all possible Arabic words with and without vowels.
9th century - Al-Kindi was the first to use frequency analysis to decipher encrypted messages and developed the first code breaking algorithm. He wrote a book entitled Manuscript on Deciphering Cryptographic Messages, containing detailed discussions on statistics and cryptanalysis. Al-Kindi also made the earliest known use of statistical inference.
13th century – An important contribution of Ibn Adlan was on sample size for use of frequency analysis.
13th century – the first known calculation of the probability for throwing 3 dices is published in the Latin poem De vetula.
1560s (published 1663) – Cardano's Liber de ludo aleae attempts to calculate probabilities of dice throws. He demonstrates the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes).
1577 – Bartolomé de Medina defends probabilism, the view that in ethics one may follow a probable opinion even if the opposite is more probable
17th century
1654 – Blaise Pascal and Pierre de Fermat create the mathematical theory of probability,
1657 – Chistiaan Huygens's De ratiociniis in ludo aleae is the first book on mathematical probability,
1662 – John Graunt's Natural and Political Observations Made upon the Bills of Mortality makes inferences from statistical data on deaths in London,
1666 – In Le Journal des Sçavans xxxi, 2 August 1666 (359–370(=364)) appears a review of the third edition (1665) of John Graunt's Observations on the Bills of Mortality. This review gives a summary of 'plusieurs reflexions curieuses', of which the second are Graunt's data on life expectancy. This review is used by Nicolaus Bernoulli in his De Usu Artis Conjectandi in Jure (1709).
1669 – Christiaan Huygens and his brother Lodewijk discuss between August and December that year Graunts mortality table (Graunt 1662, p. 62) in letters #1755
1693 – Edmond Halley prepares the first mortality tables statistically relating death rate to age,
18th century
1710 – John Arbuthnot argues that the constancy of the ratio of male to female births is a sign of divine providence,
1713 – Posthumous publication of Jacob Bernoulli's Ars Conjectandi, containing the first derivation of a law of large numbers,
1724 – Abraham de Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities upon Lives,
1733 – de Moivre introduces the normal distribution to approximate the binomial distribution in probability,
1739 – David Hume's Treatise of Human Nature argues that induc |
https://en.wikipedia.org/wiki/Timeline%20of%20number%20theory | A timeline of number theory.
Before 1000 BCE
ca. 20,000 BCE — Nile Valley, Ishango Bone: possibly the earliest reference to prime numbers and Egyptian multiplication although this is disputed.
About 300 BCE
300 BCE — Euclid proves the number of prime numbers is infinite.
1st millennium AD
250 — Diophantus writes Arithmetica, one of the earliest treatises on algebra.
500 — Aryabhata solves the general linear diophantine equation.
628 - Brahmagupta gives Brahmagupta's identity and solves the so called Pell's equation using his composition method.
ca. 650 — Mathematicians in India create the Hindu–Arabic numeral system we use, including the zero, the decimals and negative numbers.
1000–1500
ca. 1000 — Abu-Mahmud al-Khujandi first states a special case of Fermat's Last Theorem.
895 — Thabit ibn Qurra gives a theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
975 — The earliest triangle of binomial coefficients (Pascal triangle) occur in the 10th century in commentaries on the Chandas Shastra.
1150 — Bhaskara II gives first general method for solving Pell's equation
1260 — Al-Farisi gave a new proof of Thābit ibn Qurra's theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 which have also been jointly attributed to Fermat as well as Thabit ibn Qurra.
17th century
1637 — Pierre de Fermat claims to have proven Fermat's Last Theorem in his copy of Diophantus' Arithmetica.
18th century
1742 — Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture.
1770 — Joseph Louis Lagrange proves the four-square theorem, that every positive integer is the sum of four squares of integers. In the same year, Edward Waring conjectures Waring's problem, that for any positive integer k, every positive integer is the sum of a fixed number of kth powers.
1796 — Adrien-Marie Legendre conjectures the prime number theorem.
19th century
1801 — Disquisitiones Arithmeticae, Carl Friedrich Gauss's number theory treatise, is published in Latin.
1825 — Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's Last Theorem for n = 5.
1832 — Lejeune Dirichlet proves Fermat's Last Theorem for n = 14.
1835 — Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetic progressions.
1859 — Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers.
1896 — Jacques Hadamard and Charles Jean de la Vallée-Poussin independently prove the prime number theorem.
1896 — Hermann Minkowski presents Geometry of numbers.
20th century
1903 — Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem.
1909 — David Hilbert proves Waring's problem.
1912 — Josip Plemelj publish |
https://en.wikipedia.org/wiki/Timeline%20of%20geometry | The following is a timeline of key developments of geometry:
Before 1000 BC
ca. 2000 BC – Scotland, carved stone balls exhibit a variety of symmetries including all of the symmetries of Platonic solids.
1800 BC – Moscow Mathematical Papyrus, findings volume of a frustum
1800 BC – Plimpton 322 contains the oldest reference to the Pythagorean triplets.
1650 BC – Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations
1st millennium BC
800 BC – Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places
ca. 600 BC – the other Vedic "Sulba Sutras" ("rule of chords" in Sanskrit) use Pythagorean triples, contain of a number of geometrical proofs, and approximate π at 3.16
5th century BC – Hippocrates of Chios utilizes lunes in an attempt to square the circle
5th century BC – Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places
530 BC – Pythagoras studies propositional geometry and vibrating lyre strings; his group also discover the irrationality of the square root of two,
370 BC – Eudoxus states the method of exhaustion for area determination
300 BC – Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic
260 BC – Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3.
225 BC – Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola,
150 BC – Jain mathematicians in India write the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations
140 BC – Hipparchus develops the bases of trigonometry.
1st millennium
ca. 340 – Pappus of Alexandria states his hexagon theorem and his centroid theorem
500 – Aryabhata writes the "Aryabhata-Siddhanta", which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of sine and cosine, and also contains the earlie |
https://en.wikipedia.org/wiki/Timeline%20of%20calculus%20and%20mathematical%20analysis | A timeline of calculus and mathematical analysis.
500BC to 1600
5th century BC - The Zeno's paradoxes,
5th century BC - Antiphon attempts to square the circle,
5th century BC - Democritus finds the volume of cone is 1/3 of volume of cylinder,
4th century BC - Eudoxus of Cnidus develops the method of exhaustion,
3rd century BC - Archimedes displays geometric series in The Quadrature of the Parabola. Archimedes also discovers a method which is similar to differential calculus.
3rd century BC - Archimedes develops a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems using methods now termed as integral calculus. Archimedes also derives several formulae for determining the area and volume of various solids including sphere, cone, paraboloid and hyperboloid.
Before 50 BC - Babylonian cuneiform tablets show use of the Trapezoid rule to calculate of the position of Jupiter.
3rd century - Liu Hui rediscovers the method of exhaustion in order to find the area of a circle.
4th century - The Pappus's centroid theorem,
5th century - Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere.
600 - Liu Zhuo is the first person to use second-order interpolation for computing the positions of the sun and the moon.
665 - Brahmagupta discovers a second order Taylor interpolation for ,
862 - The Banu Musa brothers write the "Book on the Measurement of Plane and Spherical Figures",
9th century - Thābit ibn Qurra discusses the quadrature of the parabola and the volume of different types of conic sections.
12th century - Bhāskara II discovers a rule equivalent to Rolle's theorem for ,
14th century - Nicole Oresme proves of the divergence of the harmonic series,
14th century - Madhava discovers the power series expansion for , , and This theory is now well known in the Western world as the Taylor series or infinite series.
14th century - Parameshvara discovers a third order Taylor interpolation for ,
1445 - Nicholas of Cusa attempts to square the circle,
1501 - Nilakantha Somayaji writes the Tantrasamgraha, which contains the Madhava's discoveries,
1548 - Francesco Maurolico attempted to calculate the barycenter of various bodies (pyramid, paraboloid, etc.),
1550 - Jyeshtadeva writes the Yuktibhāṣā, a commentary to Nilakantha's Tantrasamgraha,
1560 - Sankara Variar writes the Kriyakramakari,
1565 - Federico Commandino publishes De centro Gravitati,
1588 - Commandino's translation of Pappus' Collectio gets published,
1593 - François Viète discovers the first infinite product in the history of mathematics,
17th century
1606 - Luca Valerio applies methods of Archimedes to find volumes and centres of gravity of solid bodies,
1609 - Johannes Kepler computes the integral ,
1611 - Thomas Harriot discovers an interpolation formula similar to Newton's interpolation formula,
1615 - Johannes Kepler publishes Nova stereometria doliorum,
1620 |
https://en.wikipedia.org/wiki/Representation%20theory | Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.
Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. For instance, representing a group by an infinite-dimensional Hilbert space allows methods of analysis to be applied to the theory of groups. Furthermore, representation theory is important in physics because it can describe how the symmetry group of a physical system affects the solutions of equations describing that system.
Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse. In addition to its impact on algebra, representation theory
generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.
There are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.
The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.
Definitions and concepts
Let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or compl |
https://en.wikipedia.org/wiki/Zig-zag%20lemma | In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.
Statement
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let and be chain complexes that fit into the following short exact sequence:
Such a sequence is shorthand for the following commutative diagram:
where the rows are exact sequences and each column is a chain complex.
The zig-zag lemma asserts that there is a collection of boundary maps
that makes the following sequence exact:
The maps and are the usual maps induced by homology. The boundary maps are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).
Construction of the boundary maps
The maps are defined using a standard diagram chasing argument. Let represent a class in , so . Exactness of the row implies that is surjective, so there must be some with . By commutativity of the diagram,
By exactness,
Thus, since is injective, there is a unique element such that . This is a cycle, since is injective and
since . That is, . This means is a cycle, so it represents a class in . We can now define
With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of c and b). The proof uses diagram chasing arguments similar to that above. Such arguments are also used to show that the sequence in homology is exact at each group.
See also
Mayer–Vietoris sequence
References
Homological algebra
Lemmas in category theory |
https://en.wikipedia.org/wiki/Michael%20S.%20Longuet-Higgins | Michael Selwyn Longuet-Higgins FRS (8 December 1925 – 26 February 2016) was a British mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute for Nonlinear Science, University of California, San Diego, USA. He was the younger brother of H. Christopher Longuet-Higgins.
Longuet-Higgins introduced the theory of the origin of microseisms and is the inventor of "rhombo blocks", a mathematical toy consisting of blocks whose faces are rhombuses.
Educated at The Pilgrims' School, Winchester and Winchester College from 1937 to 1941 with Freeman Dyson, his brother Christopher and James Lighthill from 1937 to 1943, he won a mathematics scholarship to Trinity College, Cambridge at the age of 17, where he graduated with a BA in mathematics in 1945 after only two years. He was awarded a doctorate in geophysics in 1951. From 1969 to 1989 he was a Royal Society Research Fellow at the University of Cambridge.
His research areas included both pure mathematics (projective geometry, polytopes, random functions and surfaces) and applied mathematics (fluid dynamics, microseisms, the generation of ocean waves by wind, the dynamics of liquid Bubbles, sonoluminescence, wave breaking, steep waves, and heat and gas exchange at the ocean surface).
Education and career
Michael was born in Lenham, Kent in England to Henry Hugh Longuet Longuet-Higgins and Albinia Cecil Bazeley. He had an elder sister Patricia (Pat) and an elder brother Hugh Christopher (Christopher). He was educated at The Pilgrims' School, Winchester and at Winchester College, together with Freeman Dyson, his brother Christopher, and James Lighthill from 1937 to 1943. In 1943, at the age of 17, he won a scholarship in mathematics to Trinity College, Cambridge, where he qualified after just two years for a BA in mathematics in 1945. He was awarded a PhD in geophysics in 1951.
Towards the end of the World War II (December 1943) he started working for the Admiralty Research Laboratory (ARL) in Teddington. He joined Group W (waves) set up on 5 June 1944 under George Deacon. Here he helped predict wave and current conditions in preparation for the Pacific landings. He worked not only on the theory of wind waves but also on the geomagnetic induction of voltages by tidal streams, and on the generation of oceanic microseisms.
In 1948 he returned to Cambridge, to read for a PhD but without a break in his research, just reporting to Sir Harold Jeffreys and later to Robert Stoneley at the end of each term. After being awarded his PhD in geophysics ("just a one hour interview which he almost forgot to attend") in 1951 at Cambridge, he was awarded a 4-year research fellowship (Title A) at Trinity College. The first year (1951–52) he spent in the US as a Commonwealth Fund Fellow, first at the Woods Hole Oceanographic Institute on Cape Cod, Massachusetts, and then at the Scripps Institution of Oceanography at La Jolla, Calif |
https://en.wikipedia.org/wiki/Exponentially%20equivalent%20measures | In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory.
Definition
Let be a metric space and consider two one-parameter families of probability measures on , say and . These two families are said to be exponentially equivalent if there exist
a one-parameter family of probability spaces ,
two families of -valued random variables and ,
such that
for each , the -law (i.e. the push-forward measure) of is , and the -law of is ,
for each , " and are further than apart" is a -measurable event, i.e.
for each ,
The two families of random variables and are also said to be exponentially equivalent.
Properties
The main use of exponential equivalence is that as far as large deviations principles are concerned, exponentially equivalent families of measures are indistinguishable. More precisely, if a large deviations principle holds for with good rate function , and and are exponentially equivalent, then the same large deviations principle holds for with the same good rate function .
References
(See section 4.2.2)
Asymptotic analysis
Probability theory
Equivalence (mathematics) |
https://en.wikipedia.org/wiki/Yuya%20Sato%20%28footballer%29 | is a Japanese football player who plays for Roasso Kumamoto.
Club statistics
Updated to 10 December 2017.
References
External links
Profile at JEF United Chiba
1986 births
Living people
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Ventforet Kofu players
Hokkaido Consadole Sapporo players
Giravanz Kitakyushu players
Tokyo Verdy players
JEF United Chiba players
Roasso Kumamoto players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg%20theorem | In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectives (or), (and), (not) and quantifiers (for all), (exists) is equivalent to a similar formula without quantifiers. An important consequence is the decidability of the theory of real-closed fields.
Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic decomposition, which allows quantifier elimination over the reals in double exponential time. This complexity is optimal, as there are examples where the output has a double exponential number of connected components. This algorithm is therefore fundamental, and it is widely used in computational algebraic geometry.
Statement
A semialgebraic set in Rn is a finite union of sets defined by a finite number of polynomial equations and inequalities, that is by a finite number of statements of the form
and
for polynomials p and q. We define a projection map π : Rn +1 → Rn by sending a point (x1, ..., xn, xn +1) to (x1, ..., xn). Then the Tarski–Seidenberg theorem states that if X is a semialgebraic set in Rn +1 for some n ≥ 1, then π(X) is a semialgebraic set in Rn.
Failure with algebraic sets
If we only define sets using polynomial equations and not inequalities then we define algebraic sets rather than semialgebraic sets. For these sets the theorem fails, i.e. projections of algebraic sets need not be algebraic. As a simple example consider the hyperbola in R2 defined by the equation
This is a perfectly good algebraic set, but projecting it down by sending (x, y) in R2 to x in R produces the set of points satisfying x ≠ 0. This is a semialgebraic set, but it is not an algebraic set as the algebraic sets in R are R itself, the empty set and the finite sets.
This example shows also that, over the complex numbers, the projection of an algebraic set may be non-algebraic. Thus the existence of real algebraic sets with non-algebraic projections does not rely on the fact that the field of real numbers is not algebraically closed.
Another example is the parabola in R2, which is defined by the equation
Its projection onto the x-axis is the half-line [0, ∞), a semialgebraic set that cannot be obtained from algebraic sets by (finite) intersections, unions, and set complements.
Relation to structures
This result confirmed that semi |
https://en.wikipedia.org/wiki/Peruvians%20in%20Spain | As of 2018, official statistics showed 201,993 Peruvian-born residents in Spain. Out of these, 129,344 were Spanish citizens and 72,649 had not yet acquired Spanish citizenship. As of 2019, the number had increased to 218,129.
Fall in population during the financial crisis
During the financial crisis, the Peruvian community decreased somewhat dramatically. In just one year, the Peruvian population in Spain decreased by 23.76%, a figure that places Peru as the country with the greatest dropout of official residents in that nation. The number went from 109,639 official residents in Spain, as of January 1, 2013, to 83,583, as of January 1, 2014. That is, 26,055 fewer Peruvians among the foreign population in the country, according to information disseminated by the National Institute of Statistics (INEI).
The population has since recovered and increased since the end of the economic downturn.
Notable people
See also
Peru–Spain relations
References
Sources
Further reading
External links
Federación de asociaciones peruanos en España
Ethnic groups in Spain
Spain |
https://en.wikipedia.org/wiki/Maass%20wave%20form | In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are eigenforms of the hyperbolic Laplace operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.
General remarks
The group
operates on the upper half plane
by fractional linear transformations:
It can be extended to an operation on by defining:
The Radon measure
defined on is invariant under the operation of .
Let be a discrete subgroup of . A fundamental domain for is an open set , so that there exists a system of representatives of with
A fundamental domain for the modular group is given by
(see Modular form).
A function is called -invariant, if holds for all and all .
For every measurable, -invariant function the equation
holds. Here the measure on the right side of the equation is the induced measure on the quotient
Classic Maass forms
Definition of the hyperbolic Laplace operator
The hyperbolic Laplace operator on is defined as
Definition of a Maass form
A Maass form for the group is a complex-valued smooth function on satisfying
If
we call Maass cusp form.
Relation between Maass forms and Dirichlet series
Let be a Maass form. Since
we have:
Therefore has a Fourier expansion of the form
with coefficient functions
It is easy to show that is Maass cusp form if and only if .
We can calculate the coefficient functions in a precise way. For this we need the Bessel function .
Definition: The Bessel function is defined as
The integral converges locally uniformly absolutely for in and the inequality
holds for all .
Therefore, decreases exponentially for . Furthermore, we have for all .
Proof: We have
By the definition of the Fourier coefficients we get
for
Together it follows that
for
In (1) we used that the nth Fourier coefficient of is for the first summation term. In the second term we changed the order of integration and differentiation, which is allowed since f is smooth in y . We get a linear differential equation of second degree:
For one can show, that for every solution there exist unique coefficients with the property
For every solution has coefficients of the form
for unique . Here and are Bessel functions.
The Bessel functions grow exponentially, while the Bessel functions decrease exponentially. Together with the polynomial growth condition 3) we get (also ) for a unique . Q.E.D.
Even and odd Maass forms: Let . Then i operates on all functions by and commutes with the hyperbolic Laplacian. A Maass form is called even, if and odd if . If f is a Maass form, then is an even Maass fo |
https://en.wikipedia.org/wiki/Tomoyuki%20Arata | is a former Japanese footballer who last played for Japanese club Nagano Parceiro. He was named J2's Rookie of the Year for 2008.
Career statistics
Updated to 2 February 2018.
References
External links
Profile at Nagano Parceiro
1985 births
Living people
Senshu University alumni
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Mito HollyHock players
Júbilo Iwata players
JEF United Chiba players
Fagiano Okayama players
Matsumoto Yamaga FC players
Oita Trinita players
AC Nagano Parceiro players
Men's association football forwards
Association football people from Shizuoka (city) |
https://en.wikipedia.org/wiki/James%20W.%20Cannon | James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
Biographical data
James W. Cannon was born on January 30, 1943, in Bellefonte, Pennsylvania. Cannon received a Ph.D. in Mathematics from the University of Utah in 1969, under the direction of C. Edmund Burgess.
He was a professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University. He held this position until his retirement in September 2012.
Cannon gave an AMS Invited address at the meeting of the American Mathematical Society in Seattle in August 1977, an invited address at the International Congress of Mathematicians in Helsinki 1978, and delivered the 1982 Mathematical Association of America Hedrick Lectures in Toronto, Canada.
Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007. In 2012 he became a fellow of the American Mathematical Society.
In 1993 Cannon delivered the 30-th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University.
James Cannon is a devout member of the Church of Jesus Christ of Latter-day Saints.
Mathematical contributions
Early work
Cannon's early work concerned topological aspects of embedded surfaces in R3 and understanding the difference between "tame" and "wild" surfaces.
His first famous result came in late 1970s when Cannon gave a complete solution to a long-standing "double suspension" problem posed by John Milnor. Cannon proved that the double suspension of a homology sphere is a topological sphere. R. D. Edwards had previously proven this in many cases.
The results of Cannon's paper were used by Cannon, Bryant and Lacher to prove (1979) an important case of the so-called characterization conjecture for topological manifolds. The conjecture says that a generalized n-manifold , where , which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established that the conjecture holds under the assumption that be a manifold except possibly at a set of dimension . Later Frank Quinn completed the proof that the characterization conjecture holds if there is even a single manifold point. In general, the conjecture is false as was proved by John Bryant, Steven Ferry, Washington Mio and Shmuel Weinberger.
1980s: Hyperbolic geometry, 3-manifolds and geometric group theory
In 1980s the focus of Cannon's work shifted to the study of 3-manifolds, hyperbolic geometry and Kleinian groups and he is considered one of the key figures in the birth of geometric group theory as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the forerunners in the development |
https://en.wikipedia.org/wiki/Arithmeum | The Arithmeum is a mathematics museum owned by the Forschungsinstitut für Diskrete Mathematik (Research Institute for Discrete Mathematics) at the University of Bonn.
It was founded in 2008 by the director of the institute, Bernhard Korte, who contributed his private collection of calculating machines.
The building's steel-glass facade – located at Lennéstrasse 2 – is meant to represent the "transparency of science".
Exhibitions
The permanent exhibit "Calculating in Olden and Modern Times" () shows the progression of mechanical calculating machines through 1,200 pieces.
It holds the very large (4,000 pieces), IJzebrand Schuitema (1929–2013) 400 year collection of slide rules.
There are also exhibits on very-large-scale integrated (VLSI) logic chips, historical arithmetic books dating back to Johannes Gutenberg's times, and the relationship between art and science.
References
External links
The Arithmeum's History
Museums in Bonn
History of mathematics
Science museums in Germany |
https://en.wikipedia.org/wiki/Symplectic%20representation | In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form
where F is the field of scalars. A representation of a group G preserves ω if
for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if
for all ξ in g and v, w in V. Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp(V,ω) or its Lie algebra sp(V,ω)
If G is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius–Schur indicator.
References
.
Representation theory
Symplectic geometry |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Belgium | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of Belgium for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for Belgium is BE and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation - the local administrative unit (LAU). In Belgium, the LAUs are municipalities.
Overall
NUTS Levels
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Belgium can be downloaded here: ''
NUTS codes
In the 2003 version, the Arrondissement of Verviers was coded BE333.
See also
Subdivisions of Belgium
ISO 3166-2 codes of Belgium
FIPS region codes of Belgium
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
BELGIQUE / BELGIË - NUTS level 2
BELGIQUE / BELGIË - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Provinces of Belgium, Statoids.com
Arrondissements of Belgium, Statoids.com
Belgium
Nuts |
https://en.wikipedia.org/wiki/Multiplicative%20character | In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication.
This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition.
Multiplicative characters are linearly independent, i.e. if are different characters on a group G then from it follows that
Examples
Consider the (ax + b)-group
Functions fu : G → C such that where u ranges over complex numbers C are multiplicative characters.
Consider the multiplicative group of positive real numbers (R+,·). Then functions fu : (R+,·) → C such that fu(a) = au, where a is an element of (R+, ·) and u ranges over complex numbers C, are multiplicative characters.
References
Lectures Delivered at the University of Notre Dame
Group theory |
https://en.wikipedia.org/wiki/David%20S.%20Moore | David Sheldon Moore is an American statistician, who is known for his leadership of statistics education for many decades.
Biography
David S. Moore received his A.B. from Princeton University and the Ph.D. from Cornell University in mathematics.
In statistics education, David S. Moore is the author of a series of influential textbooks in statistical science, which use only high school algebra: Introduction to the Practice of Statistics (with George McCabe), of An Introduction to the Basic Practice of Statistics, and of Statistics: Concepts and Controversies.
In statistical science, David S. Moore has done research in the asymptotic theory of robust and nonparametric statistics.
Professor Moore was the 1998 President of the American Statistical Association.
David S. Moore is a retired (in 2004) Shanti S. Gupta Distinguished Professor of Statistics, Emeritus at Purdue University.
Professor Moore has served as the second president of the International Association for Statistical Education. He was the content developer for the Annenberg/Corporation for Public Broadcasting college-level telecourse Against All Odds: Inside Statistics.
Publications
Introduction to the Practice of Statistics with George McCabe.
The Basic Practice of Statistics.
Statistics: Concepts and Controversies.
References
Fellows of the American Statistical Association
Purdue University faculty
Statistics educators
American statisticians
Cornell University alumni
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Uniform%20absolute-convergence | In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
Motivation
A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities.
Definition
Given a set X and functions (or to any normed vector space), the series
is called uniformly absolutely-convergent if the series of nonnegative functions
is uniformly convergent.
Distinctions
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn(x) = xn/n on the open interval (−1,0), then the series Σfn(x) converges uniformly by comparison of the partial sums to those of Σ(−1)n/n, and the series Σ|fn(x)| converges absolutely at each point by the geometric series test, but Σ|fn(x)| does not converge uniformly. Intuitively, this is because the absolute-convergence gets slower and slower as x approaches −1, where convergence holds but absolute convergence fails.
Generalizations
If a series of functions is uniformly absolutely-convergent on some neighborhood of each point of a topological space, it is locally uniformly absolutely-convergent. If a series is uniformly absolutely-convergent on all compact subsets of a topological space, it is compactly (uniformly) absolutely-convergent. If the topological space is locally compact, these notions are equivalent.
Properties
If a series of functions into C (or any Banach space) is uniformly absolutely-convergent, then it is uniformly convergent.
Uniform absolute-convergence is independent of the ordering of a series. This is because, for a series of nonnegative functions, uniform convergence is equivalent to the property that, for any ε > 0, there are finitely many terms of the series such that excluding these terms results in a series with total sum less than the constant function ε, and this property does not refer to the ordering.
See also
Modes of convergence (annotated index)
References
Mathematical analysis
Convergence (mathematics) |
https://en.wikipedia.org/wiki/Hadjicostas%27s%20formula | In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the gamma function and the Riemann zeta function. It is named after Petros Hadjicostas.
Statement
Let s be a complex number with s ≠ -1 and Re(s) > −2. Then
Here Γ is the Gamma function and ζ is the Riemann zeta function.
Background
The first instance of the formula was proved and used by Frits Beukers in his 1978 paper giving an alternative proof of Apéry's theorem. He proved the formula when s = 0, and proved an equivalent formulation for the case s = 1. This led Petros Hadjicostas to conjecture the above formula in 2004, and within a week it had been proven by Robin Chapman. He proved the formula holds when Re(s) > −1, and then extended the result by analytic continuation to get the full result.
Special cases
As well as the two cases used by Beukers to get alternate expressions for ζ(2) and ζ(3), the formula can be used to express the Euler–Mascheroni constant as a double integral by letting s tend to −1:
The latter formula was first discovered by Jonathan Sondow and is the one referred to in the title of Hadjicostas's paper.
Notes
See also
Zeta and L-functions |
https://en.wikipedia.org/wiki/Pinch%20point | Pinch point may refer to:
Pinch point (economics), the level of inventories of a commodity or product below which consumers become concerned about security of supply
Pinch point (mathematics), a type of singular point on an algebraic surface
Pinch point bar, a hand tool consisting of a long, straight metal bar
Curb extension, a traffic calming measure consisting of an angled narrowing of the roadway
The point of closest approach between the hot and cold composite curves in pinch analysis
Pinch point hazard, a mechanical hazard produced by objects coming together.
See also
Bottleneck (disambiguation), various meanings including a phenomenon where the performance or capacity of a system is limited by one component or resource
Pinch (disambiguation) |
https://en.wikipedia.org/wiki/Pinch%20point%20%28mathematics%29 | In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.
The equation for the surface near a pinch point may be put in the form
where [4] denotes terms of degree 4 or more and is not a square in the ring of functions.
For example the surface near the point , meaning in coordinates vanishing at that point, has the form above. In fact, if and then {} is a system of coordinates vanishing at then is written in the canonical form.
The simplest example of a pinch point is the hypersurface defined by the equation called Whitney umbrella.
The pinch point (in this case the origin) is a limit of normal crossings singular points (the -axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole -axis and not only the pinch point.
See also
Whitney umbrella
Singular point of an algebraic variety
References
Algebraic surfaces
Singularity theory |
https://en.wikipedia.org/wiki/Cuspidal%20point | Cuspidal point can refer to:
Cuspidal point of a curve, see Cusp (singularity)
Cuspidal point of a surface, see Pinch point (mathematics) |
https://en.wikipedia.org/wiki/Theodoros%20Natsinas | Theodoros Natsinas (; 8 July 1872 - 2 February 1949) was a Greek teacher. He was born in Siatista (), then part of the Ottoman Empire, now in Greece.
Career
He studied Physics and Mathematics at the University of Athens, where he received his degree. After the completion of his studies in 1898, he taught at secondary schools in Argos and Spetses. In 1899, the University of Athens proposed to the Greek Government that he should be posted as a teacher at Kayseri in Turkey on a "national" mission to support the local Greek population. From 1903 to 1907 he taught at Constantinople.
At 1907 he served as director of a school in Siatista and took part in the Greek Struggle for Macedonia. He was arrested by the Ottoman authorities and taken to nearby city of Lapsista in West Macedonia. He was released after the intervention of the Greek Bishop of Sisanion and Siatista.
In the interwar years, after West Macedonia became part of Greece, he taught in Kozani, founded the first Agricultural Cooperative there, and helped to set up the first Scouts team in West Macedonia. In 1918 he was appointed general Superintendent of Secondary Education of Greece.
In politics he was a supporter of the Liberal Party of Eleftherios Venizelos.
Founding a school in Thessaloniki
After 1921 he worked in Thessaloniki. Initially, together with a partner, G. Karantanis, they took over the management of a school established by Stefanos Noukas (the Marasleia School). At first, the school was known as the Natsinas - Karantanis School, but, subsequently, after the two partners split, the school was renamed the Theodoros Natsinas School.
He was one of the founding members of the Society for Macedonian Studies.
Personal life
He had one daughter and three sons: Sultana Natsina, the Lieutenant General Alexandros Natsinas, the creator and first Director (1953–1963) of the Greek Central Intelligence Service, the politician Stefanos Natsinas, and Leandros Natsinas who studied civil engineering at Ghent, Belgium.
He is the author of the book "Macedonian Itinerant Merchants in the Countries of Austria-Hungary" (first edition 1939).
The municipality of Thessaloniki honored him by naming a city street after him.
Notes
External links
Documents from National Foundation Eleftherios Venizelos
Greek Migration to Europe
Studies about Vlachs
Laboratory of Research about Modern Hellenic - Greek Philosophy
The contribution of Siatista in Greek nation's cause.
Maraslion.
1872 births
1949 deaths
People from Siatista
Greek Macedonians
National and Kapodistrian University of Athens alumni
Macedonia under the Ottoman Empire
Emigrants from the Ottoman Empire to Greece |
https://en.wikipedia.org/wiki/Australian%20cricket%20team%20in%202008%E2%80%9309 | This article contains information, results and statistics regarding the Australian national cricket team in the 2008-09 cricket season. Statisticians class the 2008–09 season as those matches played on tours that started between September 2008 and April 2009.
Player contracts
The 2008-09 list was announced on 9 April 2008. Note that uncontracted players still are available for selection for the national cricket team.
Match summary
M = Matches Played, W = Won, L = Lost, D = Drawn, T = Tied, NR = No Result * = Not all games played yet
Series Summary
Champions Trophy
Australia were due to play in the 2008 ICC Champions Trophy in Pakistan between 12 September and 28 September, however due to security concerns from several nations, the tournament was postponed until October 2009.
Tour of India
Australia arrived in India in September for a 4 Test series against the Indian cricket team. A tour match in late September starts the tour and it concludes after the final Test in November.
Tour matches
Tour Match: 27–28 September, Jaipur
Tour Match: 2–5 October, Hyderabad
Test series
First Test
Australian XI: Matthew Hayden, Simon Katich, Ricky Ponting (c), Michael Hussey, Michael Clarke, Shane Watson, Brad Haddin (wk), Cameron White, Brett Lee, Mitchell Johnson, Stuart Clark
Test debut: Cameron White
2nd Test
3rd Test
4th Test
India won the series 2-0-2
New Zealand in Australia (First Leg)
Test series
The first leg comprises two Test match series.
First Test: 20–24 November, Brisbane
Second Test: 28 November-2 December, Adelaide
Australia won the series 2-0-0.
New Zealand then goes home for a series against the West Indies and Australia will host the South African team. New Zealand returns to Australia on 29 January 2009 for the second leg.
South Africa in Australia
South Africa arrived in Australia for a 3 Tests, 2 Twenty20's and 5 ODI's against the Australian cricket team.
Test series
First Test
Second Test
Third Test
South Africa won the series 2-1-0
Twenty20 Series
1st Twenty20
Australia won the toss and elected to bat.
2nd Twenty20
South Africa won the toss and elected to bat.
Australia won the series 2-0-0
ODI Series
1st ODI
Australia won the toss and elected to bat.
2nd ODI
South Africa won the toss and elected to field.
3rd ODI
4th ODI
5th ODI
South Africa won the series 4-1-0
New Zealand in Australia (Second Leg)
New Zealand returns for the second leg, consisting of a tour match against the Prime Minister's XI, five One Day Internationals (ODIs) against Australia and a Twenty20 International also against Australia.
ODI Series
1st ODI
2nd ODI
3rd ODI
4th ODI
5th ODI
Match was a washout
Twenty20
Twenty20
Important events
See also
Australia national cricket team
International cricket in 2008-09
References
Australia in international cricket
National |
https://en.wikipedia.org/wiki/KANT%20%28software%29 | KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface. They have been developed by the Algebra and Number Theory research group of the Institute of Mathematics at Technische Universität Berlin under the project leadership of Prof. Dr Michael Pohst. Kant is free for non-commercial use.
See also
List of computer algebra systems
References
External links
Introduction to KASH3, The KANT Group
Computer algebra system software for Linux
Computer algebra systems
Proprietary freeware for Linux |
https://en.wikipedia.org/wiki/El%20Hadi%20Fay%C3%A7al%20Ouadah | El Hadi Fayçal Ouadah (born September 24, 1983 in Blida, Algeria) is an Algerian football player who is currently playing as a goalkeeper for AS Khroub in the Algerian Ligue 2.
Career statistics
Club
References
External links
1983 births
Living people
People from Blida
Algerian men's footballers
USM Blida players
USM Annaba players
MC Saïda players
Men's association football goalkeepers
21st-century Algerian people |
https://en.wikipedia.org/wiki/D%C3%A1niel%20R%C3%B3zsa | Dániel Rózsa (born 24 November 1984) is a Hungarian footballer who plays for Austrian club USC Pilgersdorf.
Honours
Hungarian Second Division: Winner: 2008
Club statistics
Updated to games played as of 19 May 2019.
References
External links
HLSZ
1984 births
Footballers from Szombathely
Living people
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's international footballers
Men's association football goalkeepers
Szombathelyi Haladás footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in Austria
Hungarian expatriate sportspeople in Austria |
https://en.wikipedia.org/wiki/Hurwitz%20quaternion%20order | The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).
Definition
Let be the maximal real subfield of where is a 7th-primitive root of unity.
The ring of integers of is , where the element can be identified with the positive real . Let be the quaternion algebra, or symbol algebra
so that and in Also let and . Let
Then is a maximal order of , described explicitly by Noam Elkies.
Module structure
The order is also generated by elements
and
In fact, the order is a free -module over
the basis . Here the generators satisfy the relations
which descend to the appropriate relations in the (2,3,7) triangle group, after quotienting by the center.
Principal congruence subgroups
The principal congruence subgroup defined by an ideal is by definition the group
mod
namely, the group of elements of reduced norm 1 in equivalent to 1 modulo the ideal . The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to PSL(2,R).
Application
The order was used by Katz, Schaps, and Vishne to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak; see systoles of surfaces.
See also
(2,3,7) triangle group
Klein quartic
Macbeath surface
First Hurwitz triplet
References
Riemann surfaces
Differential geometry of surfaces
Algebras
Algebra
Systolic geometry |
https://en.wikipedia.org/wiki/Australian%20Association%20of%20Mathematics%20Teachers | The Australian Association of Mathematics Teachers is the main representative organisation of mathematics teachers in Australia. Membership is via affiliated state organisations. The AAMT conducts a number of activities including Reach for the stars, an activity for students, as well as submissions to government bodies and reports on issues relevant to mathematics teaching.
Structure
The AAMT is a federation of 8 affiliated associations of teachers of mathematics, one from each Australian State and Territory:
The AAMT is governed by a council made up of a representatives from each of these associations, as well as an elected President, Treasurer, and either a President Elect or an Immediate Past President.
The day-to-day affairs of the association are managed by an office staff based primarily in Canberra, Australian Capital Territory.
Membership
AAMT does not direct membership; members join their local affiliated association and are then automatically a member of AAMT. AAMT has approximately 4000 individual and institutional members.
Standards
The AAMT publish Standards for Excellence in Teaching Mathematics in Australian Schools as a guide for the improvement and maintenance of teaching standards in mathematics in Australian schools.
Journals
The AAMT publishes three journals:
Australian Primary Mathematics Classroom (APMC) - primary school mathematics education
The Australian Mathematics Education Journal (AMEJ) - (Secondary school up to early tertiary level)
The AMEJ superseded AAMT's earlier publications in 2019. These were:
The Australian Mathematics Teacher (AMT) - secondary school mathematics up to age 16
Australian Senior Mathematics Journal (ASMJ) - senior secondary (Years 11 and 12) and early tertiary mathematics.
References
External links
Educational organisations based in Australia
Professional associations based in Australia
Mathematics organizations
Teaching in Australia |
https://en.wikipedia.org/wiki/Reciprocation | Reciprocation may refer to:
Reciprocating motion, a type of oscillatory motion, as in the action of a reciprocating saw
Reciprocation (geometry), an operation with circles that involves transforming each point in plane into its polar line and each line in the plane into its pole
Reciprocation, application of the reciprocal function, see multiplicative inverse
See also
Reciprocity (disambiguation)
Reciprocal (disambiguation) |
https://en.wikipedia.org/wiki/Lie%20sphere | Lie sphere may refer to:
Lie-sphere, the fourth type of classical bounded symmetric domain
Lie sphere geometry |
https://en.wikipedia.org/wiki/Pole%20and%20polar | In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole.
Properties
Pole and polar have several useful properties:
If a point P lies on the line l, then the pole L of the line l lies on the polar p of point P.
If a point P moves along a line l, its polar p rotates about the pole L of the line l.
If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points.
If a point lies on the conic section, its polar is the tangent through this point to the conic section.
If a point P lies on its own polar line, then P is on the conic section.
Each line has, with respect to a non-degenerated conic section, exactly one pole.
Special case of circles
The pole of a line L in a circle C is a point Q that is the inversion in C of the point P on L that is closest to the center of the circle. Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C.
The relationship between poles and polars is reciprocal. Thus, if a point A lies on the polar line q of a point Q, then the point Q must lie on the polar line a of the point A. The two polar lines a and q need not be parallel.
There is another description of the polar line of a point P in the case that it lies outside the circle C. In this case, there are two lines through P which are tangent to the circle, and the polar of P is the line joining the two points of tangency (not shown here). This shows that pole and polar line are concepts in the projective geometry of the plane and generalize with any nonsingular conic in the place of the circle C.
Polar reciprocation
The concepts of a pole is and its polar line were advanced in projective geometry. For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the pole, with respect to a conic. The operation of replacing every point by its polar and vice versa is known as a polarity.
A polarity is a correlation that is also an involution.
For some point P and its polar p, any other point Q on p is the pole of a line q through P. This comprises a reciprocal relationship, and is one in which incidences are preserved.
General conic sections
The concepts of pole, polar and reciprocation can be generalized from circles to other conic sections which are the ellipse, hyperbola and parabola. This generalization is possible because conic sections result from a reciprocation of a circle in another circle, and the properties involved, such as incidence and the cross-ratio, are preserved under all projective transformations.
Calculating the polar of a point
A general conic section may be written as a second-degree |
https://en.wikipedia.org/wiki/De%20Moivre%E2%80%93Laplace%20theorem | In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particular, the theorem shows that the probability mass function of the random number of "successes" observed in a series of independent Bernoulli trials, each having probability of success (a binomial distribution with trials), converges to the probability density function of the normal distribution with mean and standard deviation , as grows large, assuming is not or .
The theorem appeared in the second edition of The Doctrine of Chances by Abraham de Moivre, published in 1738. Although de Moivre did not use the term "Bernoulli trials", he wrote about the probability distribution of the number of times "heads" appears when a coin is tossed 3600 times.
This is one derivation of the particular Gaussian function used in the normal distribution.
It is a special case of the central limit theorem because a Bernoulli process can be thought of as the drawing of independent random variables from a bimodal discrete distribution with non-zero probability only for values 0 and 1. In this case, the binomial distribution models the number of successes (i.e., the number of 1s), whereas the central limit theorem states that, given sufficiently large n, the distribution of the sample means will be approximately normal. However, because in this case the fraction of successes (i.e., the number of 1s divided by the number of trials, n) is equal to the sample mean, the distribution of the fractions of successes (described by the binomial distribution divided by the constant n) and the distribution of the sample means (approximately normal with large n due to the central limit theorem) are equivalent.
Theorem
As n grows large, for k in the neighborhood of np we can approximate
in the sense that the ratio of the left-hand side to the right-hand side converges to 1 as n → ∞.
Proof
The theorem can be more rigorously stated as follows: , with a binomially distributed random variable, approaches the standard normal as , with the ratio of the probability mass of to the limiting normal density being 1. This can be shown for an arbitrary nonzero and finite point . On the unscaled curve for , this would be a point given by
For example, with at 3, stays 3 standard deviations from the mean in the unscaled curve.
The normal distribution with mean and standard deviation is defined by the differential equation (DE)
with an initial condition set by the probability axiom .
The binomial distribution limit approaches the normal if the binomial satisfies this DE. As the binomial is discrete the equation starts as a difference equation whose limit morphs to a DE. Difference equations use the discrete derivative, , the change for step size 1. As , the discrete derivative becomes the continuous derivative. Hence the proof need s |
https://en.wikipedia.org/wiki/Mathematical%20object | A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces. Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.
The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics.
List of mathematical objects by branch
Number theory
numbers, operations
Combinatorics
permutations, derangements, combinations
Set theory
sets, set partitions
functions, and relations
Geometry
points, lines, line segments,
polygons (triangles, squares, pentagons, hexagons, ...), circles, ellipses, parabolas, hyperbolas,
polyhedra (tetrahedrons, cubes, octahedrons, dodecahedrons, icosahedrons), spheres, ellipsoids, paraboloids, hyperboloids, cylinders, cones.
Graph theory
graphs, trees, nodes, edges
Topology
topological spaces and manifolds.
Linear algebra
scalars, vectors, matrices, tensors.
Abstract algebra
groups,
rings, modules,
fields, vector spaces,
group-theoretic lattices, and order-theoretic lattices.
Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In proof theory, proofs and theorems are also mathematical objects.
See also
Abstract object
Mathematical structure
References
Azzouni, J., 1994. Metaphysical Myths, Mathematical Practice. Cambridge University Press.
Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object. Oxford Univ. Press.
Davis, Philip and Reuben Hersh, 1999 [1981]. The Mathematical Experience. Mariner Books: 156–62.
Gold, Bonnie, and Simons, Roger A., 2011. Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America.
Hersh, Reuben, 1997. What is Mathematics, Really? Oxford University Press.
Sfard, A., 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P., et al., Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design. Lawrence Erlbaum.
Stewart Shapiro, 2000. Thinking about mathematics: The philosophy of mathematics. Oxford University Press.
External links
Stanford Encyclopedia of Philosophy: "Abstract Objects"—by Gideon Rosen.
Wells, Charles, "Mathematical Objects."
AMOF: The Amazing Mathematical Object Factory
Mathematical Object Exhibit |
https://en.wikipedia.org/wiki/Hermitian%20connection | In mathematics, a Hermitian connection is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric
on , meaning that
for all smooth vector fields and all smooth sections of .
If is a complex manifold, and the Hermitian vector bundle on is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator on associated to the holomorphic structure.
This is called the Chern connection on . The curvature of the Chern connection is a (1, 1)-form. For details, see Hermitian metrics on a holomorphic vector bundle.
In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection coincides with the Levi-Civita connection of the associated Riemannian metric.
References
Shiing-Shen Chern, Complex Manifolds Without Potential Theory.
Shoshichi Kobayashi, Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan, 15. Princeton University Press, Princeton, NJ, 1987. xii+305 pp. .
Complex manifolds
Structures on manifolds
Riemannian geometry |
https://en.wikipedia.org/wiki/Bismut%20connection | In mathematics, the Bismut connection is the unique connection on a complex Hermitian manifold that satisfies the following conditions,
It preserves the metric
It preserves the complex structure
The torsion contracted with the metric, i.e. , is totally skew-symmetric.
Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory.
The explicit construction goes as follows. Let denote the pairing of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. . Further let be the Levi-Civita connection. Define first a tensor such that . This tensor is anti-symmetric in the first and last entry, i.e. the new connection still preserves the metric. In concrete terms, the new connection is given by with being the Levi-Civita connection. The new connection also preserves the complex structure. However, the tensor is not yet totally anti-symmetric; the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as , with given explicitly as
still preserves the complex structure, i.e. .
So if is integrable, then above term vanishes, and the connection
gives the Bismut connection.
Complex manifolds |
https://en.wikipedia.org/wiki/Ireland%20national%20rugby%20union%20team%20tours | This article is a list of statistics from the Ireland rugby union team's 33 international tours. The article also includes details of the Ireland Wolfhounds' and Developmental sides' three international tours.
Ireland Rugby Tours
Tour Statistics
Ireland A, Emerging Ireland & Development Rugby Tours
Tour Statistics
References |
https://en.wikipedia.org/wiki/Poisson%20limit%20theorem | In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem.
Theorem
Let be a sequence of real numbers in such that the sequence converges to a finite limit . Then:
Proofs
Since
and
this leaves
Alternative proof
Using Stirling's approximation, it can be written:
Letting and :
As , so:
Ordinary generating functions
It is also possible to demonstrate the theorem through the use of ordinary generating functions of the binomial distribution:
by virtue of the binomial theorem. Taking the limit while keeping the product constant, it can be seen:
which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the exponential function.)
See also
De Moivre–Laplace theorem
Le Cam's theorem
References
Articles containing proofs
Probability theorems |
https://en.wikipedia.org/wiki/Fundamental%20theorem%20of%20calculus | The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
The first part of the theorem, the first fundamental theorem of calculus, states that for a function , an antiderivative or indefinite integral may be obtained as the integral of over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions.
Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by symbolic integration, thus avoiding numerical integration.
History
The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of continuity of functions and motion were studied by the Oxford Calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.
From the conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.
Geometric meaning
The first fundamental theorem may be interpreted as follows. Given a continuous function whose graph is plotted as a curve, one defines a corresponding "area function" such that is the area beneath the curve between and . The area may not be easily computable, but it is assumed to be well defined.
The area under the |
https://en.wikipedia.org/wiki/Plane%20%28mathematics%29 | In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.
Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional or planar space.
Euclidean plane
Embedding in three-dimensional space
Elliptic plane
Projective plane
Further generalizations
In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each level of abstraction corresponds to a specific category.
At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.
The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.
Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.
In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.
In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a mul |
https://en.wikipedia.org/wiki/Petr%20H%C3%A1jek | Petr Hájek (; 6 February 1940 – 26 December 2016) was a Czech scientist in the area of mathematical logic and a professor of mathematics. Born in Prague, he worked at the Institute of Computer Science at the Academy of Sciences of the Czech Republic and as a lecturer at the faculty of mathematics and physics at the Charles University in Prague and at the Faculty of Nuclear Sciences and Physical Engineering of the Czech Technical University in Prague.
Academics
Petr Hájek studied at the faculty of mathematics and physics of the Charles University in Prague. Influenced by Petr Vopěnka, he specialized in set theory and arithmetic, and later also in logic and artificial intelligence. He contributed to establishing the mathematical fundamentals of fuzzy logic. Following the Velvet Revolution, he was appointed a senior lecturer (1993) and a professor (1997). From 1992 to 2000 he held the position of chairman of the Institute of Computer Science at the Academy of Sciences of the Czech Republic. From 1996 to 2003 he was also president of the Kurt Gödel Society.
Later, he graduated from the Academy of Performing Arts in Prague, where he studied the pipe organ under Jiří Reinberger to become an organ player in a church.
Awards
2002, Medal of the Minister of Education of the Czech Republic
2006, Medal of Merit, third grade, in the area of sciences by President of the Czech Republic Václav Klaus
2008, doctor honoris causa from Silesian University in Opava
Papers
See also
Semiset
References
External links
Petr Hájek's former webpage
Databases of the National Library of the Czech Republic
1940 births
2016 deaths
Charles University alumni
Czech scientists
Mathematicians from Prague
Set theorists
Recipients of Medal of Merit (Czech Republic)
Academic staff of Czech Technical University in Prague
Scientists from Prague
Academy of Performing Arts in Prague alumni |
https://en.wikipedia.org/wiki/Condensation%20point | In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonymous with "-accumulation point".
Examples
If S = (0,1) is the open unit interval, a subset of the real numbers, then 0 is a condensation point of S.
If S is an uncountable subset of a set X endowed with the indiscrete topology, then any point p of X is a condensation point of X as the only neighborhood of p is X itself.
References
Walter Rudin, Principles of Mathematical Analysis, 3rd Edition, Chapter 2, exercise 27
John C. Oxtoby, Measure and Category, 2nd Edition (1980),
Lynn Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, 2nd Edition, pg. 4
Mathematical objects
Topology |
https://en.wikipedia.org/wiki/Gauss%20circle%20problem | In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius . This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area.
The first progress on a solution was made by Carl Friedrich Gauss, hence its name.
The problem
Consider a circle in with center at the origin and radius . Gauss's circle problem asks how many points there are inside this circle of the form where and are both integers. Since the equation of this circle is given in Cartesian coordinates by , the question is equivalently asking how many pairs of integers m and n there are such that
If the answer for a given is denoted by then the following list shows the first few values of for an integer between 0 and 12 followed by the list of values rounded to the nearest integer:
1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441
0, 3, 13, 28, 50, 79, 113, 154, 201, 254, 314, 380, 452
Bounds on a solution and conjecture
is roughly , the area inside a circle of radius . This is because on average, each unit square contains one lattice point. Thus, the actual number of lattice points in the circle is approximately equal to its area, . So it should be expected that
for some error term of relatively small absolute value. Finding a correct upper bound for is thus the form the problem has taken. Note that does not have to be an integer. After one has At these places increases by after which it decreases (at a rate of ) until the next time it increases.
Gauss managed to prove that
Hardy and, independently, Landau found a lower bound by showing that
using the little o-notation. It is conjectured that the correct bound is
Writing , the current bounds on are
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by Martin Huxley in 2000.
Exact forms
The value of can be given by several series. In terms of a sum involving the floor function it can be expressed as:
This is a consequence of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product.
A much simpler sum appears if the sum of squares function is defined as the number of ways of writing the number as the sum of two squares. Then
Most recent progress rests on the following Identity, which has been first discovered by Hardy:
where denotes the Bessel function of the first kind with order 1.
Generalizations
Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes, for example conics; indeed Dirichlet's divisor problem is the equivalent problem where the circle is replaced by the rectangular hyperbola. Similarly one could extend the question from two dimensions to higher dimensions, and ask for integer points within a sphere or other objects. There is an extens |
https://en.wikipedia.org/wiki/Riaz%20Ahsan | Syed Riaz Ahsan (24 December 1951 – 8 September 2008) was a Pakistani statistician and mathematician who has worked in applied statistics, applied analysis, applications of special functions. He was a noted professor of applied statistics in University of Karachi, Karachi. Previously, he has served as the president of Sindh Professors and Lecturers’ Association (SPLA). Prof. Riaz Ahsan was born on 24 December 1951 and he died on 8 September 2008.
Education and career
Born in Karachi, Ahsan had his initial education at the BYJ School, Karachi, and completed his intermediate from Adamjee Government Science College, Karachi. He received his B.Sc. (Hons) in Statistics and Mathematics. He did his M.Sc with distinctions in Statistics and M.A. in Mathematics with honors from Karachi University. Later, he received his double Ph.D. in applied statistics and applied mathematics from the same alma mater in 1974.
Academic career
In 1975, he started his career from DJ Science College, Karachi. He also served as the president of Sindh Professors and Lecturers Association (SPLA) that works for the betterment of professors and lecturers in the province of Sindh. In 1982, he went to Nigeria on deputation and taught statistics and mathematics at the University of Nigeria, Nsukka. Upon his return continued as a teacher at the Government Degree Science College, Karachi, from where he was later transferred to Saint Patrick's College, Karachi. In early 1990s, Ahsan joined Karachi University as a full professor of statistics and mathematics and continued his career until his death.
Death
Dr. Ahsan was diagnosed with cancer in one of his legs in 2004 and was on medication since then. He recovered for a while and participated in many campaigns launched by the SPLA and fell ill again in 2008, never to recover.
Registrar Prof. Raees Alvi paid him the following tribute: "He was a wise man who always guided teachers in the right direction. He always searched for development and never had any prejudice. He always worked for the promotion of merit. This great man always fought for the rights of teachers and I would call it an irreparable loss for all of us."
"The void he left behind can never be filled. It is a huge loss for SPLA and we would never get a personality like him in future", SPLA Senior Vice-president Manzoor Hussain Chishti said.
Family
His father, Professor Syed Zaheer Ahsan, was a professor of Geography at Karachi University.
References
External links
Prof. Riaz Ahsan
Academic staff of the University of Karachi
Pakistani mathematicians
Pakistani statisticians
1951 births
2008 deaths
Academic staff of the University of Nigeria |
https://en.wikipedia.org/wiki/Peter%20Giblin | Peter John Giblin (10 July 1943) is an English mathematician whose primary research involves singularity theory and its application to geometry, computer vision, and computer graphics. Giblin is an emeritus professor of mathematics at the University of Liverpool where he has served on staff for more than 40 years. His positions at Liverpool have included Head of Department (of mathematical sciences), and Head of Division (of pure mathematics).
He is the author or co-author of eight published books, some of which have been translated into Russian. The foreword for the Russian translation of his book Curves and Singularities was written by V. I. Arnold. Giblin has also authored or co-authored over a hundred peer reviewed published articles. The first of these was published in 1968.
Publications
References
Academics of the University of Liverpool
Living people
English mathematicians
1943 births |
https://en.wikipedia.org/wiki/Jordan%27s%20totient%20function | In number theory, Jordan's totient function, denoted as , where is a positive integer, is a function of a positive integer, , that equals the number of -tuples of positive integers that are less than or equal to and that together with form a coprime set of integers
Jordan's totient function is a generalization of Euler's totient function, which is the same as . The function is named after Camille Jordan.
Definition
For each positive integer , Jordan's totient function is multiplicative and may be evaluated as
, where ranges through the prime divisors of .
Properties
which may be written in the language of Dirichlet convolutions as
and via Möbius inversion as
.
Since the Dirichlet generating function of is and the Dirichlet generating function of is , the series for becomes
.
An average order of is
.
The Dedekind psi function is
,
and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of ), the arithmetic functions defined by or can also be shown to be integer-valued multiplicative functions.
.
Order of matrix groups
The general linear group of matrices of order over has order
The special linear group of matrices of order over has order
The symplectic group of matrices of order over has order
The first two formulas were discovered by Jordan.
Examples
Explicit lists in the OEIS are J2 in , J3 in , J4 in , J5 in , J6 up to J10 in up to .
Multiplicative functions defined by ratios are J2(n)/J1(n) in , J3(n)/J1(n) in , J4(n)/J1(n) in , J5(n)/J1(n) in , J6(n)/J1(n) in , J7(n)/J1(n) in , J8(n)/J1(n) in , J9(n)/J1(n) in , J10(n)/J1(n) in , J11(n)/J1(n) in .
Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in , J6(n)/J3(n) in , and J8(n)/J4(n) in .
Notes
References
External links
Modular arithmetic
Multiplicative functions |
https://en.wikipedia.org/wiki/Tomio%20Kubota | (6 December 1930 – 30 June 2020) was a Japanese mathematician working in number theory. His contributions include works on p-adic L functions and real-analytic automorphic forms.
His work on p-adic L-functions, later recognised as an aspect of Iwasawa theory, was done jointly with Leopoldt.
He extended the concept of metaplectic group, in a way significant for arithmetic applications. This opened a field for later research on associated Dirichlet series and automorphic forms, and was a major step in the solution of Kummer's conjecture.
Works
On automorphic functions and the reciprocity law in a number field. Kinokuniya, Tokyo 1969
Notes on analytic theory of numbers. University of Chicago Press, 1963
with Sigekatu Kuroda: . ("Number Theory. Foundations of Algebraic Number Theory"), Asakura Shoten, Tokyo 1963
Some arithmetical applications of an elliptic function, Journal für Reine und Angewandte Mathematik, Band 214/215, 1964/1965, 141-145
editor: Investigations in number theory. Academic Press, 1988
Notes
External links
1. Topological Linear Groupの被覆群 (代数的整数論研究会報告集)---名古屋大学理学部 久保田 富雄 (KUBOTA,TOMIO) / Proceeding of the Conference on Algebraic Number Theory
20th-century Japanese mathematicians
21st-century Japanese mathematicians
1930 births
2020 deaths
Academic staff of Nagoya University |
https://en.wikipedia.org/wiki/Tangent%20lines%20to%20circles | In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.
Tangent lines to one circle
A tangent line to a circle intersects the circle at a single point . For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed.
The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.
No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining to the center point of the circle. Thus the lengths of the segments from to the two tangent points are equal. By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle . This power equals the product of distances from to any two intersection points of the circle with a secant line passing through .
The tangent line and the tangent point have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point outside the circle and the secant line joining its two points of tangency.
If a point is exterior to a circle with center , and if the tangent lines from touch the circle at points and , then and are supplementary (sum to 180°).
If a chord is drawn from the tangency point of exterior point and then .
Cartesian equation
Suppose that the equation of the circle in Cartesian coordinates is with center at . Then the tangent line of the circle at has Cartesian equation
This can be proved by taking the implicit derivative of the circle.
Say that the circle has equation of and we are finding the slope of tangent line at where We begin by taking the implicit derivative with respect to :
Now that we have the slope of the tangent line, we can substitute the |
https://en.wikipedia.org/wiki/Mary%20Tiles | Mary Tiles (born 1946) is a philosopher and historian of mathematics and science. From 2006 until 2009, she served as chair of the philosophy department of the University of Hawaii at Manoa. She retired in 2009.
Life
At Bristol University, Tiles obtained her B.A. in philosophy and mathematics in 1967, and her Ph.D. in philosophy in 1973, followed by a B.Phil. in philosophy in 1974 at Oxford and a M.A. in 1978 at Cambridge. After positions as lecturer and visiting associate professor at different institutions, Tiles became associate professor of philosophy at University of Hawaii at Manoa in 1989, and full professor in 1992.
Work
Tiles' area of work is primarily philosophy and history of logic, mathematics and science, with a special emphasis on French contributions to this area, e.g. by Gaston Bachelard, Georges Canguilhem, Bruno Latour,
Michel Foucault, Pierre Bourdieu, Michel Serres, Jean-Claude Martzloff, Karine Chemla, Catherine Jami, and François Jullien.
One of her publications is the 1989 book The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. As the subtitle suggests, it is an example of a book that treats the philosophy of mathematics as inseparable from historical concerns. Despite some criticisms, for its lack of technical detail and correctness, and for pressing the author's philosophical agenda on its readers, it has been recommended as an introductory textbook for undergraduates interested in the philosophy of mathematics.
Bibliography
with Hans Oberdiek, Living in a Technological Culture: Human Tools and Human Values, Routledge 1995.
with Jim Tiles, An Introduction to Historical Epistemology: The Authority of Knowledge, Oxford 1993.
Mathematics and the Image of Reason, Routledge 1991.
The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, Blackwell 1989; reprinted by Dover 2004.
Bachelard: Science and Objectivity, Cambridge University Press 1984.
References
1946 births
Living people
Philosophers of science
American historians of mathematics
Mathematical logicians
Women logicians
Set theorists
British mathematicians
Alumni of the University of Oxford
Alumni of the University of Cambridge
University of Hawaiʻi faculty
20th-century American mathematicians
21st-century American mathematicians
British women mathematicians
American women mathematicians
20th-century American women
21st-century American women
Philosophers of technology |
https://en.wikipedia.org/wiki/Kensuke%20Fukuda | Kensuke Fukuda (福田 健介, born 24 July 1984 in Shizuoka) is a Japanese football player who plays for Ococias Kyoto AC.
Club career statistics
Updated to 23 February 2017.
References
External links
Profile at Ventforet Kofu
Profile at V-Varen Nagasaki
1984 births
Living people
Meiji University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Tokyo Verdy players
Ventforet Kofu players
V-Varen Nagasaki players
Tochigi SC players
Ococias Kyoto AC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Weak%20Hausdorff%20space | In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff. As a separation property, it is stronger than T1, which is equivalent to the statement that points are closed. Specifically, every weak Hausdorff space is a T1 space.
The notion was introduced by M. C. McCord to remedy an inconvenience of working with the category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces.
k-Hausdorff spaces
A is a topological space which satisfies any of the following equivalent conditions:
Each compact subspace is Hausdorff.
The diagonal is k-closed in
A subset is , if is closed in for each compact
Each compact subspace is closed and strongly locally compact.
A space is if for each and each (not necessarily open) neighborhood of there exists a compact neighborhood of such that
Properties
A k-Hausdorff space is weak Hausdorff. For if is k-Hausdorff and is a continuous map from a compact space then is compact, hence Hausdorff, hence closed.
A Hausdorff space is k-Hausdorff. For a space is Hausdorff if and only if the diagonal is closed in and each closed subset is a k-closed set.
A k-Hausdorff space is KC. A is a topological space in which every compact subspace is closed.
To show that the coherent topology induced by compact Hausdorff subspaces preserves the compact Hausdorff subspaces and their subspace topology requires that the space be k-Hausdorff; weak Hausdorff is not enough. Hence k-Hausdorff can be seen as the more fundamental definition.
Δ-Hausdorff spaces
A is a topological space where the image of every path is closed; that is, if whenever is continuous then is closed in Every weak Hausdorff space is -Hausdorff, and every -Hausdorff space is a T1 space. A space is if its topology is the finest topology such that each map from a topological -simplex to is continuous. -Hausdorff spaces are to -generated spaces as weak Hausdorff spaces are to compactly generated spaces.
See also
, a Hausdorff space where every continuous function from the space into itself has a fixed point.
References
Properties of topological spaces
Separation axioms |
https://en.wikipedia.org/wiki/Kim%20Tae-yoon%20%28footballer%29 | Kim Tae-yoon (; born 25 July 1986) is a retired South Korean football defender.
Career statistics
Honors
Club
Seongnam Ilhwa Chunma
2010 AFC Champions League Winner
2011 FA Cup Winner
External links
Kim Tae-youn at Seongnam FC
1986 births
Living people
Men's association football defenders
South Korean men's footballers
South Korean expatriate men's footballers
South Korean expatriate sportspeople in Thailand
Seongnam FC players
Gimcheon Sangmu FC players
Incheon United FC players
Kim Tae-yoon
Gwangju FC players
K League 1 players
K League 2 players
Footballers from Busan
Expatriate men's footballers in Thailand |
https://en.wikipedia.org/wiki/Ma%20Chul-jun | Ma Chul-jun (; born 16 November 1980) is a South Korean former footballer and manager.
Career statistics
External links
1980 births
Living people
Men's association football defenders
South Korean men's footballers
Goyang Zaicro FC players
Jeju United FC players
Gimcheon Sangmu FC players
Jeonbuk Hyundai Motors players
Gwangju FC players
Korea National League players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Nam%20Ik-kyung | Nam Ik-Kyung (; born 26 January 1983) is a South Korean retired football player. In March 2009 he signed a tryout agreement with Finnish team JJK.
Career statistics
As of 28 February 2011 (UTC)
References
External links
Tilastohistoria
1983 births
Living people
Men's association football forwards
South Korean men's footballers
Pohang Steelers players
Gimcheon Sangmu FC players
JJK Jyväskylä players
FC Haka players
K League 1 players
Veikkausliiga players
South Korean expatriates in Finland
Expatriate men's footballers in Finland |
https://en.wikipedia.org/wiki/Heo%20Jae-won | Heo Jae-won (Hangul: 허재원; Hanja: 許宰源; born 1 July 1984) is a South Korean football player who last played for Jeonnam Dragons.
Career statistics
As of end of 2011 season
References
External links
1984 births
Living people
Men's association football defenders
South Korean men's footballers
Suwon Samsung Bluewings players
Gimcheon Sangmu FC players
Gwangju FC players
Jeju United FC players
Al-Khor SC players
Dibba Al Fujairah FC players
Jeonnam Dragons players
K League 1 players
K League 2 players
Qatar Stars League players
UAE Pro League players
People from Uijeongbu
Footballers from Gyeonggi Province |
https://en.wikipedia.org/wiki/Choi%20Byung-do | Choi Byung-do (; born 18 January 1984) is a South Korean football defender.
Career statistics
References
External links
1984 births
Living people
Men's association football defenders
South Korean men's footballers
Incheon United FC players
Gimcheon Sangmu FC players
Goyang Zaicro FC players
Ulsan Hyundai Mipo Dockyard FC players
Bucheon FC 1995 players
Seoul E-Land FC players
K League 1 players
Korea National League players
K League 2 players |
https://en.wikipedia.org/wiki/Shin%20Soo-jin | Shin Soo-Jin (born October 26, 1982) is a South Korean football player who is currently a free agent.
He formerly played for Gwangju Sangmu and Busan I'Park in the K-League.
Career statistics
As of end of 2008 season
References
Korean FA Cup match result
1982 births
Living people
Men's association football defenders
South Korean men's footballers
Busan IPark players
Gimcheon Sangmu FC players
Ulsan Hyundai Mipo Dockyard FC players
K League 1 players
Korea National League players |
https://en.wikipedia.org/wiki/Lee%20Kwang-hyun | Lee Kwang-Hyun (born 18 July 1981) is a South Korean football player who currently plays with Penang FA in Malaysia Premier League.
Career statistics
References
Korean FA Cup match result
ifball.com
Lee Kwang-Hyun at footballmalaysia.com
1981 births
Living people
Men's association football defenders
South Korean men's footballers
Jeonbuk Hyundai Motors players
Gimcheon Sangmu FC players
Daejeon Hana Citizen players
K League 1 players
Penang F.C. players
Expatriate men's footballers in Malaysia |
https://en.wikipedia.org/wiki/Cha%20Keon-myung | Cha Keon-Myung (born December 26, 1981) is a South Korean football player who plays for Jeju United FC. He has also played for Suwon Samsung Bluewings and Gwangju Sangmu.
Career statistics
As of end of 2008 season
References
K-League player record
Korean FA Cup match result
1981 births
Living people
Men's association football defenders
South Korean men's footballers
Suwon Samsung Bluewings players
Gimcheon Sangmu FC players
Jeju United FC players
K League 1 players |
https://en.wikipedia.org/wiki/Lee%20Hyun-min | Lee Hyun-Min (born July 9, 1984) is a South Korean football player. (formerly Ulsan Hyundai FC and Gwangju Sangmu FC).
Career statistics
As of end of 2009 season
See also
Football in South Korea
List of football clubs in South Korea
References
1984 births
Living people
Men's association football defenders
South Korean men's footballers
Ulsan Hyundai FC players
Gimcheon Sangmu FC players
K League 1 players
Korea National League players |
https://en.wikipedia.org/wiki/Lee%20Wan%20%28footballer%29 | Lee Wan (; born 3 May 1984) is a South Korean football defender, who plays for Gangwon FC in K League Challenge.
Club career statistics
External links
1984 births
Living people
Men's association football defenders
South Korean men's footballers
Jeonnam Dragons players
Gimcheon Sangmu FC players
Ulsan Hyundai FC players
Gwangju FC players
Gangwon FC players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Lee%20Su-hwan%20%28footballer%29 | Lee Su-hwan (born March 3, 1984) is a South Korean football player who currently plays for Cheonan City FC.
Career statistics
As of 30 August 2009
References
K League player record
Korean FA Cup match result
1984 births
Living people
Men's association football midfielders
South Korean men's footballers
Pohang Steelers players
Gimcheon Sangmu FC players
K League 1 players
Korea National League players |
https://en.wikipedia.org/wiki/Shin%20Dong-keun | Shin Dong-keun (; born 15 February 1981) is a South Korean football midfielder.
Club career statistics
External links
1981 births
Living people
Men's association football midfielders
South Korean men's footballers
Seongnam FC players
Gimcheon Sangmu FC players
K League 1 players
Korea National League players |
https://en.wikipedia.org/wiki/Han%20Tae-you | Han Tae-you (born March 30, 1981) is a South Korean football player.
Career statistics
''As of 24 July 2014
Honors
Club
FC Seoul
K League
Winners (1): 2010, 2012
League Cup
Winners (1): 2010
References
1981 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's international footballers
FC Seoul players
Gimcheon Sangmu FC players
K League 1 players
Myongji University alumni
Footballers from Ulsan |
https://en.wikipedia.org/wiki/Han%20Seol | Han Seol (born July 15, 1983) is a South Korean football player who since November 2008 has played for Busan I'Park. (formerly Gwangju Sangmu)
Career statistics
As of end of 2008 season
References
Korean FA Cup match result
1983 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korean expatriate men's footballers
Busan IPark players
Gimcheon Sangmu FC players
Goyang KB Kookmin Bank FC players
K League 1 players
Korea National League players
Liga 1 (Indonesia) players
Expatriate men's footballers in Indonesia
South Korean expatriate sportspeople in Indonesia |
https://en.wikipedia.org/wiki/Baek%20Joo-hyun | Baek Joo-Hyun (born February 9, 1984) is a South Korea football player who since November 2008 has played for Suwon Samsung Bluewings.
Career statistics
As of end of 2008 season
References
K-League player record
Korean FA Cup match result
Suwon Samsung Bluewings players
Gimcheon Sangmu FC players
K League 1 players
Men's association football midfielders
South Korean men's footballers
1984 births
Living people |
https://en.wikipedia.org/wiki/Kim%20Myung-joong | Kim Myung-Joong (Hangul: 김명중; born 6 February 1985) is a South Korea football winger, who plays for Gangwon FC in K-League.
Career statistics
References
External links
1985 births
Living people
Men's association football midfielders
South Korean men's footballers
Pohang Steelers players
Gimcheon Sangmu FC players
Jeonnam Dragons players
Gangwon FC players
K League 1 players |
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