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https://en.wikipedia.org/wiki/B%C3%A9la%20Ker%C3%A9kj%C3%A1rt%C3%B3
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Béla Kerékjártó (1 October 1898, in Budapest – 26 June 1946, in Gyöngyös) was a Hungarian mathematician who wrote numerous articles on topology.
Kerékjártó earned his Ph.D. degree from the University of Budapest in 1920. He taught at the Faculty of Sciences of the University of Szeged starting in 1922. In 1921 he introduced his program with a talk "On topological fundamentals of analysis and geometry" where he advocated that "complex analysis should be built with instruments of topology without metric elements such as length and area."
Life and career
In 1923, Kerékjártó published one of the first books on Topology, which was reviewed by Solomon Lefschetz in 1925. Hermann Weyl wrote that this book completely changed his views of the subject.
In 1919 he published a theorem on periodic homeomorphisms of the disc and the sphere. A claim to priority to the result was made by L. E. J. Brouwer, and the subject was revisited by Samuel Eilenberg in 1934. A modern treatment of Kerékjártó's theorem has been presented by Adrian Constantin and Boris Kolev.
Kerékjártó was appointed head of the Department of Geometry and Descriptive Geometry at the János Bolyai Mathematical Institute of the University of Szeged in 1925.
In 1938 he returned to Budapest to teach at Eötvös Loránd University.
Kerékjártó proved that the sphere is the only compact surface that admits a 3-transitive topological group in 1941.
Books
1923: Vorlesungen über Topologie Bd.1 Flächentopologie, Grundlehren der mathematischen Wissenschaften, Springer Verlag
1955: Les Fondements de la Géométrie. Bd.1. La construction élémentaire de la géométrie euclidienne, Gauthier-Villars.
1966: Les Fondaments de la Géométrie Bd.2, Geometrie projective, Gauthiers-Villars.
Articles
1919: "A torus periodikus transformitioirol", Math. Term. tad. Értesitiö 39:213–9.
1930: "Geometrische Theorie der zweigliedrigen kontinuierlichen Gruppen", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 8:107–14.
1934: "Sur la character topologique des representationes conformes", Comptes rendus 198:317–20.
1934: "Über reguläre Abbildungen von Flächen auf sich", Acta Scientiarum Mathematicarum 7:65–85 & 206.
1934: "Topologische Characterisierung der linearen Abbildungen", Acta Scientiarum Mathematicarum 6:235–62, esp. 250.
1940: "Sur les inversions dans un groupe commutative", Comptes rendus 210:288.
1940: "Sur le group des homographies et des antihomographies d’une variable complexe", Commentarii Mathematici Helvetici 13:68–82.
1941: "Sur les groups compact de transformations topologique des surfaces", Acta Mathematica 74:129–73.
References
External links
20th-century Hungarian mathematicians
Topologists
Academic staff of the University of Szeged
Mathematicians from Budapest
1898 births
1946 deaths
Mathematicians from Austria-Hungary
Eötvös Loránd University alumni
Academic staff of Eötvös Loránd University
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https://en.wikipedia.org/wiki/Equating%20coefficients
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In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term. The method is used to bring formulas into a desired form.
Example in real fractions
Suppose we want to apply partial fraction decomposition to the expression:
that is, we want to bring it into the form:
in which the unknown parameters are A, B and C.
Multiplying these formulas by x(x − 1)(x − 2) turns both into polynomials, which we equate:
or, after expansion and collecting terms with equal powers of x:
At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0x2 + 0x + 1, having zero coefficients for the positive powers of x. Equating the corresponding coefficients now results in this system of linear equations:
Solving it results in:
Example in nested radicals
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that
Squaring both sides of this equation yields:
To find d and e we equate the terms not involving square roots, so and equate the parts involving radicals, so which when squared implies This gives us two equations, one quadratic and one linear, in the desired parameters d and e, and these can be solved to obtain
which is a valid solution pair if and only if is a rational number.
Example of testing for linear dependence of equations
Consider this overdetermined system of equations (with 3 equations in just 2 unknowns):
To test whether the third equation is linearly dependent on the first two, postulate two parameters a and b such that a times the first equation plus b times the second equation equals the third equation. Since this always holds for the right sides, all of which are 0, we merely need to require it to hold for the left sides as well:
Equating the coefficients of x on both sides, equating the coefficients of y on both sides, and equating the constants on both sides gives the following system in the desired parameters a, b:
Solving it gives:
The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly dependent on the first two.
Note that if the constant term in the original third equation had been anything other than –7, the values (a, b) = (1, 1) that satisfied the first two equations in the parameters would not have
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https://en.wikipedia.org/wiki/Geodesic%20manifold
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In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point , is defined on , the entire tangent space at .
Equivalently, consider a maximal geodesic . Here is an open interval of , and, because geodesics are parameterized with "constant speed", it is uniquely defined up to transversality. Because is maximal, maps the ends of to points of , and the length of measures the distance between those points. A manifold is geodesically complete if for any such geodesic , we have that .
Examples and non-examples
Euclidean space , the spheres , and the tori (with their natural Riemannian metrics) are all complete manifolds.
All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric spaces are geodesically complete.
Every finite-dimensional path-connected Riemannian manifold which is also a complete metric space (with respect to the Riemannian distance) is geodesically complete. In fact, geodesic completeness and metric completeness are equivalent for these spaces. This is the content of the Hopf–Rinow theorem.
Non-examples
A simple example of a non-complete manifold is given by the punctured plane (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line. By the Hopf–Rinow theorem, we can alternatively observe that it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. An example of this is the Clifton–Pohl torus.
In the theory of general relativity, which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged black-holes or cosmologies with a Big Bang. The fact that such incompleteness is fairly generic in general relativity is shown in the Penrose–Hawking singularity theorems.
References
Differential geometry
Geodesic (mathematics)
Manifolds
Riemannian geometry
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https://en.wikipedia.org/wiki/Core-Plus%20Mathematics%20Project
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Core-Plus Mathematics is a high school mathematics program consisting of a four-year series of print and digital student textbooks and supporting materials for teachers, developed by the Core-Plus Mathematics Project (CPMP) at Western Michigan University, with funding from the National Science Foundation. Development of the program started in 1992. The first edition, entitled Contemporary Mathematics in Context: A Unified Approach, was completed in 1995. The third edition, entitled Core-Plus Mathematics: Contemporary Mathematics in Context, was published by McGraw-Hill Education in 2015.
Key Features
The first edition of Core-Plus Mathematics was designed to meet the curriculum, teaching, and assessment standards from the National Council of Teachers of Mathematics and the broad goals outlined in the National Research Council report, Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Later editions were designed to also meet the American Statistical Association Guidelines for Assessment and Instruction in Statistics Education (GAISE) and most recently the standards for mathematical content and practice in the Common Core State Standards for Mathematics (CCSSM).
The program puts an emphasis on teaching and learning mathematics through mathematical modeling and mathematical inquiry. Each year, students learn mathematics in four interconnected strands: algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematical modeling.
First Edition (1994-2003)
The program originally comprised three courses, intended to be taught in grades 9 through 11. Later, authors added a fourth course intended for college-bound students.
Second Edition (2008-2011)
The course was re-organized around interwoven strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. Lesson structure was updated, and technology tools, including CPMP-Tools software was introduced.
CCSS Edition (2015)
The course was aligned with the Common Core State Standards (CCSS) mathematical practices and content expectations. Expanded and enhanced Teacher's Guides include a CCSS pathway and a CPMP pathway through each unit. Course 4 was split into two versions: one called Preparation for Calculus, for STEM-oriented students, and an alternative course, Transition to College Mathematics and Statistics (TCMS), for college-bound students whose intended program of study does not require calculus.
Evaluations, Research, and Reviews
Project and independent evaluations and many research studies have been conducted on Core-Plus Mathematics, including content analyses, case studies, surveys, small- and large-scale comparison studies, research reviews, and a longitudinal study.
Positive reviews
There are multiple research studies and evaluations in which students using Core-Plus Mathematics performed significantly better than comparison students on assessments of conceptual unde
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https://en.wikipedia.org/wiki/Mathematically%20Correct
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Mathematically Correct was a U.S.-based website created by educators, parents, mathematicians, and scientists who were concerned about the direction of reform mathematics curricula based on NCTM standards. Created in 1997, it was a frequently cited website in the so-called Math wars, and was actively updated until 2003.
History
Although Mathematically Correct had a national scope, much of its focus was on advocating against mathematics curricula prevalent in California in the mid-1990s. When California reversed course and adopted more traditional mathematics texts (2001 - 2002), Mathematically Correct changed its focus to reviewing the new text books. Convinced that the choices were adequate, the website went largely dormant.
Mathematically Correct maintained a large section of critical articles and reviews for a number of math programs. Most of the program opposed by Mathematically Correct had been developed from research projects funded by the National Science Foundation. Most of these programs also claimed to have been based on the 1989 Curriculum and Evaluation Standards for School Mathematics published by the National Council of Teachers of Mathematics.
Mathematically Correct's main point of contention was that, in reform textbooks, traditional methods and concepts have been omitted or replaced by new terminology and procedures. As a result, in the case of the high-school program Core-Plus Mathematics Project, for example, some reports suggest that students may be unprepared for college level courses upon completion of the program. Other programs given poor ratings include programs aimed at elementary school students, such as Dale Seymour Publications (TERC) Investigations in Numbers, Data, and Space and Everyday Learning Everyday Mathematics.
After Mathematically Correct's review of the programs, many have undergone revisions and are now with different publishers. Other programs, such as Mathland have been terminated.
Reviews by the site
Publications with poor reviews from Mathematically Correct include:
Addison-Wesley Secondary Math: An Integrated Approach: Focus on Algebra
Core-Plus Mathematics Project
Investigations in Numbers, Data, and Space
Mathland
NCTM's 1989 Standards produced by the professional association for teachers of mathematics. The Standards encouraged increased emphasis on problem solving and decreased attention to algorithms learned by rote. It was perceived by some critics to recommend elimination of standard algorithms.
Curricula not judged deficient by Mathematically Correct include:
Singapore math – Math textbooks used in Singapore
Saxon math – A program created by a retired Air Force officer who had been highly critical of mathematics teaching reforms from the late 1960s to 1990s.
State tests that were judged deficient by Mathematically Correct are:
CLAS – A defunct California test, based on NCTM standards and California mathematics standards replaces in 2002
WASL – Washington State standards
Refe
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https://en.wikipedia.org/wiki/P.%20A.%20P.%20Moran
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Patrick Alfred Pierce Moran FRS (14 July 1917 – 19 September 1988) was an Australian statistician who made significant contributions to probability theory and its application to population and evolutionary genetics.
Early years
Patrick Moran was born in Sydney and was the only child of Herbert Michael Moran (b. 1885 in Sydney, d. 1945 in Cambridge UK), a prominent surgeon and captain of the first Wallabies, and Eva Mann (b. 1887 in Sydney, d. 1977 in Sydney). Patrick did have five other siblings, but they all died at or shortly after birth. He completed his high school studies in Bathurst, in three and a half years instead of the normal five-year course. At age 16, in 1934, he commenced study at the University of Sydney where he studied chemistry, math and physics, graduating with first class honours in mathematics in 1937. Following graduation he went to study at Cambridge University from 1937 to 1939, his supervisors noted that he was not a good mathematician and the outbreak of World War II interrupted his studies. He graduated with an MA (by proxy) from St John's College, Cambridge, on 22 January 1943 and continued his studies there from 1945 to 1946. He was admitted to Balliol College, Oxford University, on 3 December 1946. He was awarded an MA, from Oxford University, by incorporation in 1947.
Career
During the war Moran worked in rocket development in the Ministry of Supply and later at the External Ballistics Laboratory in Cambridge. In late 1943 he joined the Australian Scientific Liaison Office (ASLO), run by the CSIRO. He worked on applied physics including vision, camouflage, army signals, quality control, road research, infra-red detection, metrology, UHF radio propagation, general radar, bomb-fragmentation, rockets, ASDICs and on operational research. He also wrote some papers on the Hausdorff measure during the War.
After the war, Moran returned to Cambridge where he was supervised by Frank Smithies and worked unsuccessfully on determining the nature of the set of points of divergence of Fourier integrals of functions in the class Lp, when 1 < p < 2. He gave up on this project and was employed as a senior research officer at the Institute of Statistics at Oxford University. He also gave lecture courses. Patrick Moran was appointed university lecturer in mathematics in 1951, at Oxford, without stipend, for as long as he held the post of senior research officer in the Institute of Statistics. Moran freely admitted he had difficulty with simple arithmetic and wrote that, "Arithmetic I could not do".
He married in 1946 after his appointment; he and his wife Jean Mavis Frame had three children, Louise, Michael and Hugh. At Oxford Moran wrote several papers on the nonlinear breeding cycle of the Canadian lynx. He was made a lecturer at Oxford in 1951 but left the university later that year for Australia. He never acquired a PhD, "a fact he would recall with some pride in later life", recalls Hall.
On 1 January 1951, Moran was appo
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https://en.wikipedia.org/wiki/Traditional%20mathematics
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Traditional mathematics (sometimes classical math education) was the predominant method of mathematics education in the United States in the early-to-mid 20th century. This contrasts with non-traditional approaches to math education. Traditional mathematics education has been challenged by several reform movements over the last several decades, notably new math, a now largely abandoned and discredited set of alternative methods, and most recently reform or standards-based mathematics based on NCTM standards, which is federally supported and has been widely adopted, but subject to ongoing criticism.
Traditional methods
The topics and methods of traditional mathematics are well documented in books and open source articles of many nations and languages. Major topics covered include:
Elementary arithmetic
Addition
Carry
Subtraction
Multiplication
Multiplication table
Division
Long division
Arithmetic with fractions
Lowest common denominator
Arithmetic mean
Volume
In general, traditional methods are based on direct instruction where students are shown one standard method of performing a task such as decimal addition, in a standard sequence. A task is taught in isolation rather than as only a part of a more complex project. By contrast, reform books often postpone standard methods until students have the necessary background to understand the procedures. Students in modern curricula often explore their own methods for multiplying multi-digit numbers, deepening their understanding of multiplication principles before being guided to the standard algorithm. Parents sometimes misunderstand this approach to mean that the children will not be taught formulas and standard algorithms and therefore there are occasional calls for a return to traditional methods. Such calls became especially intense during the 1990s. (See Math wars.)
A traditional sequence early in the 20th century would leave topics such as algebra or geometry entirely for high school, and statistics until college, but newer standards introduce the basic principles needed for understanding these topics very early. For example, most American standards now require children to learn to recognize and extend patterns in kindergarten. This very basic form of algebraic reasoning is extended in elementary school to recognize patterns in functions and arithmetic operations, such as the distributive law, a key principle for doing high school algebra. Most curricula today encourage children to reason about geometric shapes and their properties in primary school as preparation for more advanced reasoning in a high school geometry course. Current standards require children to learn basic statistical ideas such as organizing data with bar charts. More sophisticated concepts such as algebraic expressions with numbers and letters, geometric surface area and statistical means and medians occur in sixth grade in the newest standards.
Criticism of traditional math
Criticism of traditional mathema
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https://en.wikipedia.org/wiki/Investigations%20in%20Numbers%2C%20Data%2C%20and%20Space
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Investigations in Numbers, Data, and Space is a K–5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts, United States. The curriculum is often referred to as Investigations or simply TERC. Patterned after the NCTM standards for mathematics, it is among the most widely used of the new reform mathematics curricula. As opposed to referring to textbooks and having teachers impose methods for solving arithmetic problems, the TERC program uses a constructivist approach that encourages students to develop their own understanding of mathematics. The curriculum underwent a major revision in 2005–2007.
History
Investigations was developed between 1990 and 1998. It was just one of a number of reform mathematics curricula initially funded by a National Science Foundation grant. The goals of the project raised opposition to the curriculum from critics (both parents and mathematics teachers) who objected to the emphasis on conceptual learning instead of instruction in more recognized specific methods for basic arithmetic..
The goal of the Investigations curriculum is to help all children understand the fundamental ideas of number and arithmetic, geometry, data, measurement and early algebra. Unlike traditional methods, the original edition did not provide student textbooks to describe standard methods or provide solved examples. Instead, students were guided to develop their own invented algorithms through working with concrete representations of number such as manipulatives and drawings as well as more traditional number sentences. Additional activities include journaling, cutting and pasting, interviewing (for data collection) and playing conceptual games.
Investigations released its second edition for 2006 that continues its focus on the core value of teaching for understanding. The revised version has further emphasis on basic skills and computation to complement the development of place value concepts and number sense. It is also easier for teachers to use since the format is more user friendly, though some districts have failed to carefully implement the second edition as well, and moved back to textbooks that teach traditional arithmetic methods.
Research
A systematic review of research into Investigations was conducted by the U.S. Department of Education, Institute of Education Sciences and published as part of the What Works Clearinghouse in February 2013. This found "potentially positive effects" on mathematics achievement, supported by a "medium to large" evidence base.
A variety of measures of student achievement and learning including state-mandated standardized tests, research-based interview protocols, items from research studies published in peer-reviewed journals and specially constructed paper-and-pencil tests have been used to evaluate the effectiveness of Investigations
Research featured at the TERC website states that students who use Investigations, among other things, 'do as well or better than students using
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https://en.wikipedia.org/wiki/The%20Analyst%2C%20or%2C%20Mathematical%20Museum
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The Analyst, or, Mathematical Museum was an early American mathematics journal. Founded by Robert Adrain in 1808, it published one volume of four issues that year before discontinuing publication. Despite its extremely short life, it published papers by several notable mathematicians in the nascent American mathematical community, including Nathaniel Bowditch and Ferdinand Hassler; most importantly, Adrain himself published an independent formulation of the method of least squares.
After securing a professorship at Columbia University, Adrain attempted to revive the journal in 1814, but it published only one issue before again ceasing publication. He would later go on to found a more popularly oriented journal, The Mathematical Diary.
References
Defunct journals of the United States
Mathematics journals
Publications established in 1808
Publications disestablished in 1808
Publications established in 1814
Publications disestablished in 1814
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https://en.wikipedia.org/wiki/John%20Wood%20%28Kent%20cricketer%2C%20born%201745%29
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John Wood (1745 – July 1816 at Seal, Kent) was an English cricketer who played for Kent. His career began in the 1760s before first-class statistics began to be recorded and his known first-class career spans the 1772 to 1783 seasons.
He has often been confused with his namesake who played for Surrey at the same time. Although Wood is credited with 12 first-class appearances by CricketArchive, there are only 10 which can definitely be attributed to him. Using the data in Scores and Biographies, there were 12 matches in which a player known only as "Wood" took part, with Wood of Surrey specifically recorded in 13.
According to John Nyren, Wood of Kent was a "change bowler who was tall, stout, bony and a very good general player". According to H T Waghorn, he suffered a serious knee injury in 1773 and there were fears of amputation being necessary. However, he was playing again in 1774 so things cannot have been as bad as they first seemed.
The first time a John Wood is mentioned in the sources is when one plays for Caterham against Hambledon in 1769. This was probably the Surrey-based player. In the same season, a player called Wood played for the Duke of Dorset's XI against Wrotham in the minor match that featured John Minshull's century. Given Dorset's strong Kent connection, this was probably John Wood of Kent.
References
Bibliography
Arthur Haygarth, Scores & Biographies, Volume 1 (1744-1826), Lillywhite, 1862
John Nyren, The Cricketers of my Time (ed. Ashley Mote), Robson, 1998
H T Waghorn, Cricket Scores, Notes, etc. (1730–1773), Blackwood, 1899
English cricketers
English cricketers of 1701 to 1786
Kent cricketers
1745 births
1816 deaths
Non-international England cricketers
West Kent cricketers
People from Seal, Kent
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https://en.wikipedia.org/wiki/Vertical%20line%20test
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In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function. If all vertical lines intersect a curve at most once then the curve represents a function.
See also
Horizontal line test
Notes
Functions and mappings
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https://en.wikipedia.org/wiki/Bootstrapping%20%28statistics%29
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Bootstrapping is any test or metric that uses random sampling with replacement (e.g. mimicking the sampling process), and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.
Bootstrapping estimates the properties of an estimand (such as its variance) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data. In the case where a set of observations can be assumed to be from an independent and identically distributed population, this can be implemented by constructing a number of resamples with replacement, of the observed data set (and of equal size to the observed data set).
It may also be used for constructing hypothesis tests. It is often used as an alternative to statistical inference based on the assumption of a parametric model when that assumption is in doubt, or where parametric inference is impossible or requires complicated formulas for the calculation of standard errors.
History
The bootstrap was published by Bradley Efron in "Bootstrap methods: another look at the jackknife" (1979), inspired by earlier work on the jackknife. Improved estimates of the variance were developed later. A Bayesian extension was developed in 1981. The bias-corrected and accelerated (BCa) bootstrap was developed by Efron in 1987, and the ABC procedure in 1992.
Approach
The basic idea of bootstrapping is that inference about a population from sample data (sample → population) can be modeled by resampling the sample data and performing inference about a sample from resampled data (resampled → sample). As the population is unknown, the true error in a sample statistic against its population value is unknown. In bootstrap-resamples, the 'population' is in fact the sample, and this is known; hence the quality of inference of the 'true' sample from resampled data (resampled → sample) is measurable.
More formally, the bootstrap works by treating inference of the true probability distribution J, given the original data, as being analogous to an inference of the empirical distribution Ĵ, given the resampled data. The accuracy of inferences regarding Ĵ using the resampled data can be assessed because we know Ĵ. If Ĵ is a reasonable approximation to J, then the quality of inference on J can in turn be inferred.
As an example, assume we are interested in the average (or mean) height of people worldwide. We cannot measure all the people in the global population, so instead, we sample only a tiny part of it, and measure that. Assume the sample is of size N; that is, we measure the heights of N individuals. From that single sample, only one estimate of the mean can be obtained. In orde
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https://en.wikipedia.org/wiki/Vertex%20normal
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In the geometry of computer graphics, a vertex normal at a vertex of a polyhedron is a directional vector associated with a vertex, intended as a replacement to the true geometric normal of the surface. Commonly, it is computed as the normalized average of the surface normals of the faces that contain that vertex. The average can be weighted for example by the area of the face or it can be unweighted. Vertex normals can also be computed for polygonal approximations to surfaces such as NURBS, or specified explicitly for artistic purposes. Vertex normals are used in Gouraud shading, Phong shading and other lighting models. Using vertex normals, much smoother shading than flat shading can be achieved; however, without some modifications to topology such a support loops, it cannot produce a sharper edge.
See also
Specular highlight
Per-pixel lighting
References
Shading
pt:Normal de Vértice
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https://en.wikipedia.org/wiki/2001%E2%80%9302%20in%20Portuguese%20football
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List of Portuguese football statistics for the 2001 to 2002 Season.
Primeira Liga
References
Portuguese League Association website
Seasons in Portuguese football
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https://en.wikipedia.org/wiki/Compu-Math%20series
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The Compu-Math series are mathematics tutorials developed and published by Edu-Ware Services in the 1980s. Each program in the Compu-Math series begins with a diagnostic Pre-Test, which presents learners with mathematics problems to determine their current skill level in the subject and then recommends the appropriate learning module. Each learning module begins by specifying the instructional objectives for that module and proceeds to teach those specific goals using shaping and cueing methods, and finishes by testing to verify that learners have indeed learned the skills being taught by the module.
After learners progress through all recommended learning modules and successfully solve the minimum number of randomly generated problems, the program provides a Post-Test, so that learners can see how much their mathematics skills have improved.
The Compu-Math series also provided the learner with controls for modifying the instructional environment, such as a special remedia learner setting, pass/fail levels, and allowable error rate prior to remediation.
Fractions
Compu-Math: Fractions was the first program created in the Compu-Math series, being introduced in Edu-Ware's March 1, 1980 catalog. Fractions six learning modules include tutorials on definitions, common and lowest denominators, fraction addition, fraction subtraction, fraction multiplication, and fraction division. Each module includes the use of both common fractions and mixed numbers.
Originally developed by Edu-Ware founders Sherwin Steffin and Steven Pederson as a text-based program, Edu-Ware upgraded it to high-resolution graphics using its EWS3 engine in 1982, renaming it Edu-Ware Fractions, and later, simply Fractions. The program was featured in the company's catalogs until its closure in 1985.
Decimals
Compu-Math: Decimals was the second program created in the Compu-Math series, being introduced in Edu-Ware's August 1, 1980 catalog. Decimals seven learning modules include tutorials on conversion, addition, subtraction, rounding off, multiplication, division and percentage.
Originally developed by Edu-Ware founder Sherwin Steffin and programmer David Mullich as a text-based program, Edu-Ware upgraded it to hi-res graphics in 1982, renaming it Edu-Ware Decimals, and later, simply Decimals. The program was featured in the company's catalogs until its closure in 1985.
Arithmetic SkillsCompu-Math: Arithmetic Skills was the third and final program created in the Compu-Math series, being introduced in Edu-Ware's December 1, 1980 catalog. Arithmetic Skills seven learning modules include tutorials on counting, addition, subtraction, mulciplication and addition. The program also includes a learner management system for changing the system parameters, instructional parameters, and performance criteria. Originally developed by Edu-Ware founder Sherwin Steffin as a text-based program, Edu-Ware upgraded it to hi-res graphics in 1982; however, its name was not changed, unl
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https://en.wikipedia.org/wiki/Feller%20process
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In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.
Definitions
Let X be a locally compact Hausdorff space with a countable base. Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||. From analysis, we know that C0(X) with the sup norm is a Banach space.
A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that
||Ttf || ≤ ||f || for all t ≥ 0 and f in C0(X), i.e., it is a contraction (in the weak sense);
the semigroup property: Tt + s = Tt ∘Ts for all s, t ≥ 0;
limt → 0||Ttf − f || = 0 for every f in C0(X). Using the semigroup property, this is equivalent to the map Ttf from t in [0,∞) to C0(X) being right continuous for every f.
Warning: This terminology is not uniform across the literature. In particular, the assumption that Tt maps C0(X) into itself
is replaced by some authors by the condition that it maps Cb(X), the space of bounded continuous functions, into itself.
The reason for this is twofold: first, it allows including processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of
spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.
A Feller transition function is a probability transition function associated with a Feller semigroup.
A Feller process is a Markov process with a Feller transition function.
Generator
Feller processes (or transition semigroups) can be described by their infinitesimal generator. A function f in C0 is said to be in the domain of the generator if the uniform limit
exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.
A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the Hille–Yosida theorem. This uses the resolvent of the Feller semigroup, defined below.
Resolvent
The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by
It can be shown that it satisfies the identity
Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and
Examples
Brownian motion and the Poisson process are examples of Feller processes. More generally, every Lévy process is a Feller process.
Bessel processes are Feller processes.
Solutions to stochastic differential equations with Lipschitz continuous coefficients are Feller processes.
Every adapted right continuous Feller process on a filtered probability space satisfies the strong Markov property with respect to the filtration , i.e., for each -stopping time , conditioned on the event , we have that for each , is independent of given .
See also
Markov process
Markov chain
Hunt process
Infinitesimal generator (stochastic processes)
References
Markov process
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https://en.wikipedia.org/wiki/Clifford%20Taubes
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Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taubes.
Early career
Taubes received his PhD in physics in 1980 under the direction of Arthur Jaffe, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex equations and the Bogomol'nyi monopole equations.
Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space of solutions to the Yang-Mills equations was used by Simon Donaldson in his proof of Donaldson's theorem. He proved in that R4 has an uncountable number of smooth structures (see also exotic R4), and (with Raoul Bott in ) proved Witten's rigidity theorem on the elliptic genus.
Work based on Seiberg–Witten theory
In a series of four long papers in the 1990s (collected in ), Taubes proved that, on a closed symplectic four-manifold, the (gauge-theoretic) Seiberg–Witten invariant is equal to an invariant which enumerates certain pseudoholomorphic curves and is now known as Taubes's Gromov invariant. This fact improved mathematicians' understanding of the topology of symplectic four-manifolds.
More recently (in ), by using Seiberg–Witten Floer homology as developed by Peter Kronheimer and Tomasz Mrowka together with some new estimates on the spectral flow of Dirac operators and some methods from , Taubes proved the longstanding Weinstein conjecture for all three-dimensional contact manifolds, thus establishing that the Reeb vector field on such a manifold always has a closed orbit. Expanding both on this and on the equivalence of the Seiberg–Witten and Gromov invariants, Taubes has also proven (in a long series of preprints, beginning with ) that a contact 3-manifold's embedded contact homology is isomorphic to a version of its Seiberg–Witten Floer cohomology. More recently, Taubes, C. Kutluhan and Y-J. Lee proved that Seiberg–Witten Floer homology is isomorphic to Heegaard Floer homology.
Honors and awards
Four-time speaker at International Congress of Mathematicians (1986, 1994 (plenary), 1998, 2010 (plenary; selected, but did not speak))
Veblen Prize (AMS) (1991)
Elie Cartan Prize (Académie des Sciences) (1993)
Elected as a fellow of the American Academy of Arts and Sciences in 1995.
Elected to the National Academy of Sciences in 1996.
Clay Research Award (2008)
NAS Award in Mathematics (2008) from the National Academy of Sciences.
Shaw Prize in Mathematics (2009) jointly with Simon Donaldson
Books
1980: (with Arthur Jaffe) Vortices and Monopoles: The Structure of Static Gauge Theories, Progress in Physics, volume 2, Birkhäuser
1993: The L2 Moduli Spaces on Four Manifold With Cylindrical Ends (Monographs in Geometry and Topology)
1996: Metrics, Connections and Gluing Theorems (CBMS Regional Conference Series in Mathematics)
2008 [
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https://en.wikipedia.org/wiki/Concept%20mining
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Concept mining is an activity that results in the extraction of concepts from artifacts. Solutions to the task typically involve aspects of artificial intelligence and statistics, such as data mining and text mining. Because artifacts are typically a loosely structured sequence of words and other symbols (rather than concepts), the problem is nontrivial, but it can provide powerful insights into the meaning, provenance and similarity of documents.
Methods
Traditionally, the conversion of words to concepts has been performed using a thesaurus, and for computational techniques the tendency is to do the same. The thesauri used are either specially created for the task, or a pre-existing language model, usually related to Princeton's WordNet.
The mappings of words to concepts are often ambiguous. Typically each word in a given language will relate to several possible concepts. Humans use context to disambiguate the various meanings of a given piece of text, where available machine translation systems cannot easily infer context.
For the purposes of concept mining, however, these ambiguities tend to be less important than they are with machine translation, for in large documents the ambiguities tend to even out, much as is the case with text mining.
There are many techniques for disambiguation that may be used. Examples are linguistic analysis of the text and the use of word and concept association frequency information that may be inferred from large text corpora. Recently, techniques that base on semantic similarity between the possible concepts and the context have appeared and gained interest in the scientific community.
Applications
Detecting and indexing similar documents in large corpora
One of the spin-offs of calculating document statistics in the concept domain, rather than the word domain, is that concepts form natural tree structures based on hypernymy and meronymy. These structures can be used to generate simple tree membership statistics, that can be used to locate any document in a Euclidean concept space. If the size of a document is also considered as another dimension of this space then an extremely efficient indexing system can be created. This technique is currently in commercial use locating similar legal documents in a 2.5 million document corpus.
Clustering documents by topic
Standard numeric clustering techniques may be used in "concept space" as described above to locate and index documents by the inferred topic. These are numerically far more efficient than their text mining cousins, and tend to behave more intuitively, in that they map better to the similarity measures a human would generate.
See also
Formal concept analysis
Information extraction
Compound term processing
References
Natural language processing
Applications of artificial intelligence
Data mining
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https://en.wikipedia.org/wiki/Secondary%20calculus%20and%20cohomological%20physics
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In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
Secondary calculus
Secondary calculus acts on the space of solutions of a system of partial differential equations (usually non-linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus.
All objects in secondary calculus are cohomology classes of differential complexes growing on diffieties. The latter are, in the framework of secondary calculus, the analog of smooth manifolds.
Cohomological physics
Cohomological physics was born with Gauss's theorem, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a de Rham cohomology class. It is not by chance that formulas of this kind, such as the well known Stokes formula, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.
Classical analogues
All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of differential forms in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.
The simplest diffieties are infinite prolongations of partial differential equations, which are subvarieties of infinite jet spaces. The latter are infinite dimensional varieties that can not be studied by means of standard functional analysis. On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.
Theoretical physics
Recent developments of particle physics, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, i
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https://en.wikipedia.org/wiki/Paley%20construction
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In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley.
The Paley construction uses quadratic residues in a finite field GF(q) where q is a power of an odd prime number. There are two versions of the construction depending on whether q is congruent to 1 or 3 (mod 4).
Quadratic character and Jacobsthal matrix
Let q be a power of an odd prime. In the finite field GF(q) the quadratic character χ(a) indicates whether the element a is zero, a non-zero perfect square, or a non-square:
For example, in GF(7) the non-zero squares are 1 = 12 = 62, 4 = 22 = 52, and 2 = 32 = 42. Hence χ(0) = 0, χ(1) = χ(2) = χ(4) = 1, and χ(3) = χ(5) = χ(6) = −1.
The Jacobsthal matrix Q for GF(q) is the q×q matrix with rows and columns indexed by finite field elements such that the entry in row a and column b is χ(a − b). For example, in GF(7), if the rows and columns of the Jacobsthal matrix are indexed by the field elements 0, 1, 2, 3, 4, 5, 6, then
The Jacobsthal matrix has the properties Q QT = q I − J and Q J = J Q = 0 where I is the q×q identity matrix and J is the q×q all 1 matrix. If q is congruent to 1 (mod 4) then −1 is a square in GF(q)
which implies that Q is a symmetric matrix. If q is congruent to 3 (mod 4) then −1 is not a square, and Q is a
skew-symmetric matrix. When q is a prime number and rows and columns are indexed by field elements in the usual 0, 1, 2, … order, Q is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation.
Paley construction I
If q is congruent to 3 (mod 4) then
is a Hadamard matrix of size q + 1. Here j is the all-1 column vector of length q and I is the (q+1)×(q+1) identity matrix. The matrix H is a skew Hadamard matrix, which means it satisfies H+HT = 2I.
Paley construction II
If q is congruent to 1 (mod 4) then the matrix obtained by replacing all 0 entries in
with the matrix
and all entries ±1 with the matrix
is a Hadamard matrix of size 2(q + 1). It is a symmetric Hadamard matrix.
Examples
Applying Paley Construction I to the Jacobsthal matrix for GF(7), one produces the 8×8 Hadamard matrix,
11111111
-1--1-11
-11--1-1
-111--1-
--111--1
-1-111--
--1-111-
---1-111.
For an example of the Paley II construction when q is a prime power rather than a prime number, consider GF(9). This is an extension field of GF(3) obtained
by adjoining a root of an irreducible quadratic. Different irreducible quadratics produce equivalent fields. Choosing x2+x−1 and letting a be a root of this polynomial, the nine elements of GF(9) may be written 0, 1, −1, a, a+1, a−1, −a, −a+1, −a−1. The non-zero squares are 1 = (±1)2, −a+1 = (±a)2, a−1 = (±(a+1))2, and −1 = (±(a−1))2. The Jacobsthal matrix is
It is a symmetric matrix consisting of nine 3×3 circulant blocks. Paley Construction II produces the symmetric 20×20 Hadamard matrix,
1- 111111 111111 111111
-
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https://en.wikipedia.org/wiki/David%20Klein%20%28mathematician%29
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David Klein is a professor of Mathematics at California State University in Northridge. He is an advocate of increasingly rigorous treatment of mathematics in school curricula and a frequently cited opponent of reforms based on the NCTM standards. One of the participants in the founding of Mathematically Correct, Klein appears regularly in the Math Wars.
Klein, who is a member of the U.S. Campaign for the Academic and Cultural Boycott of Israel, supports the BDS movement which seeks to impose comprehensive boycotts against Israel until it meets its obligations under international law. Klein hosts a webpage supportive of the BDS movement on his university website and, starting in 2011, it became the target of numerous complaints from the pro-Israel groups AMCHA Initiative, Shurat HaDin, and the Global Frontier Justice Center who claimed that it constituted a misuse of state resources. The complaints were dismissed both by the university's staff and by legal authorities as baseless.
Concordant with his support for the BDS movement, Klein defended University of Michigan associate professor John Cheney-Lippold's decision to decline to write a letter of recommendation to a student who planned to study in Israel.
Klein is the director of CSUNs Climate Science Program.
References
Citations
Sources
External links
Open Letter to Governor Schwarzenegger and Members of the California Legislature in support of California's Standards System for K-12 Education
Why Johnny Can't Calculate by David Klein and Jennifer Marple, Los Angeles Times, September 26, 2005
Living people
Traditional mathematics
Mathematics educators
20th-century American mathematicians
21st-century American mathematicians
1953 births
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https://en.wikipedia.org/wiki/Claude%20Meisch
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Claude Meisch (born 27 November 1971, in Pétange) is a Luxembourg politician with a degree in financial mathematics from Trier university. Meisch was appointed Minister of Education in 2013 in the government of Xavier Bettel. He has been a member of the Chamber of Deputies from 1999 to 2013 and Mayor of Differdange since 2002. He was President of the Democratic Party (DP) from 2004 until 2013, of which he has been a member since 1994.
Born in Pétange, in the south-west of the country, Meisch attended the town's Lycée technique Mathias-Adam, before studying at the University of Trier, in Germany. After graduating, he worked for the private Banque de Luxembourg. Meisch was Vice-President of the Democratic and Liberal Youth, the DP's youth wing, from 1995 until 2000.
Meisch ran for the Chamber of Deputies, to represent Sud, in the 1999 election. Meisch finished sixth amongst DP candidates, with the top four being elected. However, the election saw the DP become kingmakers, giving them enough leverage over the Christian Social People's Party (CSV) to allow them to appoint seven Democratic deputies, including Henri Grethen and Eugène Berger, to the new government. Grethen insisted that Berger be appointed along with him, specifically so that Meisch could enter the Chamber. With Grethen and Berger required to vacate their seats to take up their government positions, Meisch filled in the gap and entered the Chamber of Deputies on 12 August 1999.
In the 2004 legislative election, Meisch was re-elected to the Chamber directly, placing second amongst DP candidates in an election that saw the party's representation from Sud reduced from four to two. The result was bad for the DP across the country, losing five seats and seeing them replaced as the Christian Social People's Party's (CSV) coalition partners by the LSAP. After the election, Lydie Polfer resigned as DP President, having served the term limit imposed by the party's statutes. Meisch was the only candidate put forward to replace her, and recorded a 90% vote in his favour (between him and none of the above), holding the position since 10 October 2004.
The 2005 election to Differdange communal council saw Meisch score an 'historic' victory, in leading the DP to buck the national trend and greatly increase their vote: winning 43% of the vote and winning eight seats. Meisch thus remained as mayor, heading a coalition with the Greens, although the size of the victory allowed Meisch to choose his coalition partner from any of the other three parties.
In the 2009 legislative election, Meisch was re-elected, winning more votes that any other Democratic candidate in the entire country, and winning more than twice as many votes as Eugène Berger, who placed second on the DP list in Sud. The party nationwide fell 1.1% of the vote and lost a seat. Immediately after the election, Meisch ruled out a coalition with the CSV, so the DP continued in opposition.
In 2020, Meisch was at the centre of c
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https://en.wikipedia.org/wiki/Mansfield%20Ski%20Club
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Mansfield Ski Club is a ski resort near the village of Mansfield, Ontario, northwest of Toronto, Ontario, Canada.
Statistics
Vertical drop:
Number of runs: 15
Number of lifts: 7
Snowmaking coverage: 100%
Number of eateries: 4
Number of bars: 2
Lifts
Handle Tow
Chalet Magic Carpet (longest in North America)
Javelin Chairlift
Low's Chairlift
Devil's Staircase t-bar "Banana Bar" (actually two t-bars side by side)
Summit Chairlift
Runs
Awesome (green)
Chalet Run (green)
Hemlock (green)
Hector's Hill (blue)
Javelin (blue)
Boomerang (blue)
Gilly's Glades (blue)
Glades (black)
Low's Run (blue)
Big Tree (black)
Devil's Staircase (black)
Breenger (black)
Mouse Trap (black)
Shortcut Glades (blue)
Sully's Dream (black)
Outer Limits (black)
External links
Mansfield Ski Club
Ski areas and resorts in Ontario
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https://en.wikipedia.org/wiki/NCHS
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NCHS may refer to:
National Center for Health Statistics, a center of the US Centers for Disease Control and Prevention (CDC)
Nan Chiau High School, Singapore
Naperville Central High School
Natrona County High School in Casper, Wyoming
Nebraska City High School in Nebraska City, Nebraska
New Canaan High School in New Canaan, Connecticut
New Castle High School (disambiguation), a disambiguation page listing multiple High Schools of that name.
Newcomer Charter High School, now Liberty High School (Houston, Texas)
Newton-Conover High School, in Newton, North Carolina
Normal Community High School in Normal, Illinois
North County High School (disambiguation), a disambiguation page listing multiple High Schools of that name.
Northwest Christian High School (Lacey, Washington)
North Central High School (disambiguation), a disambiguation page listing multiple High Schools of that name.
The Northern California Herpetological Society
North Cobb High School
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https://en.wikipedia.org/wiki/Konstantin%20Andreev
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Konstantin Alekseevich Andreev (14 March 1848 – 29 October 1921) was a Russian mathematician, best known for his work on geometry, especially projective geometry. He was one of the founders of the Kharkov Mathematical Society. This society is one of the early mathematics societies in Russia and was founded in 1879.
Andreev was born in Moscow in a merchant family specialized in fur trading. When he was young, their business went into decline, and the family had to endure severe hardship. During that time, he also lost one eye in an accident that had delayed his studies – he entered gymnasium only in 1860, at the age of 12. However, he rapidly progressed, especially in mathematics, and by the age of 14 started giving private lessons to earn money for his subsistence. In 1867, Andreev enrolled to the Mathematics Department of the Moscow University. As the fourth year he wrote an essay "On the tables of mortality" which was awarded gold medal by the faculty and published in the Scientific Memoirs of the Moscow University, thereby becoming his first scientific work. Andreev graduated in 1871 but remained at the faculty and within two years obtained a Master Diploma.
Around that time, by recommendation of one of his teachers, Andreev was invited for PhD studies to the University of Kharkiv. He accepted and from January 1874 began teaching university courses there. In February 1875, he defended his PhD "On a geometric formation of planar curves" and was promoted to a full-time lecturer. At the end of 1876, Andreev was sent for practice to Europe for one and half years. He spent that time mostly in Berlin and Paris, where he prepared his habilitation work "On the geometric correspondences, as applied to the problem of constructing curves". He defended that work in Moscow in February 1879 and was soon appointed as full professor of the Kharkiv University, as well as of Kharkiv Technology Institute.
In 1884, he was elected as a correspondent member of the Russian Academy of Sciences and in the summer of that year reported his work "On Poncelet polygons" at a conference in La Rochelle, France. In 1898 Andreev returned to Moscow, to assume a post of professor at the Department of Mathematics of Moscow University. Simultaneously, he became director of Alexander School of Business (at Basman), which post he held until 1907, and spent much time working for secondary education system. At Moscow University, Andreev became the first dean elect of the Physics and Mathematics Faculty (from 1905 to 1911), where he introduced the standard lecture cycle system. In 1911, he had to resign as a dean and stop lecturing due to a throat tumor, which he had operated in 1913 in Europe. He then resumed teaching at Moscow University until 1917, when other health problems urged him to abandon most activities and moved to the health resorts of Crimea. He died near Sevastopol in October 1921.
References
External links
20th-century Russian mathematicians
Corresponding memb
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https://en.wikipedia.org/wiki/Algebraic%20Geometry%20%28book%29
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Algebraic Geometry is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.
Importance
It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students.
Contents
The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.
Notes
References
Graduate Texts in Mathematics
1977 non-fiction books
Algebraic geometry
Monographs
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https://en.wikipedia.org/wiki/Tight%20closure
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In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by .
Let be a commutative noetherian ring containing a field of characteristic . Hence is a prime number.
Let be an ideal of . The tight closure of , denoted by , is another ideal of containing . The ideal is defined as follows.
if and only if there exists a , where is not contained in any minimal prime ideal of , such that for all . If is reduced, then one can instead consider all .
Here is used to denote the ideal of generated by the 'th powers of elements of , called the th Frobenius power of .
An ideal is called tightly closed if . A ring in which all ideals are tightly closed is called weakly -regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of -regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.
found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly -regular ring is -regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?
References
Commutative algebra
Ideals (ring theory)
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https://en.wikipedia.org/wiki/Ivan%20%C5%BDivanovi%C4%87%20%28footballer%2C%20born%201981%29
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Ivan Živanović (; born 10 December 1981) is a Serbian former football defender.
Živanović made one appearance in the Coppa Italia while playing for Sampdoria in the 2006–07 season.
Career statistics
External links
1981 births
Living people
Footballers from Šabac
Serbian men's footballers
Serbian expatriate men's footballers
Men's association football defenders
FK Mačva Šabac players
FK Smederevo 1924 players
UC Sampdoria players
FC Rostov players
Russian Premier League players
Expatriate men's footballers in Italy
Expatriate men's footballers in Russia
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https://en.wikipedia.org/wiki/Sigma-ring
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In mathematics, a nonempty collection of sets is called a -ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Formal definition
Let be a nonempty collection of sets. Then is a -ring if:
Closed under countable unions: if for all
Closed under relative complementation: if
Properties
These two properties imply:
whenever are elements of
This is because
Every -ring is a δ-ring but there exist δ-rings that are not -rings.
Similar concepts
If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then is a ring but not a -ring.
Uses
-rings can be used instead of -fields (-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every -field is also a -ring, but a -ring need not be a -field.
A -ring that is a collection of subsets of induces a -field for Define Then is a -field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal -field containing since it must be contained in every -field containing
See also
References
Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses -rings in development of Lebesgue theory.
Measure theory
Families of sets
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https://en.wikipedia.org/wiki/Conoid
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In geometry a conoid () is a ruled surface, whose rulings (lines) fulfill the additional conditions:
(1) All rulings are parallel to a plane, the directrix plane.
(2) All rulings intersect a fixed line, the axis.
The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis.
Because of (1) any conoid is a Catalan surface and can be represented parametrically by
Any curve with fixed parameter is a ruling, describes the directrix and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by
.
If the directrix is a circle, the conoid is called a circular conoid.
The term conoid was already used by Archimedes in his treatise On Conoids and Spheroides.
Examples
Right circular conoid
The parametric representation
describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line
Special features:
The intersection with a horizontal plane is an ellipse.
is an implicit representation. Hence the right circular conoid is a surface of degree 4.
Kepler's rule gives for a right circular conoid with radius and height the exact volume: .
The implicit representation is fulfilled by the points of the line , too. For these points there exist no tangent planes. Such points are called singular.
Parabolic conoid
The parametric representation
describes a parabolic conoid with the equation . The conoid has a parabola as directrix, the y-axis as axis and a plane parallel to the x-z-plane as directrix plane. It is used by architects as roof surface (s. below).
The parabolic conoid has no singular points.
Further examples
hyperbolic paraboloid
Plücker conoid
Whitney Umbrella
helicoid
Applications
Mathematics
There are a lot of conoids with singular points, which are investigated in algebraic geometry.
Architecture
Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid (s. parabolic conoid).
External links
mathworld: Plücker conoid
References
A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, FL:CRC Press, 2006. ()
Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE ()
Surfaces
Geometric shapes
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https://en.wikipedia.org/wiki/Jornal%20Nacional
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; ) is the flagship television newscast of TV Globo. First airing on September 1, 1969, according to IBOPE (Brazilian Institute of Public Opinion and Statistics), in the week of September 28October 4, 2015, it was the second most watched program in Brazilian television, with an average of 26,007,251 viewers per minute (roughly 12.5% of the Brazilian population) and for 5.5 million people worldwide via Globo International.
History
premiered on September 1, 1969, hosted by Hilton Gomes and Cid Moreira, the first Rio de Janeiro-produced newscast to be shown nationwide. Months later, the program featured the network's first female weekend presenter Márcia Mendes.
During the 1970s, preferred to emphasize international news and sports. The British documentary Beyond Citizen Kane suggests that this happened so that Globo wouldn't have to report the repression of the Brazilian military government, which would have provided a substantial part of the network's growth. Despite this, the program introduced some innovations (color broadcasts in 1970, via satellite reports in 1973, live reports in 1976 and videotape footage in 1977).
Through the 1980s, three episodes involving the program caused controversy. In 1982, coverage of the state elections of Rio de Janeiro was accused of participating in a plot to fraud the elections. According to former Rede Globo employee Roméro da Costa Machado, Leonel Brizola, a candidate of the opposition to the military regime, was a politician historically persecuted by Rede Globo owner Roberto Marinho. Two years later, the program was accused of omitting information about the Diretas Já, a popular campaign for resuming the direct election for president, near the end of the dictatorship. Finally, in 1989, was accused of editing a presidential debate between runoff candidates Fernando Collor and Luis Inacio Lula da Silva in order to favor Collor. This episode is also extensively debated on Beyond Citizen Kane.
In the 1990s, the quality of Rede Globo's journalism increased dramatically. presented its viewers breaking stories such as police brutality at favelas, an interview with Paulo César Farias when he was on the run from the law, corruption cases on the social security, the kicking of the saint incident among several others.
In recent days, after the death of Marinho, Rede Globo's journalism again declined in quality. has preferred to broadcast stories produced on the Southeast, in spite of Globo having affiliates in every Brazilian state. During the Mensalão scandal and the 2006 general elections, was once again accused of airing anti-Lula biased news. Even worse, it lost a significant part of its viewership to Rede Record's Jornal da Record, which copied its style and also features former anchors, Celso Freitas and Marcos Hummel (as a relief presenter for Jornal da Record), and was known to be widely preferred as a more credible newscast than . In November 2005, host William Bonner caused controversy after h
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https://en.wikipedia.org/wiki/Invariant%20differential%20operator
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In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.
In an invariant differential operator , the term differential operator indicates that the value of the map depends only on and the derivatives of in . The word invariant indicates that the operator contains some symmetry. This means that there is a group with a group action on the functions (or other objects in question) and this action is preserved by the operator:
Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.
Invariance on homogeneous spaces
Let M = G/H be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation gives rise to a vector bundle
Sections can be identified with
In this form the group G acts on sections via
Now let V and W be two vector bundles over M. Then a differential operator
that maps sections of V to sections of W is called invariant if
for all sections in and elements g in G. All linear invariant differential operators on homogeneous parabolic geometries, i.e. when G is semi-simple and H is a parabolic subgroup, are given dually by homomorphisms of generalized Verma modules.
Invariance in terms of abstract indices
Given two connections and and a one form , we have
for some tensor . Given an equivalence class of connections , we say that an operator is invariant if the form of the operator does not change when we change from one connection in the equivalence class to another. For example, if we consider the equivalence class of all torsion free connections, then the tensor Q is symmetric in its lower indices, i.e. . Therefore we can compute
where brackets denote skew symmetrization. This shows the invariance of the exterior derivative when acting on one forms.
Equivalence classes of connections arise naturally in differential geometry, for example:
in conformal geometry an equivalence class of connections is given by the Levi Civita connections of all metrics in the conformal class;
in projective geometry an equivalence class of connection is given by all connections that have the same geodesics;
in CR geometry an equivalence class of connections is given by the Tanaka-Webster connections for each choice of pseudohermitian structure
Examples
The usual gradient operator acting on real valued functions on Euclidean space is invariant with respect to all Euclidean transformations.
The differential acting on functions on a manifold with values in 1-forms (its expression is in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on differential f
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https://en.wikipedia.org/wiki/Weight%20space
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In mathematics, weight space may refer to:
Weight space (representation theory)
Parameter space in artificial neural networks, where the parameters are weights on graph edges.
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https://en.wikipedia.org/wiki/AMCS
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AMCS may refer to:
The Australian Marine Conservation Society
The International Journal of Applied Mathematics and Computer Science
pl:AMCS
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https://en.wikipedia.org/wiki/Delta-ring
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In mathematics, a non-empty collection of sets is called a -ring (pronounced "") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a -ring which is closed under countable unions.
Definition
A family of sets is called a -ring if it has all of the following properties:
Closed under finite unions: for all
Closed under relative complementation: for all and
Closed under countable intersections: if for all
If only the first two properties are satisfied, then is a ring of sets but not a -ring. Every -ring is a -ring, but not every -ring is a -ring.
-rings can be used instead of σ-algebras in the development of measure theory if one does not wish to allow sets of infinite measure.
Examples
The family is a -ring but not a -ring because is not bounded.
See also
References
Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html
Measure theory
Families of sets
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https://en.wikipedia.org/wiki/STST
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STST may refer to:
Single-trunk Steiner tree (STST) a design and topology, see Rectilinear Steiner tree
Strassenbahn Stansstad-Stans (StSt) a defunct rail company, see List of railway companies in Switzerland
Argon ST (NASDAQ stock ticker: STST) a subsidiary of Boeing
Store Status (STST) a command code for the TI-990
Scott Trimble, aka STST, from full name Scott Thomas Suggs Trimble
Sounding the Seventh Trumpet, the first studio album by Avenged Sevenfold.
Subjective total sleep time, a measure of sleep quality
See also
ST (disambiguation)
ST2 (disambiguation)
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https://en.wikipedia.org/wiki/Sirous%20Dinmohammadi
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Sirous Dinmohammadi (, born 2 July 1970 in Tabriz) is a retired Iranian football player.
Club career
He is most notably for playing for Tractor and Esteghlal.
Career statistics
International career
Dinmohammadi made 40 appearances for the Iran national football team and participated in the 1998 FIFA World Cup.
International goals
Honours
Esteghlal
Iranian Football League: 2000–01
Hazfi Cup: 2001–02
References
External links
1970 births
Living people
Footballers from Tabriz
Iranian men's footballers
Iranian expatriate men's footballers
Iran men's international footballers
Men's association football midfielders
Expatriate men's footballers in Germany
Tractor S.C. players
Shahrdari Tabriz F.C. players
Esteghlal F.C. players
1. FSV Mainz 05 players
Pegah F.C. players
Persian Gulf Pro League players
2. Bundesliga players
1996 AFC Asian Cup players
1998 FIFA World Cup players
Iranian expatriate sportspeople in Germany
20th-century Iranian people
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https://en.wikipedia.org/wiki/Gregorio%20Fontana
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Gregorio Fontana, born Giovanni Battista Lorenzo Fontana (7 December 1735 – 24 August 1803) was an Italian mathematician and a religious of the Piarist order. He was chair of mathematics at the university of Pavia succeeding Roger Joseph Boscovich. He has been credited with the introduction of polar coordinates.
His brother was the physicist Felice Fontana (1730–1805).
Works
References
Academic staff of the University of Pavia
18th-century Italian mathematicians
1735 births
1803 deaths
Fellows of the Royal Society
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https://en.wikipedia.org/wiki/Specificity
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Specificity may refer to:
Being specific (disambiguation)
Specificity (statistics), the proportion of negatives in a binary classification test which are correctly identified
Sensitivity and specificity, in relation to medical diagnostics
Specificity (linguistics), whether a noun phrase has a particular referent as opposed to referring to any member of a class
Specificity (symbiosis), the taxonomic range an organism associates with in a symbiosis
Particular, as opposed to abstract, in philosophy
Asset specificity, the extent that investments supporting a particular transaction have a higher value than if they were redeployed for any other purpose
Domain specificity, theory that many aspects of cognition are supported by specialized learning devices
Specificity theory, theory that pain is "a specific sensation, with its own sensory apparatus independent of touch and other senses"
, determines which styles are applied to an html element when more than one rule could apply.
Chemical specificity, in chemistry and biochemistry, with regard to enzymes or catalysts and their substrates
See also
Species (disambiguation)
Specification (disambiguation)
Specialty (disambiguation)
Site-specific (disambiguation)
Language for specific purposes
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https://en.wikipedia.org/wiki/Arf%20invariant
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In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.
In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to , even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field.
The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of -dimensional manifolds (singly even-dimensional manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4k-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.
Definitions
The Arf invariant is defined for a quadratic form q over a field K of characteristic 2 such that q is nonsingular, in the sense that the associated bilinear form is nondegenerate. The form is alternating since K has characteristic 2; it follows that a nonsingular quadratic form in characteristic 2 must have even dimension. Any binary (2-dimensional) nonsingular quadratic form over K is equivalent to a form with in K. The Arf invariant is defined to be the product . If the form is equivalent to , then the products and differ by an element of the form with in K. These elements form an additive subgroup U of K. Hence the coset of modulo U is an invariant of , which means that it is not changed when is replaced by an equivalent form.
Every nonsingular quadratic form over K is equivalent to a direct sum of nonsingular binary forms. This was shown by Arf, but it had been earlier observed by Dickson in the case of finite fields of characteristic 2. The Arf invariant Arf() is defined to be the sum of the Arf invariants of the . By definition, this is a coset of K modulo U. Arf showed that indeed does not change if is replaced by an equivalent quadratic form, which is to say that it is an invariant of .
The Arf invariant is additive; in other words, the Arf invariant of an orthogonal sum of two quadratic forms is the sum of their Arf invariants.
For a field K of characteristic 2, Artin–Schreier theory identifies the
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https://en.wikipedia.org/wiki/Municipality%20of%20the%20District%20of%20St.%20Mary%27s
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St. Mary's, officially named the Municipality of the District of St. Mary's, is a district municipality in Guysborough County, Nova Scotia, Canada. Statistics Canada classifies the district municipality as a municipal district.
The district municipality occupies the western half of the county and its administrative seat is in the village of Sherbrooke.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, the Municipality of the District of St. Mary's had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Electoral districts
The district municipality is divided into seven electoral districts for municipal representation:
Sherbrooke & Area
Ecum Secum & Area
Caledonia & Area
Sonora - Port Hilford
Goshen & Area
Liscomb & Area
Port Bickerton & Area
See also
List of municipalities in Nova Scotia
References
External links
District municipalities in Nova Scotia
Communities in Guysborough County, Nova Scotia
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https://en.wikipedia.org/wiki/Minnesota%20Vikings%20statistics
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The Minnesota Vikings is an American football franchise based in Minneapolis, Minnesota. The team was established in 1961 and is part of the National Football League's NFC North division. Since then, the team has taken part in the NFL playoffs 31 times, reaching four Super Bowls in 1970, 1974, 1975 and 1977.
This list encompasses the major records set by the team, its coaches and its players. The players section of this page lists the individual records for passing, rushing and receiving, as well as selected defensive records. The team has had three full-time home stadiums since its establishment – Metropolitan Stadium, Hubert H. Humphrey Metrodome, and U.S. Bank Stadium; attendance records, both home and away, are included on this page.
All-time series
The Vikings have played against every other team in the NFL at least five times each in the regular season. They have a winning record against 20 teams, a losing record against 10 and an even record against one: the Cincinnati Bengals, whom they have played 14 times, winning seven each. The Vikings' best record is against the Houston Texans – they have won each of the five meetings between the two teams – and their worst record is against the New York Jets, whom they have beaten just three times in 11 meetings for a win percentage of .273; that is a team record for the fewest wins against another franchise, tied with their three wins in seven meetings with the Baltimore Ravens (.429). The Vikings have recorded the most wins in their history against the Detroit Lions (80), as well as the most points (2,690). The Vikings have conceded the most points to the Green Bay Packers (2,689), who are also the team against whom they have suffered the most losses (63) and ties (3).
In the postseason, the Vikings have faced 18 different opponents, but they only have a win percentage of .500 or higher against six of them. The Vikings' best postseason win percentage is against the Arizona Cardinals and the Cleveland Browns, whom they have beaten every time they have met – the Cardinals twice and the Browns once; meanwhile, they have faced eight teams without recording a single win, the worst being against the Philadelphia Eagles, whom they have met four times without winning. The Vikings have recorded their most postseason wins against the Los Angeles Rams, whom they have beaten five times in seven meetings, and their most losses against the San Francisco 49ers, to whom they have lost five times in six meetings. The Vikings have also conceded their most postseason points against the 49ers (181), and scored their most against the New Orleans Saints (161). As well as meeting the Rams seven times in the postseason, the Vikings have also met the Dallas Cowboys seven times, recording three wins and four losses.
Last updated: As of week 3 of the 2023 NFL season
Team records
Firsts
First home game: Chicago Bears 13–37 Minnesota Vikings, September 17, 1961
First road game: Minnesota Vikings 7–21 Dallas Cowboys, Sep
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https://en.wikipedia.org/wiki/Kvant%20%28magazine%29
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Kvant ( for "quantum") is a popular science magazine in physics and mathematics for school students and teachers, issued in print between 1970 and 2011. The magazine became an online-only publication in 2011. Translation of selected articles from Kvant had been published in Quantum Magazine in 1990–2001, which in turn had been translated and published in Greece in 1994–2001.
History
Kvant was started as a joint project of the USSR Academy of Sciences and USSR Academy of Pedagogical Sciences. In Soviet time, it was published by Nauka publisher with circulation about 200,000.
The idea of the magazine was introduced by Pyotr Kapitsa. Its first chief editors were physicist Isaak Kikoin and mathematician Andrei Kolmogorov. In 1985, its editorial board had 18 Academicians and Corresponding Members of the USSR Academy of Sciences and USSR Academy of Pedagogical Sciences, 14 Doctors of Sciences and 20 Candidates of Science.
The last print issue of Kvant was published at the beginning of 2011. Then the print edition was closed making the magazine an online publication.
Availability
All published issues of Kvant were freely available online.
Translations
Quantum Magazine
Quantum Magazine was a US-based bimonthly magazine published by the National Science Teachers Association (NSTA) from 1990 to 2001. Some of its articles were translations from Kvant.
Kvant Selecta
In 1999, American Mathematical Society published translation of selected articles from Kvant on algebra and mathematical analysis as two volumes in the Mathematical World series. Yet another volume, published in 2002, included translation of selected articles on combinatorics.
Other translations
There were two books with selected articles from Kvant published in France by Jean-Michel Kantor
References
External links
Kvant archive website
Kvant website
The official website of Quantum Magazine
The Greek version of Quantum Magazine.
1970 establishments in the Soviet Union
2011 disestablishments in Russia
Education in the Soviet Union
Education magazines
Magazines established in 1970
Magazines disestablished in 2011
Magazines published in Moscow
Magazines published in the Soviet Union
Online magazines with defunct print editions
Russian-language magazines
Science education materials
Science and technology in the Soviet Union
Science and technology magazines published in Russia
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https://en.wikipedia.org/wiki/Ian%20Cullimore
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Ian H. S. Cullimore is an English-born mathematician and computer scientist who has been influential in the pocket PC arena.
Biography
Cullimore has a degree in mathematics from King's College London, and a PhD in cognitive and computer science from the University of Sussex.
He was the original founder (in 1985) and main inventor of the pocket PC which became the Atari Portfolio (originally known as the "DIP Pocket PC") in 1989. DIP Research Ltd. was acquired by Phoenix Technologies in 1994.
In 1988 Cullimore was also one of the founders and Vice President of Software at Poqet Computer Corporation in Silicon Valley, where he developed the Poqet PC.
His interest in PDAs was sparked from his early times at Psion, working on the first Organiser products.
He was also the original instigator of the PC Card (formerly "PCMCIA Card") movement. This came about from his decision to use the then-emerging credit card memories in the design of the Atari Portfolio. On founding Poqet, and with major investment from Fujitsu, a decision was made to use the 68-pin JEIDA card. He successfully persuaded the board of Poqet to set up an industry standards organization, PCMCIA, to promote this as a standard.
Cullimore wrote parts of the PCMCIA driver stack for (NetWare) PalmDOS 1.0, a variant of Digital Research's DR DOS, tailored specifically at battery powered mobile PCs in 1992.
Publications
References
English computer scientists
Computer hardware researchers
Alumni of King's College London
Alumni of the University of Sussex
Living people
People educated at The College of Richard Collyer
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/John%20Hajnal
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John Hajnal FBA (born Hajnal-Kónyi, ; 26 November 1924 – 30 November 2008), was a Hungarian-British academic in the fields of mathematics and economics (statistics).
Hajnal is best known for identifying, in a landmark 1965 paper, the historical pattern of marriage of northwest Europe in which people married late and many adults remained single. The geographical boundary of this unusual marriage pattern is now known as the Hajnal line.
Biography
Hajnal was born in Darmstadt, at the time the capital of the People's State of Hesse in Weimar Germany, to a Hungarian Jewish family. In 1936 his parents left Nazi Germany, and placed him in a Quaker school in the Dutch countryside while they arranged to settle in Britain. In 1937, John was reunited with his parents in London, where he attended University College School, Hampstead.
At age 16, he entered Balliol College, Oxford. He gained a first there in economics, philosophy and politics in 1943. His skills in academic-level mathematics were mostly autodidactical.
After the war, Hajnal worked on demography for the United Nations in New York, and later for the Office of Population Research, Princeton University.
He met Berlin-born Nina Lande in New York. They were married from 1950 until her death in 2008 and had three daughters and a son.
Returning to the United Kingdom, he worked at Manchester University as a statistician from 1953. The family moved to London in 1956, when John was assured a lectureship at the London School of Economics. He was Professor of Statistics at the London School of Economics from 1975 until his retirement in 1986.
Career
Royal Commission on Population, 1944–48
United Nations, New York, 1948–51
Office of Population Research, Princeton University, 1951–53
Manchester University, 1953–57
London School of Economics, 1957–86. Reader, 1966–75, Professor of Statistics 1975–1986
Visiting Fellow Commoner, Trinity College, Cambridge, 1974–75
Visiting Professor, Rockefeller University, 1981
He was a member of the International Statistical Institute and was elected FBA in 1966.
References
Who's Who (2006)
The Palgrave Dictionary of Anglo-Jewish History, ed. W. D. Rubinstein, Palgrave Macmillan, 2011, p. 387.
Obituary, The Jewish Chronicle, 5 February 2009.
1924 births
Academics of the London School of Economics
Academics of the Victoria University of Manchester
Alumni of Balliol College, Oxford
British Jews
20th-century British mathematicians
21st-century British mathematicians
British statisticians
Hungarian statisticians
Fellows of the British Academy
People educated at University College School
2008 deaths
Elected Members of the International Statistical Institute
German emigrants to the United Kingdom
Demographers
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https://en.wikipedia.org/wiki/Locally%20connected%20space
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In topology and other branches of mathematics, a topological space X is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness and compactness have been two of the most
widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of compact subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, connected subsets of (for n > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below).
This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space. As an example, the notion of weak local connectedness at a point and its relation to local connectedness will be considered later on in the article.
In the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifolds, which are locally well understood (being locally homeomorphic to Euclidean space) but have complicated global behavior. By this it is meant that although the basic point-set topology of manifolds is relatively simple (as manifolds are essentially metrizable according to most definitions of the concept), their algebraic topology is far more complex. From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a universal cover it must be connected and locally path connected. Local path connectedness will be discussed as well.
A space is locally connected if and only if for every open set U, the connected components of U (in the subspace topology) are open. It follows, for instance, that a continuous function from a locally connected space to a totally disconnected space must be locally constant. In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance Cantor space is totally disconnected but not discrete.
Definitions
Let be a topological space, and let be a point of
A space is called locally connected at if every neighborhood of contains a connected open neighborhood of , that is, if the point has a neighborhood base consisting of connected open sets. A locally connected space is a space that is locally connected at each of its points.
Local connectedness does not imply connectedness (consider two disjoint open intervals in for example)
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https://en.wikipedia.org/wiki/Double%20helix%20%28disambiguation%29
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Double helix may refer to:
Science and engineering
Double helix, the structure of DNA
Double helix (geometry), two helices with the same axis differing by a translation along the axis
Double Helix Nebula, a gaseous nebula in the Milky Way Galaxy
Double helical gear, also known as a herringbone gear
The Helix Bridge, a bridge shaped as the double helix located at the Marina Bay, Singapore, Singapore
Gaming and animation
Double Helix Games, a video game developer
Soldier of Fortune II: Double Helix, a video game
Double-Helix, an alien race in the Wing Commander (franchise)
Doublehelix, a character in the animated film Asterix and the Vikings
Literature
The Double Helix, a book about the discovery of the double-helical structure of DNA
Double Helix (novel), a 2004 novel by Nancy Werlin
Star Trek: Double Helix, a six-book miniseries and a spinoff of the Star Trek: The Next Generation book series
Double Helix: Double or Nothing, a book in the aforementioned series
Other uses
Double Helix (music composition), a 1991 piece for jazz orchestra by Jack Cooper
Double Helix (database), a database management system for the Apple Macintosh computer system
"Double Helix" (The Outer Limits), a 1997 television episode
Double Helix Corporation, a non-profit media organization in St. Louis, Missouri, US
"Double Helix", a song by Death Grips from The Money Store
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https://en.wikipedia.org/wiki/Andrew%20Casson
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Andrew John Casson FRS (born 1943) is a mathematician, studying geometric topology. Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University.
Education and Career
Casson was educated at Latymer Upper School and Trinity College, Cambridge, where he graduated with a BA in the Mathematical Tripos in 1965. His doctoral advisor at the University of Liverpool was C. T. C. Wall, but he never completed his doctorate; instead what would have been his Ph.D. thesis became his fellowship dissertation as a research fellow at Trinity College.
Casson was Professor of Mathematics at the University of Texas at Austin between 1981 and 1986, at the University of California, Berkeley, from 1986 to 2000, and has been at Yale since 2000.
Work
Casson has worked in both high-dimensional manifold topology and 3- and 4-dimensional topology,
using both geometric and algebraic techniques. Among other discoveries, he contributed
to the disproof of the manifold Hauptvermutung, introduced the Casson invariant, a modern invariant for 3-manifolds, and Casson handles, used in Michael Freedman's proof of the 4-dimensional Poincaré conjecture.
Awards
In 1991, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1998, he was elected to Fellowship of the Royal Society.
References
External links
Official Home Page
The Hauptvermutung book (including Casson's 1967 Trinity College fellowship dissertation)
Proceedings of the Casson Fest (Arkansas and Texas 2003) A conference to celebrate Casson's 60th birthday, with biographical information.
Photos from conference, including the `honorary degree' presented to Casson by the participants
1943 births
Living people
Fellows of the Royal Society
Alumni of Trinity College, Cambridge
University of Texas at Austin faculty
University of California, Berkeley College of Letters and Science faculty
Yale University faculty
Topologists
Alumni of the University of Liverpool
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https://en.wikipedia.org/wiki/Total%20correlation
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In probability theory and in particular in information theory, total correlation (Watanabe 1960) is one of several generalizations of the mutual information. It is also known as the multivariate constraint (Garner 1962) or multiinformation (Studený & Vejnarová 1999). It quantifies the redundancy or dependency among a set of n random variables.
Definition
For a given set of n random variables , the total correlation is defined as the Kullback–Leibler divergence from the joint distribution to the independent distribution of ,
This divergence reduces to the simpler difference of entropies,
where is the information entropy of variable , and is the joint entropy of the variable set . In terms of the discrete probability distributions on variables , the total correlation is given by
The total correlation is the amount of information shared among the variables in the set. The sum represents the amount of information in bits (assuming base-2 logs) that the variables would possess if they were totally independent of one another (non-redundant), or, equivalently, the average code length to transmit the values of all variables if each variable was (optimally) coded independently. The term is the actual amount of information that the variable set contains, or equivalently, the average code length to transmit the values of all variables if the set of variables was (optimally) coded together. The difference between
these terms therefore represents the absolute redundancy (in bits) present in the given
set of variables, and thus provides a general quantitative measure of the
structure or organization embodied in the set of variables
(Rothstein 1952). The total correlation is also the Kullback–Leibler divergence between the actual distribution and its maximum entropy product approximation .
Total correlation quantifies the amount of dependence among a group of variables. A near-zero total correlation indicates that the variables in the group are essentially statistically independent; they are completely unrelated, in the sense that knowing the value of one variable does not provide any clue as to the values of the other variables. On the other hand, the maximum total correlation (for a fixed set of individual entropies ) is given by
and occurs when one of the variables determines all of the other variables. The variables are then maximally related in the sense that knowing the value of one variable provides complete information about the values of all the other variables, and the variables can be figuratively regarded as cogs, in which the position of one cog determines the positions of all the others (Rothstein 1952).
It is important to note that the total correlation counts up all the redundancies among a set of variables, but that these redundancies may be distributed throughout the variable set in a variety of complicated ways (Garner 1962). For example, some variables in the set may be totally inter-redundant while others in the set are c
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https://en.wikipedia.org/wiki/Path%20space
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In mathematics, the term path space refers to any topological space of paths from one specified set into another. In particular, it may refer to:
The classical Wiener space of continuous paths
The Skorokhod space of càdlàg paths
For the usage in algebraic topology, see path space (algebraic topology). For Moore's path space, see path space fibration#Moore's path space.
See also: loop space, the space of loops in a topological space
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https://en.wikipedia.org/wiki/Principal%20ideal%20ring
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In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.
If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains.
A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
General properties
If R is a principal right ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian.
Principal right ideal rings are closed under finite direct products. If , then each right ideal of R is of the form , where each is a right ideal of Ri. If all the Ri are principal right ideal rings, then Ai=xiRi, and then it can be seen that . Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.
Principal right ideal rings and right Bézout rings are also closed under quotients, that is, if I is a proper ideal of principal right ideal ring R, then the quotient ring R/I is also principal right ideal ring. This follows readily from the isomorphism theorems for rings.
All properties above have left analogues as well.
Commutative examples
1. The ring of integers:
2. The integers modulo n: .
3. Let be rings and . Then R is a principal ring if and only if Ri is a principal ring for all i.
4. The localization of a principal ring at any multiplicative subset is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
5. Let R be a Dedekind domain and I be a nonzero ideal of R. Then the quotient R/I is a principal ring. Indeed, we may factor I as a product of prime
powers: , and by the Chinese Remainder Theorem
, so it suffices to see that each
is a principal ring. But is isomorphic to the quotient of the discrete valuation ring
and, being a quotient of a principal ring, is itself a principal ring.
6. Let k be a finite field and put , and . Then R is a finite local ring which is not principal.
7. Let X be a finite set. Then forms a commutative principal ideal ring with unity, where represents set symmetric difference and represents the powerset of X. If X has at least two elements, then the ring also has zero divisors. If I is an ideal,
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https://en.wikipedia.org/wiki/Association%20of%20Track%20and%20Field%20Statisticians
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The Association of Track and Field Statisticians (ATFS) was founded in 1950. It is an international organization run by volunteers whose goal is to collect and disseminate the statistics of track and field athletics.
Foundation
On 26 August 1950 at the Café de la Madeleine in the Rue de la Montagne, Brussels, Belgium, whilst attending the 1950 European Athletics Championships, the Association of Track and Field Statisticians was founded.
Two of the founding members, Roberto Quercetani and Donald Potts, had published in 1948 the first study of world all-time lists, A Handbook on Olympic Games Track and Field Athletics. Readers of this book were inspired to found an international association of track statisticians. One of the founding members, the Belgian journalist André Greuze, organised the first meeting on 26 August. One of the meeting's first acts was to elect Harold Abrahams as honorary president.
Founding members
President: Roberto Quercetani (Italy)
Secretary: Fulvio Regli (Switzerland)
Committee: Norris McWhirter (UK), Donald Potts (USA)
Members: Harold Abrahams (UK) - Hon. President, Bruno Bonomelli (Italy), André Greuze (Belgium), Erich Kamper (Austria), Ekkehard zur Megede (Germany), André Senay (France), Björn-Johan Weckman (Finland), Wolfgang Wünsche (Germany)
Note: two of the founders, Potts and Wünsche, were not actually present at the first meeting but are considered as founding members.
Yearbook
The association has published an annual yearbook since 1951. The first edition was published in Lugano, Switzerland, titled The 1951 A.T.F.S. International Athletic Annual. The editors were Fulvio Regli and Roberto Quercetani. It is an authoritative compilation of international athletics statistics and has been known as The International Athletics Annual and The ATFS Annual. It is currently published under the title of Athletics: The International Track and Field Annual and its editor is Peter Matthews.
Roberto Quercetani
Roberto Luigi Quercetani (known as RLQ to other track and field statisticians)was one of the eleven founding members of ATFS and was also its first president, remaining in the position for 18 years. He was also the editor of the first ATFS Annual. and is a renowned historian and writer on athletics
Quercetani was born in Florence, Italy on May 3, 1922. He was interested in both foreign languages and athletics from an early age, his study in both helped by reading foreign newspapers sent by friends from Switzerland. After World War II, he served as a technical interpreter for the allied forces in Italy. Now fluent in English, French and German, he started writing articles for the foreign sporting publications Leichathletik of Germany, World Sports of the UK, and Track and Field News of the United States.
In 1948, he wrote his first athletics statistical publication (with Don Potts).
From 1951, Quercetani started as a long-time contributor to the Italian newspapers La Gazzetta dello Sport and La Nazion
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https://en.wikipedia.org/wiki/Heinz%20mean
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In mathematics, the Heinz mean (named after E. Heinz) of two non-negative real numbers A and B, was defined by Bhatia as:
with 0 ≤ x ≤ .
For different values of x, this Heinz mean interpolates between the arithmetic (x = 0) and geometric (x = 1/2) means such that for 0 < x < :
The Heinz means appear naturally when symmetrizing
-divergences.
It may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.
See also
Mean
Muirhead's inequality
Inequality of arithmetic and geometric means
References
Means
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https://en.wikipedia.org/wiki/Carl%20Morris
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Carl Morris may refer to:
Carl Morris (painter) (1911–1993), American painter
Carl E. Morris (1887–1951), American boxer
Carl Morris (statistician), professor of statistics at Harvard University
Carl Morris, a fictional character in the British TV series, Moving Wallpaper
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https://en.wikipedia.org/wiki/Kirkwood%20approximation
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The Kirkwood superposition approximation was introduced in 1935 by John G. Kirkwood as a means of representing a discrete probability distribution. The Kirkwood approximation for a discrete probability density function is given by
where
is the product of probabilities over all subsets of variables of size i in variable set . This kind of formula has been considered by Watanabe (1960) and, according to Watanabe, also by Robert Fano. For the three-variable case, it reduces to simply
The Kirkwood approximation does not generally produce a valid probability distribution (the normalization condition is violated). Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure. The Kirkwood approximation is the probabilistic counterpart of the interaction information.
Judea Pearl (1988 §3.2.4) indicates that an expression of this type can be exact in the case of a decomposable model, that is, a probability distribution that admits a graph structure whose cliques form a tree. In such cases, the numerator contains the product of the intra-clique joint distributions and the denominator contains the product of the clique intersection distributions.
References
Jakulin, A. & Bratko, I. (2004), Quantifying and visualizing attribute interactions: An approach based on entropy, Journal of Machine Learning Research, (submitted) pp. 38–43.
Discrete distributions
Statistical approximations
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https://en.wikipedia.org/wiki/List%20of%20Indonesia-related%20topics
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This is a list of topics related to Indonesia.
Cities in Indonesia
List of regencies and cities of Indonesia
List of cities in Indonesia including population statistics
Jakarta
Lists
Colonial buildings and structures in Jakarta
Governors of Jakarta
Radio stations in Jakarta
Areas of Jakarta
Districts of Jakarta
List of districts of Jakarta
Buildings and structures in Jakarta
See Architecture of Indonesia
Transport in Jakarta
Communications in Indonesia
Communications in Indonesia
.id
Internet in Indonesia
Palapa
Mobile phone companies of Indonesia
Indosat Ooredoo Hutchison
Smartfren
Telkomsel
XL Axiata
Indonesian culture
Indonesian architecture
Indonesian architecture
Traditional architecture
Buildings and structures in Indonesia
Kelong
Villa Isola
Buildings and structures in Jakarta
Palaces in Indonesia
Istana Bogor
Istana Luwu
Istana Maimun
Istana Merdeka
Istana Negara, Jakarta
Istana Wakil Presiden
Prisons
Kambangan Island
Shopping malls
#Shopping malls in Jakarta
#Shopping malls in Bandung
#Shopping malls in Surabaya
#Shopping malls in Batam
Towers
Monumen Nasional
Wisma 46
Architects
Albert Aalbers
Thomas Karsten
Gunadharma
Indonesian art and culture
Bisj Pole
Artists
List of Indonesian painters
Photographers
Isidore van Kinsbergen
Indonesian culture
Indonesian folklore and Balinese mythology
Ameta
Hainuwele
Malin Kundang
Balinese mythology
Cinema
Useful links
Cinema of Indonesia
Indonesian Film Festival
Jakarta International Film Festival
Films of the Dutch East Indies
Sejarah Film 1900-1950
Indonesian films
Arisan!
Gie
Joni's Promise
Long Road to Heaven
What's Up with Love?
Whispering Sands
The Mirror Never Lies
Indonesian film directors
Joko Anwar
Rudy Soedjarwo
Actors
Barry Prima
Mariana Renata
Pierre Roland
Nora Samosir
Dian Sastrowardoyo
Tora Sudiro
Suzzanna
Yati Octavia
Christine Hakim
Evan Sanders
Deddy Mizwar
Comedians
Jojon
Dorce Gamalama
Tukul Arwana
Peppy
Mandra
Tora Sudiro
Didi Petet
Sule (comedian)
Indonesian clothing
Indonesian cuisine
List of Indonesian dishes
Languages of Indonesia
Indonesian language
Libraries and museums
Literature and writers
Old Sundanese Literature
Bujangga Manik
Sanghyang Siksakanda ng Karesian
Old Javanese Literature
Kidung Sunda
Kakawin
Kakawin Bhāratayuddha
Bhinneka Tunggal Ika
Kakawin Hariwangsa
Kakawin
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Malay literature
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Indonesian writers
Music and dance
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Composers
List of Indonesian composers
Groups
Kekal
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Peterpan (band)
Sajama cut
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Albums
Peterpan albums
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Musicians
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Dance
Economy
Businesspeople
Companies and banks
List of companies
List of airlines
List
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https://en.wikipedia.org/wiki/Polyhex
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Polyhex can mean:
Polyhex (mathematics), a class of mathematical shapes
Polyhex (Transformers), a fictional city in the Transformers stories
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https://en.wikipedia.org/wiki/440%20%28number%29
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440 (four hundred [and] forty) is the natural number following 439 and preceding 441.
In mathematics
440 has the factorization
440 is:
Even
The sum of the first 17 prime numbers
A harshad number
An abundant number
A happy number
References
Integers
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https://en.wikipedia.org/wiki/The%2085%20Ways%20to%20Tie%20a%20Tie
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The 85 Ways to Tie a Tie is a book by Thomas Fink and Yong Mao about the history of the knotted neckcloth, the modern necktie, and how to tie each. It is based on two mathematics papers published by the authors in Nature and Physica A while they were research fellows at Cambridge University’s Cavendish Laboratory. The authors prove that, assuming both the tie and the wearer to be of typical size, there are exactly 85 ways of tying a necktie using the conventional method of wrapping the wide end of the tie around the narrow end. They describe each and highlight those that they determine to be historically notable or aesthetically pleasing.
It was published by Fourth Estate on November 4, 1999, and subsequently published in nine other languages.
The mathematics
The discovery of all possible ways to tie a tie depends on a mathematical formulation of the act of tying a tie. In their papers (which are technical) and book (which is for a lay audience, apart from an appendix), the authors show that necktie knots are equivalent to persistent random walks on a triangular lattice, with some constraints on how the walks begin and end. Thus enumerating tie knots of n moves is equivalent to enumerating walks of n steps. Imposing the conditions of symmetry and balance reduces the 85 knots to 13 aesthetic ones.
Knot representation
The basic idea is that tie knots can be described as a sequence of five
different possible moves, although not all moves can follow each other. These are summarized as follows. All diagrams are as the tie would appear were you wearing it and looking in a mirror.
L: left; C: centre; R: right; these must change every move.
i: into the diagram; o: out of the diagram; these must alternate.
T: through the loop just made.
With this shorthand, traditional and new knots can be compactly expressed, as below. Note that any knot that begins with an o move must start with the tie turned inside out around the neck.
Knots
Selection criteria
Of the 85 knots possible with a typical necktie, Fink and Mao selected thirteen as "aesthetic knots" suitable for use. They made their selection based on three criteria: shape, symmetry, and balance.
Shape
In Fink and Mao's classification, each of the 85 tie knots belongs to a particular "class", which is defined by its total number of moves and its number of centering moves. For example, the four-in-hand is a four-move, one-center knot, while the half-Windsor is a six-move, two-center knot. Knots with fewer centering moves, less than one-third of the total, appear narrower and more elongated, while knots with more centering moves appear wider and more squat. Due to the triangular nature of tie knots, the number of centering moves must necessarily be less than half the total number of moves.
There are a total of 16 classes, ranging from three moves with one center to nine moves with four centers, but only classes in which the ratio of centering moves to total moves is 1:6 or greater contain a
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https://en.wikipedia.org/wiki/Bunching
|
Bunching can refer to:
Bunching (mathematics), also known as Muirhead's inequality.
Bunching (animals), the practice of stealing pets for laboratories.
Bus bunching, two or more transit vehicles running together despite evenly spaced scheduling
Photon bunching, in physics, the statistical tendency for photons to arrive simultaneously at a detector
Within the Wikipedia community it can also refer to:
A term for section edit buttons showing up after images or textboxes—see Wikipedia:How to fix bunched-up edit links.
See also
Bunch (disambiguation)
Bunch (surname)
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https://en.wikipedia.org/wiki/LiveMath
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LiveMath is a computer algebra system available on a number of platforms including Mac OS, macOS (Carbon), Microsoft Windows, Linux (x86) and Solaris (SPARC). It is the latest release of a system that originally emerged as Theorist for the "classic" Mac in 1989, became MathView and MathPlus in 1997 after it was sold to Waterloo Maple, and finally LiveMath after it was purchased by members of its own userbase in 1999. The application is currently owned by MathMonkeys of Cambridge, Massachusetts. The overall LiveMath suite contains LiveMath Maker, the main application, as well as LiveMath Viewer for end-users, and LiveMath Plug-In, an ActiveX plugin for browsers, which was discontinued in 2014.
Description
LiveMath uses a worksheet-based approach, similar to products like Mathematica or MathCAD. The user enters equations into the worksheet and then uses the built-in functions to help solve them, or reduce them numerically. Workbooks typically contain a number of equations separated into sections, along with data tables, graphs, and similar outputs. Unlike most CAS applications, LiveMath uses a full GUI with high-quality graphical representations of the equations at every step, including input.
LiveMath also allows the user to interact with the equation in the sheet; for instance, one can drag an instance of to the left hand side of the equation, at which point LiveMath will re-arrange the equation to solve for . LiveMath's algebraic solving systems are relatively simple compared to better known systems like Mathematica, and does not offer the same sort of automated single-step solving of these packages.
See also
Comparison of computer algebra systems
References
External links
Computer algebra system software for Linux
Computer algebra system software for Windows
Computer algebra system software for macOS
Proprietary commercial software for Linux
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https://en.wikipedia.org/wiki/M.%20S.%20Narasimhan
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Mudumbai Seshachalu Narasimhan (7 June 1932 – 15 May 2021) was an Indian mathematician. His focus areas included number theory, algebraic geometry, representation theory, and partial differential equations. He was a pioneer in the study of moduli spaces of holomorphic vector bundles on projective varieties. His work is considered the foundation for Kobayashi–Hitchin correspondence that links differential geometry and algebraic geometry of vector bundles over complex manifolds. He was also known for his collaboration with mathematician C. S. Seshadri, for their proof of the Narasimhan–Seshadri theorem which proved the necessary conditions for stable vector bundles on a Riemann surface.
He was a recipient of the Padma Bhushan, India's third highest civilian honor, in 1990, and the Ordre national du Mérite from France in 1989. He was an elected Fellow of the Royal Society, London. He was also the recipient of Shanti Swarup Bhatnagar Prize in 1975 and was the only Indian to receive the King Faisal International Prize in the field of science.
Early life
Narasimhan was born on 7 June 1932 into a rural family in Tandarai in present day Tamil Nadu, as the eldest among five children. His family hailed from the North Arcot district. After his early education in rural part of the country, he joined Loyola College in Madras for his undergraduate education. Here he studied under Father Charles Racine, a French Jesuit professor, who in turn had studied under the French mathematician and geometer Élie Cartan. He joined the Tata Institute of Fundamental Research (TIFR), Bombay, for his graduate studies in 1953. He obtained his Ph.D. from the University of Mumbai in 1960 where his advisor was the mathematician K. S. Chandrasekharan, who was known for his work on number theory.
Career
Narasimhan started his career in 1960 when he joined the faculty of the Tata Institute of Fundamental Research (TIFR); he later went on to become an honorary fellow. His areas of focus while at TIFR included studying partial differential operators and elliptic operators. During this time, he visited France under the invitation of Laurent Schwartz and was exposed to the works of other French mathematicians including Jean-Pierre Serre, Claude Chevalley, Élie Cartan, and Jean Leray. He contracted pleurisy during his time in France and was hospitalized. He would later recount the incident as exposing him to the "real France" and further strengthening his leftist sympathies which were already triggered by his interactions with the Trotskyist Schwartz.
During his time in France he also collaborated with Japanese mathematician Takeshi Kotake working on the analyticity theorems for determining specific types of elliptic operators that satisfied Cauchy–Schwarz inequalities. His work with Kotake was known as the Kotake–Narasimhan theorem for elliptic operators in the setting of ultradifferentiable functions.
He collaborated with Indian mathematician C. S. Seshadri for the ground-breaki
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https://en.wikipedia.org/wiki/Charles%20G.%20Callard
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Charles "Chuck" Gordon Callard (2 June 1923 – 1 May 2004) was a prominent figure in the financial community due to his innovative application of mathematics and statistics to stock analysis. Born in Lansing, Michigan, he was a Corsair fighter pilot on an aircraft carrier while serving in the United States Navy during World War II. After his military service, Callard earned his MBA at the University of Chicago Graduate School of Business in 1943. He then taught statistics at Miami University in Oxford, Ohio for several years.
Callard worked as a securities analyst in Chicago. He then held marketing and planning positions at Armour & Co. and Ball Brothers. In 1969, Callard left the corporate sector and started Callard, Madden & Associates. The insights that he developed became a bridge between academic finance and the worlds of corporate finance and asset management. Callard was the first in the inflationary 1970s to adjust standard accounting data so that they would conform with the financial concepts then being developed at the Graduate School of Business. He recognized the flaws in the traditional accounting measures and developed alternative economic measures of corporate performance. He used this approach to demonstrate that the effective corporate tax rates were much higher than the legislated rates and differed greatly among firms that otherwise appeared to be subject to the same tax rates.
One of Callard’s greatest contributions was to develop a systematic explanation of the cost of capital for individual firms. The cost of capital for all firms varied in response to changes in the national inflation rate and to changes in corporate tax rates. The cost of capital for individual firms was affected by the levels and the rates of growth of their anticipated profits and by the variability of their cash flows. Similarly a cost of capital could be assigned to each of the business units within a firm.
Callard’s insights extended to capital structure; he distinguished between the costs of debt capital and the costs of equity capital, which varied extensively relative to each other over the phases of the business and economic cycle. While each firm was a price-taker in the capital market, each could affect its aggregate cost of capital by altering the shares of debt and of equity in its capital structure. Some firms were too extensively leveraged and could reduce their costs of capital by reducing the debt component of their capital structure while other firms could reduce their costs of capital by increasing the debt component.
Chuck Callard died in 2004.
A group study room was dedicated in his memory at the Booth School of Business.
External links
References
20th-century American educators
American financial businesspeople
United States Navy pilots of World War II
Businesspeople from Lansing, Michigan
University of Chicago Booth School of Business alumni
1923 births
2004 deaths
Miami University faculty
Place of death missing
Economists fro
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https://en.wikipedia.org/wiki/Arithmetic%20and%20geometric%20Frobenius
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In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R.
The image of φ is then Rp, the subring of R consisting of p-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring automorphism.
The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping
φ*: Spec(Rp) → Spec(R)
of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field.
Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.
References
, p. 5
Mathematical terminology
Algebraic geometry
Algebraic number theory
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https://en.wikipedia.org/wiki/Tom%C3%A1%C5%A1ovce%2C%20Rimavsk%C3%A1%20Sobota%20District
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Tomášovce () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Equations%20defining%20abelian%20varieties
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In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields.
Complete intersections
The only 'easy' cases are those for d = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In P3, an elliptic curve can be obtained as the intersection of two quadrics.
In general abelian varieties are not complete intersections. Computer algebra techniques are now able to have some impact on the direct handling of equations for small values of d > 1.
Kummer surfaces
The interest in nineteenth century geometry in the Kummer surface came in part from the way a quartic surface represented a quotient of an abelian variety with d = 2, by the group of order 2 of automorphisms generated by x → −x on the abelian variety.
General case
Mumford defined a theta group associated to an invertible sheaf L on an abelian variety A. This is a group of self-automorphisms of L, and is a finite analogue of the Heisenberg group. The primary results are on the action of the theta group on the global sections of L. When L is very ample, the linear representation can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of nilpotent group, a central extension of a group of torsion points on A, and the extension is known (it is in effect given by the Weil pairing). There is a uniqueness result for irreducible linear representations of the theta group with given central character, or in other words an analogue of the Stone–von Neumann theorem. (It is assumed for this that the characteristic of the field of coefficients doesn't divide the order of the theta group.)
Mumford showed how this abstract algebraic formulation could account for the classical theory of theta functions with theta characteristics, as being the case where the theta group was an extension of the two-torsion of A.
An innovation in this area is to use the Mukai–Fourier transform.
The coordinate ring
The goal of the theory is to prove results on the homogeneous coordinate ring of the embedded abelian variety A, that is, set in a projective space according to a very ample L and its global sections. The graded commutative ring that is formed by the direct sum of the global sections of the
meaning the n-fold tensor product of itself, is re
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https://en.wikipedia.org/wiki/Extra%20special%20group
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In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two (up to isomorphism) extraspecial groups of order p1+2n. Extraspecial groups often occur in centralizers of involutions. The ordinary character theory of extraspecial groups is well understood.
Definition
Recall that a finite group is called a p-group if its order is a power of a prime p.
A p-group G is called extraspecial if its center Z is cyclic of order p, and the quotient G/Z is a non-trivial elementary abelian p-group.
Extraspecial groups of order p1+2n are often denoted by the symbol p1+2n. For example, 21+24 stands for an extraspecial group of order 225.
Classification
Every extraspecial p-group has order p1+2n for some positive integer n, and conversely for each such number there are exactly two extraspecial groups up to isomorphism. A central product of two extraspecial p-groups is extraspecial, and every extraspecial group can be written as a central product of extraspecial groups of order p3. This reduces the classification of extraspecial groups to that of extraspecial groups of order p3. The classification is often presented differently in the two cases p odd and p = 2, but a uniform presentation is also possible.
p odd
There are two extraspecial groups of order p3, which for p odd are given by
The group of triangular 3x3 matrices over the field with p elements, with 1's on the diagonal. This group has exponent p for p odd (but exponent 4 if p = 2).
The semidirect product of a cyclic group of order p2 by a cyclic group of order p acting non-trivially on it. This group has exponent p2.
If n is a positive integer there are two extraspecial groups of order p1+2n, which for p odd are given by
The central product of n extraspecial groups of order p3, all of exponent p. This extraspecial group also has exponent p.
The central product of n extraspecial groups of order p3, at least one of exponent p2. This extraspecial group has exponent p2.
The two extraspecial groups of order p1+2n are most easily distinguished by the fact that one has all elements of order at most p and the other has elements of order p2.
p = 2
There are two extraspecial groups of order 8 = 23, which are given by
The dihedral group D8 of order 8, which can also be given by either of the two constructions in the section above for p = 2 (for p odd they give different groups, but for p = 2 they give the same group). This group has 2 elements of order 4.
The quaternion group Q8 of order 8, which has 6 elements of order 4.
If n is a positive integer there are two extraspecial groups of order 21+2n, which are given by
The central product of n extraspecial groups of order 8, an odd number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 1.
The central product of n extraspecial groups of order 8, an even n
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https://en.wikipedia.org/wiki/Hypercomplex%20manifold
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In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle
equipped with an action by the algebra of quaternions
in such a way that the quaternions
define integrable almost complex structures.
If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.
Examples
Every hyperkähler manifold is also hypercomplex.
The converse is not true. The Hopf surface
(with acting
as a multiplication by a quaternion , ) is
hypercomplex, but not Kähler,
hence not hyperkähler either.
To see that the Hopf surface is not Kähler,
notice that it is diffeomorphic to a product
hence its odd cohomology
group is odd-dimensional. By Hodge decomposition,
odd cohomology of a compact Kähler manifold
are always even-dimensional. In fact Hidekiyo Wakakuwa proved
that on a compact hyperkähler manifold .
Misha Verbitsky has shown that any compact
hypercomplex manifold admitting a Kähler structure is also hyperkähler.
In 1988, left-invariant hypercomplex structures on some compact Lie groups
were constructed by the physicists Philippe Spindel, Alexander Sevrin, Walter Troost, and Antoine Van Proeyen. In 1992, Dominic Joyce
rediscovered this construction, and gave a complete classification of
left-invariant hypercomplex structures on compact Lie groups.
Here is the complete list.
where denotes an -dimensional compact torus.
It is remarkable that any compact Lie group becomes
hypercomplex after it is multiplied by a sufficiently
big torus.
Basic properties
Hypercomplex manifolds as such were studied by Charles Boyer in 1988. He also proved that in real dimension 4, the only compact hypercomplex
manifolds are the complex torus , the Hopf surface and
the K3 surface.
Much earlier (in 1955) Morio Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann of almost quaternionic structures). His construction leads to what Edmond Bonan called the Obata connection which is torsion free, if and only if, "two" of the almost complex structures are integrable and in this case the manifold is hypercomplex.
Twistor spaces
There is a 2-dimensional sphere of quaternions
satisfying .
Each of these quaternions gives a complex
structure on a hypercomplex manifold M. This
defines an almost complex structure on the manifold
, which is fibered over
with fibers identified with .
This complex structure is integrable, as follows
from Obata's theorem (this was first explicitly proved by
Dmitry Kaledin). This complex manifold
is called the twistor space of .
If M is , then its twistor space
is isomorphic to .
See also
Quaternionic manifold
Hyperkähler manifold
References
.
.
.
.
Complex manifolds
Structures on manifolds
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https://en.wikipedia.org/wiki/Betty%20Gibson
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Betty Gibson (1911–2001) was a Canadian educator considered by the community to be instrumental in developing and implementing the Mathematics and Language Arts Curriculum in Manitoba. Celebrated as "an exemplary educator", the Betty Gibson School in Brandon, Manitoba was named in her honour.
Biography
Over Gibson's twenty year teaching career (1929-1949) she taught in rural Manitoba, the city of Brandon, Manitoba and South Africa. She served as Principal of Fleming School between 1949 and 1959. At the same time she worked on her Bachelor of Arts at Brandon University, which she completed in 1959. Gibson was a professor at Brandon University between 1956-1975 and served as Assistant Superintendent for the Brandon School Division briefly between 1967 and 1968. In 1981 she authored a children's book, Pride of the Golden Bear. Gibson also penned The Story of Little Quack in 1991.
Awards
Received Centennial Medal, 1967
Received J.M. Brown Award, 1974 for contributions to education in Manitoba
Inducted into Brandon University Wall of Fame on November 14, 2003
External links
Brandon University Hall of Fame
Betty Gibson School
Book Review of The Story of Little Quack
1911 births
2001 deaths
Brandon University alumni
Academic staff of Brandon University
Canadian expatriates in South Africa
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https://en.wikipedia.org/wiki/Ordered%20Bell%20number
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In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from , these numbers are
The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron.
History
The ordered Bell numbers appear in the work of , who used them to count certain plane trees with totally ordered leaves. In the trees considered by Cayley, each root-to-leaf path has the same length, and the number of nodes at distance from the root must be strictly smaller than the number of nodes at distance , until reaching the leaves. In such a tree, there are pairs of adjacent leaves, that may be weakly ordered by the height of their lowest common ancestor; this weak ordering determines the tree. call the trees of this type "Cayley trees", and they call the sequences that may be used to label their gaps (sequences of positive integers that include at least one copy of each positive integer between one and the maximum value in the sequence) "Cayley permutations".
traces the problem of counting weak orderings, which has the same sequence as its solution, to the work of . These numbers were called Fubini numbers by Louis Comtet, because they count the number of different ways to rearrange the ordering of sums or integrals in Fubini's theorem, which in turn is named after Guido Fubini. The Bell numbers, named after Eric Temple Bell, count the number of partitions of a set, and the weak orderings that are counted by the ordered Bell numbers may be interpreted as a partition together with a total order on the sets in the partition.
Formulas
Summation
The th ordered Bell number may be given by a summation formula involving the Stirling numbers of the second kind, which count the number of partitions of an -element set into nonempty subsets. A weak ordering may be described as a permutation of the subsets in this partition, and so the ordered Bell numbers (the number of weak orderings) may be calculated by summing these numbers, multiplied by a factorial, , that counts the number of these permutations:
An alternative interpretation of the terms of this sum is that they count the features of each dimension in a permutohedron of dimension , with the th term counting the features of dimension . For instance, the three-dimensional permutohedron, the truncated octahedron,
has one volume (), 14 two-dimensional faces (), 36 edges (), and 24 vertices (). The total number of these faces is 1 + 14 + 36 + 24 = 75, an ordered Bell number, correspon
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https://en.wikipedia.org/wiki/Immersion%20%28mathematics%29
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In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, is an immersion if
is an injective function at every point of (where denotes the tangent space of a manifold at a point in ). Equivalently, is an immersion if its derivative has constant rank equal to the dimension of :
The function itself need not be injective, only its derivative must be.
A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that is diffeomorphic to its image in . An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion.
If is compact, an injective immersion is an embedding, but if is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
Regular homotopy
A regular homotopy between two immersions and from a manifold to a manifold is defined to be a differentiable function such that for all in the function defined by for all is an immersion, with , . A regular homotopy is thus a homotopy through immersions.
Classification
Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for every map of an -dimensional manifold to an -dimensional manifold is homotopic to an immersion, and in fact to an embedding for ; these are the Whitney immersion theorem and Whitney embedding theorem.
Stephen Smale expressed the regular homotopy classes of immersions as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence.
Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any -dimensional manifold in any -dimensional manifold .
The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.
Existence
The primary obstruction to the existence of an immersion is the stable normal bundle of , as detected by its characteristic classes, notably its Stiefel–Whitney classes. That is, since is parallelizable, the pullback of its tangent bundle to is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle on , , which has dimension , and of the normal bundle of the immersion , which has dimension , for there to be a codimension immersion of , there must be a vector bundle of dimension , , standing in for the normal bundle , such that is trivial. Conversely, given such a bundle, an immersion of with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold.
The stable normal bundle is the class of normal
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https://en.wikipedia.org/wiki/Fractional%20factorial%20design
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In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset is chosen so as to exploit the sparsity-of-effects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources. In other words, it makes use of the fact that many experiments in full factorial design are often redundant, giving little or no new information about the system.
Notation
Fractional designs are expressed using the notation lk − p, where l is the number of levels of each factor investigated, k is the number of factors investigated, and p describes the size of the fraction of the full factorial used. Formally, p is the number of generators, assignments as to which effects or interactions are confounded, i.e., cannot be estimated independently of each other (see below). A design with p such generators is a 1/(lp)=l−p fraction of the full factorial design.
For example, a 25 − 2 design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 25 factorial experiment, this experiment requires only eight runs.
In practice, one rarely encounters l > 2 levels in fractional factorial designs, since response surface methodology is a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels. In addition, the methodology to generate such designs for more than two levels is much more cumbersome.
The levels of a factor are commonly coded as +1 for the higher level, and −1 for the lower level. For a three-level factor, the intermediate value is coded as 0.
To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minus signs. The strings have as many symbols as factors, and their values dictate the level of each factor: conventionally, for the first (or low) level, and for the second (or high) level. The points in this experiment can thus be represented as , , , and .
The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that the specified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at its low (or first) level (for example, "a" indicates that factor A is on its high setting, while all other factors are at their low (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) values.
Generation
In practice, experimenters typically rely on statistical reference books to supply the "standard" fractional factorial designs, consisting of the principal fraction. The principal fraction is the set of treatment combinations for which the generators evaluate to + under the treatment combination algebr
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https://en.wikipedia.org/wiki/Good%E2%80%93Turing%20frequency%20estimation
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Good–Turing frequency estimation is a statistical technique for estimating the probability of encountering an object of a hitherto unseen species, given a set of past observations of objects from different species. In drawing balls from an urn, the 'objects' would be balls and the 'species' would be the distinct colors of the balls (finite but unknown in number). After drawing red balls, black balls and green balls, we would ask what is the probability of drawing a red ball, a black ball, a green ball or one of a previously unseen color.
Historical background
Good–Turing frequency estimation was developed by Alan Turing and his assistant I. J. Good as part of their methods used at Bletchley Park for cracking German ciphers for the Enigma machine during World War II. Turing at first modelled the frequencies as a multinomial distribution, but found it inaccurate. Good developed smoothing algorithms to improve the estimator's accuracy.
The discovery was recognized as significant when published by Good in 1953, but the calculations were difficult so it was not used as widely as it might have been. The method even gained some literary fame due to the Robert Harris novel Enigma.
In the 1990s, Geoffrey Sampson worked with William A. Gale of AT&T to create and implement a simplified and easier-to-use variant of the Good–Turing method described below. Various heuristic justifications and a simple combinatorial derivation have been provided.
The method
The Good–Turing estimator is largely independent of the distribution of species frequencies.
Notation
Assuming that distinct species have been observed, enumerated
Then the frequency vector, has elements that give the number of individuals that have been observed for species
The frequency of frequencies vector, shows how many times the frequency r occurs in the vector (i.e. among the elements ):
For example, is the number of species for which only one individual was observed. Note that the total number of objects observed, can be found from
Calculation
The first step in the calculation is to estimate the probability that a future observed individual (or the next observed individual) is a member of a thus far unseen species. This estimate is:
The next step is to estimate the probability that the next observed individual is from a species which has been seen times. For a single species this estimate is:
Here, the notation means the smoothed, or adjusted value of the frequency shown in parentheses. An overview of how to perform this smoothing follows in the next section (see also empirical Bayes method).
To estimate the probability that the next observed individual is from any species from this group (i.e., the group of species seen times) one can use the following formula:
Smoothing
For smoothing the erratic values in for large , we would like to make a plot of versus but this is problematic because for large many will be zero. Instead a revised quantity, is plotted versus
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https://en.wikipedia.org/wiki/Symplectic%20filling
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In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n-dimensional topological manifold X is the boundary of an (n + 1)-dimensional manifold W. Perhaps the most active area of current research is when n = 3, where one may consider certain types of fillings.
There are many types of fillings, and a few examples of these types (within a probably limited perspective) follow.
An oriented filling of any orientable manifold X is another manifold W such that the orientation of X is given by the boundary orientation of W, which is the one where the first basis vector of the tangent space at each point of the boundary is the one pointing directly out of W, with respect to a chosen Riemannian metric. Mathematicians call this orientation the outward normal first convention.
All the following cobordisms are oriented, with the orientation on W given by a symplectic structure. Let ξ denote the kernel of the contact form α.
A weak symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with such that .
A strong symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with such that ω is exact near the boundary (which is X) and α is a primitive for ω. That is, ω = dα in a neighborhood of the boundary .
A Stein filling of a contact manifold (X,ξ) is a Stein manifold W which has X as its strictly pseudoconvex boundary and ξ is the set of complex tangencies to X – that is, those tangent planes to X that are complex with respect to the complex structure on W. The canonical example of this is the 3-sphere where the complex structure on is multiplication by in each coordinate and W is the ball {|x| < 1} bounded by that sphere.
It is known that this list is strictly increasing in difficulty in the sense that there are examples of contact 3-manifolds with weak but no strong filling, and others that have strong but no Stein filling. Further, it can be shown that each type of filling is an example of the one preceding it, so that a Stein filling is a strong symplectic filling, for example. It used to be that one spoke of semi-fillings in this context, which means that X is one of possibly many boundary components of W, but it has been shown that any semi-filling can be modified to be a filling of the same type, of the same 3-manifold, in the symplectic world (Stein manifolds always have one boundary component).
References
Y. Eliashberg, A Few Remarks about Symplectic Filling, Geometry and Topology 8, 2004, p. 277–293
J. Etnyre, On Symplectic Fillings Algebr. Geom. Topol. 4 (2004), p. 73–80 online
H. Geiges, An Introduction to Contact Topology, Cambridge University Press, 2008
Geometric topology
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https://en.wikipedia.org/wiki/Giovanni%20Antonio%20Medrano
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Giovanni Antonio Medrano (11 December, 1703–1760) was the "Major Regius Praefectus Mathematicis Regni Neapolitani" (Major Royal Governor of Mathematics of the Kingdom of Naples), chief engineer of the kingdom, architect, brigadier, and teacher of Charles III of Spain and his brothers the infantes. Giovanni was born in Sciacca in the Kingdom of Sicily. Giovanni designed the Obelisk of Bitonto, the Palace of Capodimonte and the Teatro di San Carlo in Italy for Charles III of Spain. Medrano’s career is particularly studied, from his stay in Seville as a teacher for the royal princes, and his influence on Prince Charles’ architectural taste, to his projects in the Kingdom of Naples and the royal palace at Capodimonte.
Education of Charles III and the Infantes
The Medrano family had close ties to the Spanish Monarchs. In fact, the Regency of the Royal Council of Navarre from May 9th 1702 to 1705 under Don Pedro Antonio de Medrano for King Philip V confirms this.
During this Andalusian period, Giovanni Antonio Medrano began to deal with the military and architectural education of the Infante Don Carlos and his brothers; of these tasks, for "instruction and amusement of the Most Serene Prince our Lord and Lords Infantes", there are two plans of a Fort, erected between 1729 and 1730 in Buenavista, on the outskirts of Seville, which included a ravelin dedicated to the Infante don Carlos himself.
In December 1731 he followed Philip V's sixteen-year-old son, Charles of Bourbon, Duke of Parma and Piacenza, to Livorno, as an ordinary engineer and with the rank of lieutenant. From 1732 to 1734 he remained in the service of the Infante, teaching him geography, history and mathematics, as well as military art and architecture during his stay in the cities of Florence, Parma and Piacenza.
Rise in Ranks
The fact that Giovanni Antonio Medrano was promoted in 1733 to lieutenant and ordinary engineer and, later and already in Naples, in 1737, to brigadier and chief engineer, testifies to his efforts and work. After the coronation of Charles as king of Naples and Sicily in 1734, probably due to his close bond with the young sovereign, but more generally for reasons related to the need for the government to have more direct control over the entire local system of public works, Medrano was invested with some of the most prestigious and strategic positions of a public nature initiated by the Bourbons in the capital.
While still a teenager, he moved with his family to Spain, where he embarked on a military career within the royal corps of engineers created in 1711 by King Philip V of Bourbon. Giovanni had entered the service of Spain in 1719 as a Military Architect, although it is probable that Giovanni had already joined the army of the Marquess of Verboom Jorge Próspero de Verboom in the Sicilian campaign of 1718, since in December of that same year he appeared as extraordinary engineer and sub-lieutenant of this body. The Marquess of Verboom was one of the b
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https://en.wikipedia.org/wiki/Jon%C3%ADlson
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Jonilson Clovis Nascimento Breves or Jonilson (born in Pinheiral on November 28, 1978) is an association football player, currently playing for Goiás on loan from Botafogo (SP).
Club statistics
Achievements
Rio de Janeiro's Cup: 2000, 2002
Rio de Janeiro State League (2nd division): 2004
Guanabara Cup: 2005
Minas Gerais State League: 2006
References
External links
netvasco.com
netvasco 2008 stats
1978 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
J2 League players
Ceará Sporting Club players
Volta Redonda FC players
Botafogo de Futebol e Regatas players
Cruzeiro Esporte Clube players
Vegalta Sendai players
CR Vasco da Gama players
Clube Atlético Mineiro players
Goiás Esporte Clube players
Men's association football midfielders
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https://en.wikipedia.org/wiki/Green%E2%80%93Tao%20theorem
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In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.
Statement
Let denote the number of primes less than or equal to . If is a subset of the prime numbers such that
then for all positive integers , the set contains infinitely many arithmetic progressions of length . In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions.
In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula
for the number of k tuples of primes in arithmetic progression. Here, is the constant
The result was made unconditional by Green–Tao
and Green–Tao–Ziegler.
Overview of the proof
Green and Tao's proof has three main components:
Szemerédi's theorem, which asserts that subsets of the integers with positive upper density have arbitrarily long arithmetic progressions. It does not a priori apply to the primes because the primes have density zero in the integers.
A transference principle that extends Szemerédi's theorem to subsets of the integers which are pseudorandom in a suitable sense. Such a result is now called a relative Szemerédi theorem.
A pseudorandom subset of the integers containing the primes as a dense subset. To construct this set, Green and Tao used ideas from Goldston, Pintz, and Yıldırım's work on prime gaps. Once the pseudorandomness of the set is established, the transference principle may be applied, completing the proof.
Numerous simplifications to the argument in the original paper have been found. provide a modern exposition of the proof.
Numerical work
The proof of the Green–Tao theorem does not show how to find the arithmetic progressions of primes; it merely proves they exist. There has been separate computational work to find large arithmetic progressions in the primes.
The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 · k; k = 0, 1, . . ., 22.'.
On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression:
468,395,662,504,823 + 205,619 · 223,092,870 · n, for n = 0 to 23.
The constant 223,092,870 here is the product of the prime numbers up to 23, more compactly written 23# in primorial notation.
On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primes:
6,171,054,912,832,631 + 366,384 · 23# · n, for n = 0 to 24.
On April 12, 2010, Benoît Perichon with software by Wróblewski and Geoff Reynold
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https://en.wikipedia.org/wiki/Prefix%20order
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In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical systems as a set of functions from time (a totally-ordered set) to some phase space. In this case, the elements of the set are usually referred to as executions of the system.
The name prefix order stems from the prefix order on words, which is a special kind of substring relation and, because of its discrete character, a tree.
Formal definition
A prefix order is a binary relation "≤" over a set P which is antisymmetric, transitive, reflexive, and downward total, i.e., for all a, b, and c in P, we have that:
a ≤ a (reflexivity);
if a ≤ b and b ≤ a then a = b (antisymmetry);
if a ≤ b and b ≤ c then a ≤ c (transitivity);
if a ≤ c and b ≤ c then a ≤ b or b ≤ a (downward totality).
Functions between prefix orders
While between partial orders it is usual to consider order-preserving functions, the most important type of functions between prefix orders are so-called history preserving functions. Given a prefix ordered set P, a history of a point p∈P is the (by definition totally ordered) set p− = {q | q ≤ p}. A function f: P → Q between prefix orders P and Q is then history preserving if and only if for every p∈P we find f(p−) = f(p)−. Similarly, a future of a point p∈P is the (prefix ordered) set p+ = {q | p ≤ q} and f is future preserving if for all p∈P we find f(p+) = f(p)+.
Every history preserving function and every future preserving function is also order preserving, but not vice versa.
In the theory of dynamical systems, history preserving maps capture the intuition that the behavior in one system is a refinement of the behavior in another. Furthermore, functions that are history and future preserving surjections capture the notion of bisimulation between systems, and thus the intuition that a given refinement is correct with respect to a specification.
The range of a history preserving function is always a prefix closed subset, where a subset S ⊆ P is prefix closed if for all s,t ∈ P with t∈S and s≤t we find s∈S.
Product and union
Taking history preserving maps as morphisms in the category of prefix orders leads to a notion of product that is not the Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an arbitrary interleaving of the original prefix orders. The union of two prefix orders is the disjoint union, as it is with partial orders.
Isomorphism
Any bijective history preserving function is an order isomorphism. Furthermore, if for a given prefix ordered set P we construct the set P- ≜ { p- | p∈ P} we find that this set is prefix ordered by the subset relation ⊆, and furthermore, that the function max: P- → P is an isomorphism, where max(S) returns for each set S∈P- the maximum element in terms of the order on P (i.e.
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https://en.wikipedia.org/wiki/Inverted%20bell%20%28disambiguation%29
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Inverted bell may refer to one of the following:
Inverted bell, a shape
Inverted bell (music), a musical instrument
Inverted bell curve, in statistics, a bimodal distribution
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https://en.wikipedia.org/wiki/Decrement%20table
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Decrement tables, also called life table methods, are used to calculate the probability of certain events.
Birth control
Life table methods are often used to study birth control effectiveness. In this role, they are an alternative to the Pearl Index.
As used in birth control studies, a decrement table calculates a separate effectiveness rate for each month of the study, as well as for a standard period of time (usually 12 months). Use of life table methods eliminates time-related biases (i.e. the most fertile couples getting pregnant and dropping out of the study early, and couples becoming more skilled at using the method as time goes on), and in this way is superior to the Pearl Index.
Two kinds of decrement tables are used to evaluate birth control methods. Multiple-decrement (or competing) tables report net effectiveness rates. These are useful for comparing competing reasons for couples dropping out of a study. Single-decrement (or noncompeting) tables report gross effectiveness rates, which can be used to accurately compare one study to another.
See also
Survival analysis
Footnotes
Birth control
Actuarial science
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https://en.wikipedia.org/wiki/Ramanujan%20prime
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In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
Origins and definition
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
where is the prime-counting function, equal to the number of primes less than or equal to x.
The converse of this result is the definition of Ramanujan primes:
The nth Ramanujan prime is the least integer Rn for which for all x ≥ Rn. In other words: Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all x ≥ Rn.
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Note that the integer Rn is necessarily a prime number: and, hence, must increase by obtaining another prime at x = Rn. Since can increase by at most 1,
Bounds and an asymptotic formula
For all , the bounds
hold. If , then also
where pn is the nth prime number.
As n tends to infinity, Rn is asymptotic to the 2nth prime, i.e.,
Rn ~ p2n (n → ∞).
All these results were proved by Sondow (2009), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.
References
Srinivasa Ramanujan
Classes of prime numbers
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https://en.wikipedia.org/wiki/At%20bats%20per%20home%20run
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In baseball statistics, at bats per home run (AB/HR) is a way to measure how frequently a batter hits a home run. It is determined by dividing the number of at bats by the number of home runs hit. Mark McGwire possesses the MLB record for this statistic with a career ratio of 10.61 at bats per home run and Babe Ruth is second, with 11.76 at bats per home run. Aaron Judge has the best career ratio among active players with 11.99 at bats per home run, as of October 5, 2022.
Major League Baseball leaders
Career
Totals are current , minimum 3,000 plate appearances.
Mark McGwire - 10.61
Babe Ruth - 11.76
Aaron Judge - 11.99
Barry Bonds - 12.92
Jim Thome - 13.76
Season
Single-season statistics are current .
Barry Bonds - 6.52
Mark McGwire - 7.27
Josh Gibson - 7.80
Mark McGwire - 8.02
Mark McGwire - 8.13
Babe Ruth was the first batter to average fewer than nine at-bats per home run over a season, hitting his 54 home runs of the 1920 season in 457 at-bats; an average of 8.463. Seventy-eight years later, Mark McGwire became the first batter to average fewer than eight AB/HR, hitting his 70 home runs of the 1998 season in 509 at-bats (an average of 7.2714). In 2001, Barry Bonds became the first batter to average fewer than seven AB/HR, setting the Major League record by hitting his 73 home runs of the 2001 season in 476 at-bats for an average of 6.5205.
Ruth led the American League every year from 1918 until 1931, except for 1925.
Ruth, Josh Gibson, McGwire and Bonds are the only batters in history to average nine or fewer AB/HR over a season, having done so a combined ten times:
Aaron Judge's 62 HR season in 2022 came at a rate of 9.19 AB/HR.
References
Batting statistics
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https://en.wikipedia.org/wiki/Hardy%27s%20inequality
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Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if is a sequence of non-negative real numbers, then for every real number p > 1 one has
If the right-hand side is finite, equality holds if and only if for all n.
An integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then
If the right-hand side is finite, equality holds if and only if f(x) = 0 almost everywhere.
Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above.
General one-dimensional version
The general weighted one dimensional version reads as follows:
If , then
If , then
Multidimensional version
In the multidimensional case, Hardy's inequality can be extended to -spaces, taking the form
where , and where the constant is known to be sharp.
Fractional Hardy inequality
If and , , there exists a constant such that for every satisfying , one has
Proof of the inequality
Integral version
A change of variables gives , which is less or equal than by Minkowski's integral inequality. Finally, by another change of variables, the last expression equals .
Discrete version: from the continuous version
Assuming the right-hand side to be finite, we must have as . Hence, for any positive integer j, there are only finitely many terms bigger than . This allows us to construct a decreasing sequence containing the same positive terms as the original sequence (but possibly no zero terms). Since for every n, it suffices to show the inequality for the new sequence. This follows directly from the integral form, defining if and otherwise. Indeed, one has and, for , there holds (the last inequality is equivalent to , which is true as the new sequence is decreasing) and thus .
Discrete version: Direct proof
Let and let be positive real numbers. Set First we prove the inequality ,
Let and let be the difference between the -th terms in the RHS and LHS of , that is, . We have:
or
According to Young's inequality we have:
from which it follows that:
By telescoping we have:
proving . By applying Hölder's inequality to the RHS of we have:
from which we immediately obtain:
Letting we obtain Hardy's inequality.
See also
Carleman's inequality
Notes
References
.
External links
Inequalities
Theorems in real analysis
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https://en.wikipedia.org/wiki/Hopf%20manifold
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In complex geometry, a Hopf manifold is obtained
as a quotient of the complex vector space
(with zero deleted)
by a free action of the group of
integers, with the generator
of acting by holomorphic contractions. Here, a holomorphic contraction
is a map
such that a sufficiently big iteration
maps any given compact subset of
onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.
Examples
In a typical situation, is generated
by a linear contraction, usually a diagonal matrix
, with
a complex number, . Such manifold
is called a classical Hopf manifold.
Properties
A Hopf manifold
is diffeomorphic to .
For , it is non-Kähler. In fact, it is not even
symplectic because the second cohomology group is zero.
Hypercomplex structure
Even-dimensional Hopf manifolds admit
hypercomplex structure.
The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.
References
Complex manifolds
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https://en.wikipedia.org/wiki/Hopf%20surface
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In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) by a free action of a discrete group. If this group is the integers the Hopf surface is called primary, otherwise it is called secondary. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".) The first example was found by , with the discrete group isomorphic to the integers, with a generator acting on by multiplication by 2; this was the first example of a compact complex surface with no Kähler metric.
Higher-dimensional analogues of Hopf surfaces are called Hopf manifolds.
Invariants
Hopf surfaces are surfaces of class VII and in particular all have Kodaira dimension , and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal central infinite cyclic subgroup of finite index. The Hodge diamond is
In particular the first Betti number is 1 and the second Betti number is 0.
Conversely showed that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface.
Primary Hopf surfaces
In the course of classification of compact complex surfaces, Kodaira classified the primary Hopf surfaces.
A primary Hopf surface is obtained as
where is a group generated by
a polynomial contraction .
Kodaira has found a normal form for .
In appropriate coordinates,
can be written as
where are complex numbers
satisfying , and either
or .
These surfaces contain an elliptic curve (the image of the x-axis) and if the image of the y-axis is a second elliptic curve. When , the Hopf surface is an elliptic fiber space over the projective line if
for some positive integers m and n, with the map to the projective line given by , and otherwise the only curves are the two images of the axes.
The Picard group of any primary Hopf surface is isomorphic to the non-zero complex numbers .
has proven that a complex surface
is diffeomorphic to if and only if it is a primary Hopf surface.
Secondary Hopf surfaces
Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. classified them by finding the finite groups acting without fixed points on primary Hopf surfaces.
Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a spherical space forms and a circle.
References
Complex surfaces
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https://en.wikipedia.org/wiki/Phase%20line
|
A phase line may refer to:
Phase line (mathematics), used to analyze autonomous ordinary differential equations
Phase line (cartography), used to identify phases of military operations or changing borders over time
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https://en.wikipedia.org/wiki/International%20Mathematical%20Olympiad%20selection%20process
|
This article describes the selection process, by country, for entrance into the International Mathematical Olympiad.
The International Mathematical Olympiad (IMO) is an annual mathematics olympiad for students younger than 20 who have not started at university.
Each year, participating countries send at most 6 students. The selection process varies between countries, but typically involves several rounds of competition, each progressively more difficult, after which the number of candidates is repeatedly reduced until the final 6 are chosen.
Many countries also run training events for IMO potentials, with the aim of improving performance as well as assisting with team selection.
IMO Selection process by country
Argentina
In Argentina, the Olimpíada Matemática Argentina is organized each year by Fundación Olimpíada Matemática Argentina. All students that took and passed the National Finals (fifth and last round of the competition) exams, usually held in November; and were born before July 1 21 years ago, are allowed to take two new written tests to be selected for IMO, usually in May. From the results of that tests, six titular students and a number of substitutes are selected to represent Argentina at the International Mathematical Olympiad.
Australia
In Australia, selection into the IMO team is determined by the Australian Mathematics Trust and is based on the results from four exams:
The Australian Mathematics Olympiad
The Asian Pacific Mathematics Olympiad
two IMO selection exams
The Australian Mathematics Olympiad (AMO) is held annually in the second week of February. It is composed of two four-hour papers held over two consecutive days. There are four questions in each exam for a total of eight questions. Entry is by invitation only with approximately 100 candidates per year.
A month after the AMO, the Asian Pacific Mathematics Olympiad is held (APMO) and the top 25 from the AMO are invited to sit the exam. It is a four and a half hour exam with five questions.
The top 12 students from the AMO and APMO (along with another 12 or so junior students) are then invited to a ten-day camp held in Sydney in the April school holidays. During this camp,3 times (four-and-a-half hour
selection exams) are held, each with three questions. The top six candidates along with a reserve are then announced as part of the team mostly based on their results in the three exams.
Bangladesh
The selection process is organised by Bangladesh Mathematical Olympiad.
There are four levels of selection in Bangladesh. The students can participate in four academic categories: primary, junior, secondary and higher secondary.
Preliminary Selection: This is the first step of the selection. This is mainly done in district level. The selected contestants then go for the divisional round. Students are given 7 problems and have to solve them with in 1hr 15mins.
Divisional: Currently(2011) the country is divided in 13 regions for divisional Olympiad. The number of
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https://en.wikipedia.org/wiki/Affine%20action
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Let be the Weyl group of a semisimple Lie algebra (associate to fixed choice of a Cartan subalgebra ). Assume that a set of simple roots in is chosen.
The affine action (also called the dot action) of the Weyl group on the space is
where is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.
References
.
Representation theory of Lie algebras
|
https://en.wikipedia.org/wiki/Bayesian%20inference%20in%20phylogeny
|
Bayesian inference of phylogeny combines the information in the prior and in the data likelihood to create the so-called posterior probability of trees, which is the probability that the tree is correct given the data, the prior and the likelihood model. Bayesian inference was introduced into molecular phylogenetics in the 1990s by three independent groups: Bruce Rannala and Ziheng Yang in Berkeley, Bob Mau in Madison, and Shuying Li in University of Iowa, the last two being PhD students at the time. The approach has become very popular since the release of the MrBayes software in 2001, and is now one of the most popular methods in molecular phylogenetics.
Bayesian inference of phylogeny background and bases
Bayesian inference refers to a probabilistic method developed by Reverend Thomas Bayes based on Bayes' theorem. Published posthumously in 1763 it was the first expression of inverse probability and the basis of Bayesian inference. Independently, unaware of Bayes' work, Pierre-Simon Laplace developed Bayes' theorem in 1774.
Bayesian inference or the inverse probability method was the standard approach in statistical thinking until the early 1900s before RA Fisher developed what's now known as the classical/frequentist/Fisherian inference. Computational difficulties and philosophical objections had prevented the widespread adoption of the Bayesian approach until the 1990s, when Markov Chain Monte Carlo (MCMC) algorithms revolutionized Bayesian computation.
The Bayesian approach to phylogenetic reconstruction combines the prior probability of a tree P(A) with the likelihood of the data (B) to produce a posterior probability distribution on trees P(A|B). The posterior probability of a tree will be the probability that the tree is correct, given the prior, the data, and the correctness of the likelihood model.
MCMC methods can be described in three steps: first using a stochastic mechanism a new state for the Markov chain is proposed. Secondly, the probability of this new state to be correct is calculated. Thirdly, a new random variable (0,1) is proposed. If this new value is less than the acceptance probability the new state is accepted and the state of the chain is updated. This process is run thousands or millions of times. The number of times a single tree is visited during the course of the chain is an approximation of its posterior probability. Some of the most common algorithms used in MCMC methods include the Metropolis–Hastings algorithms, the Metropolis-Coupling MCMC (MC³) and the LOCAL algorithm of Larget and Simon.
Metropolis–Hastings algorithm
One of the most common MCMC methods used is the Metropolis–Hastings algorithm, a modified version of the original Metropolis algorithm. It is a widely used method to sample randomly from complicated and multi-dimensional distribution probabilities. The Metropolis algorithm is described in the following steps:
An initial tree, Ti, is randomly selected.
A neighbour tree, Tj, is selected f
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https://en.wikipedia.org/wiki/Casson%20handle
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In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifolds.
Motivation
In the proof of the h-cobordism theorem, the following construction is used.
Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle. If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position" it becomes an embedding. The number 5 appears for the following reason: submanifolds of dimension m and n in general position do not intersect provided the dimension of the manifold containing them has dimension greater than . In particular, a disc (of dimension 2) in general position will have no self intersections inside a manifold of dimension greater than 2+2.
If the manifold is 4 dimensional, this does not work: the problem is that a disc in general position may have double points where two points of the disc have the same image. This is the main reason why the usual proof of the h-cobordism theorem only works for cobordisms whose boundary has dimension at least 5. We can try to get rid of these double points as follows. Draw a line on the disc joining two points with the same image. If the image of this line is the boundary of an embedded disc (called a Whitney disc), then it is easy to remove the double point. However this argument seems to be going round in circles: in order to eliminate a double point of the first disc, we need to construct a second embedded disc, whose construction involves exactly the same problem of eliminating double points.
Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit.
Construction
A Casson handle has a 2-dimensional skeleton, which can be constructed as follows.
Start with a 2-disc .
Identify a finite number of pairs of points in the disc.
For each pair of identified points, choose a path in the disc joining these points, and construct a new disc with boundary this path. (So we add a disc for each pair of identified points.)
Repeat steps 2–3 on each new disc.
We can represent these skeletons by rooted trees such that each point is joined to only a finite number of other points: the tree has a point for each disc, and a line joining points if the corresponding discs intersect in the skeleton.
A Casson handle is constructed by "thickening" the 2-dimension
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https://en.wikipedia.org/wiki/Statistical%20parsing
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Statistical parsing is a group of parsing methods within natural language processing. The methods have in common that they associate grammar rules with a probability. Grammar rules are traditionally viewed in computational linguistics as defining the valid sentences in a language. Within this mindset, the idea of associating each rule with a probability then provides the relative frequency of any given grammar rule and, by deduction, the probability of a complete parse for a sentence. (The probability associated with a grammar rule may be induced, but the application of that grammar rule within a parse tree and the computation of the probability of the parse tree based on its component rules is a form of deduction.) Using this concept, statistical parsers make use of a procedure to search over a space of all candidate parses, and the computation of each candidate's probability, to derive the most probable parse of a sentence. The Viterbi algorithm is one popular method of searching for the most probable parse.
"Search" in this context is an application of search algorithms in artificial intelligence.
As an example, think about the sentence "The can can hold water". A reader would instantly see that there is an object called "the can" and that this object is performing the action 'can' (i.e. is able to); and the thing the object is able to do is "hold"; and the thing the object is able to hold is "water". Using more linguistic terminology, "The can" is a noun phrase composed of a determiner followed by a noun, and "can hold water" is a verb phrase which is itself composed of a verb followed by a verb phrase. But is this the only interpretation of the sentence? Certainly "The can can" is a perfectly valid noun-phrase referring to a type of dance, and "hold water" is also a valid verb-phrase, although the coerced meaning of the combined sentence is non-obvious. This lack of meaning is not seen as a problem by most linguists (for a discussion on this point, see Colorless green ideas sleep furiously) but from a pragmatic point of view it is desirable to obtain the first interpretation rather than the second and statistical parsers achieve this by ranking the interpretations based on their probability.
(In this example various assumptions about the grammar have been made, such as a simple left-to-right derivation rather than head-driven, its use of noun-phrases rather than the currently fashionable determiner-phrases, and no type-check preventing a concrete noun being combined with an abstract verb phrase. None of these assumptions affect the thesis of the argument and a comparable argument can be made using any other grammatical formalism.)
There are a number of methods that statistical parsing algorithms frequently use. While few algorithms will use all of these they give a good overview of the general field. Most statistical parsing algorithms are based on a modified form of chart parsing. The modifications are necessary to suppo
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https://en.wikipedia.org/wiki/Brocard%20points
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In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.
Definition
In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in counterclockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that
Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle. This angle has the property that
where are the vertex angles respectively.
There is also a second Brocard point, Q, in triangle ABC such that line segments AQ, BQ, and CQ form equal angles with sides b, c, and a respectively. In other words, the equations apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle is the same as
The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken. So for example, the first Brocard point of triangle ABC is the same as the second Brocard point of triangle ACB.
The two Brocard points of a triangle ABC are isogonal conjugates of each other.
Construction
The most elegant construction of the Brocard points goes as follows. In the following example the first Brocard point is presented, but the construction for the second Brocard point is very similar.
As in the diagram above, form a circle through points A and B, tangent to edge BC of the triangle (the center of this circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle through points B and C, tangent to edge AC, and a circle through points A and C, tangent to edge AB. These three circles have a common point, the first Brocard point of triangle ABC. See also Tangent lines to circles.
The three circles just constructed are also designated as epicycles of triangle ABC. The second Brocard point is constructed in similar fashion.
Trilinears and barycentrics of the first two Brocard points
Homogeneous trilinear coordinates for the first and second Brocard points are and respectively. Thus their barycentric coordinates are respectively and
The segment between the first two Brocard points
The Brocard points are an example of a bicentric pair of points, but they are not triangle centers because neither Brocard point is invariant under similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other. However, the unordered pair formed by both points is invariant under similarities. The midpoint of the two Brocard points, called the Brocard midpoint, has trilinear coordinates
and is a triangle center; it is center X(39) in the Encyclopedia of Triangle Centers. The third Brocard p
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https://en.wikipedia.org/wiki/Schreier%20conjecture
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In finite group theory, the Schreier conjecture asserts that the outer automorphism group of every finite simple group is solvable. It was proposed by Otto Schreier in 1926, and is now known to be true as a result of the classification of finite simple groups, but no simpler proof is known.
References
.
Theorems about finite groups
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Rokhlin%27s%20theorem
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In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group , is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.
Examples
The intersection form on M
is unimodular on by Poincaré duality, and the vanishing of implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature.
A K3 surface is compact, 4 dimensional, and vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
A complex surface in of degree is spin if and only if is even. It has signature , which can be seen from Friedrich Hirzebruch's signature theorem. The case gives back the last example of a K3 surface.
Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing and intersection form of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure. This manifold shows that Rokhlin's theorem fails for the set of merely topological (rather than smooth) manifolds.
If the manifold M is simply connected (or more generally if the first homology group has no 2-torsion), then the vanishing of is equivalent to the intersection form being even. This is not true in general: an Enriques surface is a compact smooth 4 manifold and has even intersection form II1,9 of signature −8 (not divisible by 16), but the class does not vanish and is represented by a torsion element in the second cohomology group.
Proofs
Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres is cyclic of order 24; this is Rokhlin's original approach.
It can also be deduced from the Atiyah–Singer index theorem. See  genus and Rochlin's theorem.
gives a geometric proof.
The Rokhlin invariant
Since Rokhlin's theorem states that the signature of a spin smooth manifold is divisible by 16, the definition of the Rokhlin invariant is deduced as follows:
For 3-manifold and a spin structure on , the Rokhlin invariant in is defined to be the signature of any smooth compact spin 4-manifold with spin boundary .
If N is a spin 3-manifold then it bounds a spin 4-manifold M. The signature of M is divisible by 8, and an easy application of Rokhlin's theorem shows that its value mod 16 depends only on N and not on the choice of M. Homology 3-spheres have a unique spin structure so we can define the Rokhlin invariant of a homology 3-sphere to be the element of , where M any spin 4-manifold bounding the homology sphere.
For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form , so its Rokhlin invariant is 1. This result has some elementary conse
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https://en.wikipedia.org/wiki/Dodecahedral%20conjecture
|
The dodecahedral conjecture in geometry is intimately related to sphere packing.
László Fejes Tóth, a 20th-century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit spheres. He conjectured in 1943 that the minimal volume of any cell in the resulting Voronoi decomposition was at least as large as the volume of a regular dodecahedron circumscribed to a unit sphere.
Thomas Callister Hales and Sean McLaughlin proved the conjecture in 1998, following the same strategy that led Hales to his proof of the Kepler conjecture. The proofs rely on extensive computations. McLaughlin was awarded the 1999 Morgan Prize for his contribution to this proof.
References
Theorems in geometry
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Half-logistic%20distribution
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In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for
where Y is a logistic random variable, X is a half-logistic random variable.
Specification
Cumulative distribution function
The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F(k) is the cdf for the logistic distribution, then G(k) = 2F(k) − 1 is the cdf of a half-logistic distribution. Specifically,
Probability density function
Similarly, the probability density function (pdf) of the half-logistic distribution is g(k) = 2f(k) if f(k) is the pdf of the logistic distribution. Explicitly,
References
Continuous distributions
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https://en.wikipedia.org/wiki/Arthur%20P.%20Dempster
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Arthur Pentland Dempster (born 1929) is a Professor Emeritus in the Harvard University Department of Statistics. He was one of four faculty when the department was founded in 1957.
Biography
Dempster received his B.A. in mathematics and physics (1952) and M.A. in mathematics (1953), both from the University of Toronto. He obtained his Ph.D. in mathematical statistics from Princeton University in 1956. His thesis, titled The two-sample multivariate problem in the degenerate case, was written under the supervision of John Tukey.
Academic works
Among his contributions to statistics are the Dempster–Shafer theory and the expectation-maximization (EM) algorithm.
Selected publications
Honors and awards
Dempster was a Putnam Fellow in 1951. He was elected as an American Statistical Association Fellow in 1964, an Institute of Mathematical Statistics Fellow in 1963, and an American Academy of Arts and Sciences Fellow in 1997.
References
External links
Homepage on Harvard University
American statisticians
20th-century American mathematicians
21st-century American mathematicians
Princeton University alumni
Harvard University faculty
Putnam Fellows
Fellows of the American Statistical Association
Living people
1929 births
University of Toronto alumni
Mathematical statisticians
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https://en.wikipedia.org/wiki/Hanna%20Neumann
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Johanna (Hanna) Neumann (née von Caemmerer; 12 February 1914 – 14 November 1971) was a German-born mathematician who worked on group theory.
Biography
Neumann was born on 12 February 1914 in Lankwitz, Steglitz-Zehlendorf (today a district of Berlin), Germany. She was the youngest of three children of Hermann and Katharina von Caemmerer. As a result of her father's death in the first days of the First World War, the family income was small, and from the age of thirteen she was coaching school children.
After two years at a private school she entered the Auguste-Viktoria-Schule, a girls' grammar school (Realgymnasium), in 1922. She graduated in early 1932 and then entered the University of Berlin. The lecture courses in mathematics that she took in her first year were: Introduction to Higher Mathematics given by Georg Feigl; Analytical Geometry and Projective Geometry both given by Ludwig Bieberbach, Differential and Integral Calculus given by Erhard Schmidt, and the Theory of Numbers given by Friedrich Schur. She also took formal courses in physics, and attended lectures in psychology, literature and law.
As a result of her first year work, Hanna was awarded three-quarters' remission of fees and a position as a part-time assistant in the Mathematical Institute's library.
A friendship between Hanna and Bernhard Neumann began in January 1933. In March 1933, the Nazis came to power and in August 1933, Bernhard, who was Jewish, moved to Cambridge, England. She visited Bernhard in London at Easter 1934 and they became secretly engaged. After this she returned to Germany to continue her studies.
In her second year Neumann was part of a group of students who tried to prevent Nazi disruption of Jewish academics' lectures by ensuring that only genuine students attended. She lost her job in the Mathematical Institute, presumably as a result of such activities. However she had by then been awarded, and continued to earn for the rest of her course, full remission of fees.
In the remainder of her undergraduate degree, she studied mathematics, physics and philosophy. She completed her studies in 1936 with distinctions in the Staatsexamen in mathematics and physics. She began studying for her Ph.D. at the University of Göttingen in 1937, under the supervision of Helmut Hasse.
During this time Bernhard and Hanna corresponded anonymously through friends, and were only able to meet once, in Denmark in 1936 when Bernhard was travelling to the International Congress of Mathematicians in Oslo. In July 1938, Hanna moved to England. She married Bernhard in December 1938 in Cardiff. They went on to have five children. The Neumanns moved to Oxford in 1940.
Neumann completed her D Phil. in group theory at the Society of Oxford Home-Students in 1944 under Olga Taussky-Todd. Her thesis was entitled 'Sub-group Structure of Free Products of Groups with an Amalgamated Subgroup'. The University of Oxford later awarded her a D.Sc. for her publications.
Fol
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https://en.wikipedia.org/wiki/Generalized%20inverse
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In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix .
A matrix is a generalized inverse of a matrix if A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.
Motivation
Consider the linear system
where is an matrix and the column space of . If is nonsingular (which implies ) then will be the solution of the system. Note that, if is nonsingular, then
Now suppose is rectangular (), or square and singular. Then we need a right candidate of order such that for all
That is, is a solution of the linear system .
Equivalently, we need a matrix of order such that
Hence we can define the generalized inverse as follows: Given an matrix , an matrix is said to be a generalized inverse of if The matrix has been termed a regular inverse of by some authors.
Types
Important types of generalized inverse include:
One-sided inverse (right inverse or left inverse)
Right inverse: If the matrix has dimensions and , then there exists an matrix called the right inverse of such that , where is the identity matrix.
Left inverse: If the matrix has dimensions and , then there exists an matrix called the left inverse of such that , where is the identity matrix.
Bott–Duffin inverse
Drazin inverse
Moore–Penrose inverse
Some generalized inverses are defined and classified based on the Penrose conditions:
where denotes conjugate transpose. If satisfies the first condition, then it is a generalized inverse of . If it satisfies the first two conditions, then it is a reflexive generalized inverse of . If it satisfies all four conditions, then it is the pseudoinverse of , which is denoted by and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an -inverse of as an inverse that satisfies the subset of the Penrose conditions listed above. Relations, such as , can be established between these different classes of -inverses.
When is non-singular, any generalized inverse and is therefore unique. For a singular , some generalised inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined.
Examples
Reflexive generalized inverse
Let
Since , is singular and has no regular inverse. However, and satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence, is
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https://en.wikipedia.org/wiki/David%20Lane
|
David Lane may refer to:
Academics
David J. Lane (astronomer) (born 1963), Canadian astronomer at Saint Mary's University
David A. Lane (born 1945), American professor of statistics and economics at the University of Modena and Reggio Emilia
David C. Lane (born 1956), American professor of philosophy
David Lane (oncologist) (born 1952), British researcher and discoverer of the p53 gene
Entertainment
David Lane (director) (born 1940), British television and film director
David Lane (musician) (born 1981), Australian musician with You Am I
David Ian (David Ian Lane, born 1961), English stage actor and producer
Politics
David Campbell Lane, state legislator in Florida
David J. Lane (ambassador) (born 1960), U.S. ambassador to the United Nations Agencies for Food and Agriculture
David Lane (British politician) (1922–1998), British Conservative Party politician
David Lane (activist) (born c. 1955), American social conservative Christian activist
David Lane (Massachusetts politician) (1927–2020), Massachusetts state representative
Other
David Lane (cricketer) (born 1965), Montserratian cricketer
David Lane (white supremacist) (1938–2007), American white supremacist
David Lane tram stop, a tram stop on the Nottingham Express Transit
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