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https://en.wikipedia.org/wiki/Fully%20normalized%20subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjective.
In symbols, a subgroup is fully normalized in if, given an automorphism of , there is a such that the map , when restricted to is equal to .
Some facts:
Every group can be embedded as a normal and fully normalized subgroup of a bigger group. A natural construction for this is the holomorph, which is its semidirect product with its automorphism group.
A complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner.
Every fully normalized subgroup has the automorphism extension property.
Subgroup properties
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https://en.wikipedia.org/wiki/CEP%20subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a congruence of the whole group. Equivalently, every normal subgroup of the subgroup arises as the intersection with the subgroup of a normal subgroup of the whole group.
In symbols, a subgroup is a CEP subgroup in a group if every normal subgroup of can be realized as where is normal in .
The following facts are known about CEP subgroups:
Every retract has the CEP.
Every transitively normal subgroup has the CEP.
References
.
.
Subgroup properties
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https://en.wikipedia.org/wiki/William%20S.%20Massey
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William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology ().
Life
William Massey was born in Granville, Illinois, in 1920, the son of Robert and Alma Massey, and grew up in Peoria. He was an undergraduate student at the University of Chicago. After serving as a meteorologist aboard aircraft carriers in the United States Navy for 4 years during World War II, he received a Ph.D. degree from Princeton University in 1949. His dissertation, entitled Classification of mappings of an -dimensional space into an n-sphere, was written under the direction of Norman Steenrod. He spent two additional years at Princeton as a post-doctoral research assistant. He then taught for ten years on the faculty of Brown University. In 1958 he was elected to the American Academy of Arts and Sciences. From 1960 till his retirement he was a professor at Yale University. He died on June 17, 2017, in Hamden, Connecticut. He had 23 PhD students, including Donald Kahn, Larry Smith, and Robert Greenblatt.
Selected works
See also
Blakers–Massey theorem
Exact couple
Massey product
External links
Address at Yale
References
1920 births
2017 deaths
20th-century American mathematicians
21st-century American mathematicians
Topologists
Brown University faculty
Yale University faculty
University of Chicago alumni
Princeton University alumni
United States Navy personnel of World War II
People from Granville, Illinois
Mathematicians from Illinois
Fellows of the American Academy of Arts and Sciences
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https://en.wikipedia.org/wiki/Retract%20%28group%20theory%29
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In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is the identity on the subgroup. In symbols, is a retract of if and only if there is an endomorphism such that for all and for all .
The endomorphism is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism or a retraction.
The following is known about retracts:
A subgroup is a retract if and only if it has a normal complement. The normal complement, specifically, is the kernel of the retraction.
Every direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
Every retract has the congruence extension property.
Every regular factor, and in particular, every free factor, is a retract.
See also
Retraction (category theory)
Retraction (topology)
References
Group theory
Subgroup properties
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https://en.wikipedia.org/wiki/Norm%20%28group%29
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In mathematics, in the field of group theory, the norm of a group is the intersection of the normalizers of all its subgroups. This is also termed the Baer norm, after Reinhold Baer.
The following facts are true for the Baer norm:
It is a characteristic subgroup.
It contains the center of the group.
It is contained inside the second term of the upper central series.
It is a Dedekind group, so is either abelian or has a direct factor isomorphic to the quaternion group.
If it contains an element of infinite order, then it is equal to the center of the group.
References
Group theory
Functional subgroups
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https://en.wikipedia.org/wiki/Demography%20of%20London
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The demography of London is analysed by the Office for National Statistics and data is produced for each of the Greater London wards, the City of London and the 32 London boroughs, the Inner London and Outer London statistical sub-regions, each of the Parliamentary constituencies in London, and for all of Greater London as a whole. Additionally, data is produced for the Greater London Urban Area. Statistical information is produced about the size and geographical breakdown of the population, the number of people entering and leaving country and the number of people in each demographic subgroup. The total population of London as of 2021 is 8,799,800.
History
Creation of Greater London - 1965
Through the London Government Act of 1963, the Greater London region was established officially in 1965.
Migration boom - 1997 to today
From 1997 onwards, London has experienced a drastic change in the composition of the city's population, which has off set the decline of the population which had been occurring. In 1991, 21.7% of the city was foreign born but by 2011 this had risen to 36.7%.
In 2011, a historic tipping point occurred with the release of the 2011 census indicating that the White British population, which had before been the majority, was now no longer a majority of the city's population, although it remained by far the largest single ethnic group.
Population
The historical population for the current area of Greater London, divided into the statistical areas of Inner and Outer London is as follows:
Age
Fertility
In 2021, a total of 110,961 live births occurred within the city. The fertility rate of London in 2021 was 1.52, which is below replacement.
Population density
The population density of London was 5,727 per km2 in 2011.
Urban and metropolitan area
At the 2001 census, the population of the Greater London Urban Area was 8,278,251. This area does not include some outliers within Greater London, but does extend into the adjacent South East England and East of England regions. In 2004 the London Plan of the Mayor of London defined a metropolitan region with a population of 18 million. Eurostat has developed a harmonising standard for comparing metropolitan areas in the European Union and the population of the London Larger Urban Zone is 11,917,000; it occupies an area of . Another definition gives the population of the metropolitan area as 13,709,000.
Ethnicity
For the overwhelming majority of London's history, the population of the city was ethnically homogenous with the population being of White British ethnic origin, with small clusters of minority groups such as Jewish people, most notably in areas of the East End. From 1948 onwards and especially since the Blair government in the late 1990s and 2000s, the population has diversified in international terms at an increased rate. In 2011, it was reported for the first time that White British people had become a minority within the city, establishing it was a majority-minor
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https://en.wikipedia.org/wiki/AP%20Statistics
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Advanced Placement (AP) Statistics (also known as AP Stats) is a college-level high school statistics course offered in the United States through the College Board's Advanced Placement program. This course is equivalent to a one semester, non-calculus-based introductory college statistics course and is normally offered to sophomores, juniors and seniors in high school.
One of the College Board's more recent additions, the AP Statistics exam was first administered in May 1996 to supplement the AP program's math offerings, which had previously consisted of only AP Calculus AB and BC. In the United States, enrollment in AP Statistics classes has increased at a higher rate than in any other AP class.
Students may receive college credit or upper-level college course placement upon passing the three-hour exam ordinarily administered in May. The exam consists of a multiple-choice section and a free-response section that are both 90 minutes long. Each section is weighted equally in determining the students' composite scores.
History
The Advanced Placement program has offered students the opportunity to pursue college-level courses while in high school. Along with the Educational Testing Service, the College Board administered the first AP Statistics exam in May 1997. The course was first taught to students in the 1996-1997 academic year. Prior to that, the only mathematics courses offered in the AP program included AP Calculus AB and BC. Students who didn't have a strong background in college-level math, however, found the AP Calculus program inaccessible and sometimes declined to take a math course in their senior year. Since the number of students required to take statistics in college is almost as large as the number of students required to take calculus, the College Board decided to add an introductory statistics course to the AP program. Since the prerequisites for such a program doesn't require mathematical concepts beyond those typically taught in a second-year algebra course, the AP program's math offerings became accessible to a much wider audience of high school students. The AP Statistics program addressed a practical need as well since the number of students enrolling in majors that use statistics has grown. A total of 7,667 students took the exam during the first administration, the highest number of students to take an AP exam in its first year. Since then, the number of students taking the exam rapidly grew to 98,033 in 2007, making it one of the 10 most popular AP exams.
Course
If the course is provided by their school, students normally take AP Statistics in their junior or senior year and may decide to take it concurrently with a pre-calculus course. This offering is intended to imitate a one-semester, non-calculus based college statistics course, but high schools can decide to offer the course over one semester, two trimesters, or a full academic year.
The six-member AP Statistics Test Development Committee is responsible for deve
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https://en.wikipedia.org/wiki/Mathematics%20%28disambiguation%29
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Mathematics is a field of knowledge.
Mathematics may also refer to:
Music
Mathematics (album), a 1985 album by Melissa Manchester
"Mathematics" (Cherry Ghost song), a song by Cherry Ghost
"Mathematics" (Mos Def song), a song by Mos Def
Mathematics, an EP by The Servant
"Mathematics", a song by bbno$
"Mathematics", a song by Little Boots from Hands
"Mathematics", a song by Macintosh Plus from Floral Shoppe
Other uses
Mathematics (producer), a hip hop producer
Mathematics (UIL), an American student mathematics competition
Microsoft Mathematics, an educational program designed for Microsoft Windows
Mathematics Magazine, a publication of the Mathematical Association of America
See also
Math (disambiguation)
Mathematica (disambiguation)
:Category:Mathematics
Portal:Mathematics
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https://en.wikipedia.org/wiki/Proceedings%20of%20the%20American%20Mathematical%20Society
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Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages.
According to the Journal Citation Reports, the journal has a 2018 impact factor of 0.813.
Scope
Proceedings of the American Mathematical Society publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics.
Abstracting and indexing
This journal is indexed in the following databases:
Mathematical Reviews
Zentralblatt MATH
Science Citation Index
Science Citation Index Expanded
ISI Alerting Services
CompuMath Citation Index
Current Contents / Physical, Chemical & Earth Sciences.
Other journals from the American Mathematical Society
Bulletin of the American Mathematical Society
Memoirs of the American Mathematical Society
Notices of the American Mathematical Society
Journal of the American Mathematical Society
Transactions of the American Mathematical Society
References
External links
Proceedings of the American Mathematical Society on JSTOR
American Mathematical Society academic journals
Mathematics journals
Monthly journals
Academic journals established in 1950
1950 establishments in the United States
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https://en.wikipedia.org/wiki/American%20Journal%20of%20Mathematics
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The American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.
History
The American Journal of Mathematics is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898.
Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen Smale, Jun-Ichi Igusa, and Joseph A. Shalika.
Fields medalist Cédric Villani has speculated that "the most famous article in its long history" may be a 1958 paper by John Nash, "Continuity of solutions of parabolic and elliptic equations".
Scope and impact factor
The American Journal of Mathematics is a general-interest (i.e., non-specialized) mathematics journal covering all the major areas of contemporary mathematics. According to the Journal Citation Reports, its 2009 impact factor is 1.337, ranking it 22nd out of 255 journals in the category "Mathematics".
Editors
As of June, 2012, the editors are Christopher D. Sogge, editor-in-chief (Johns Hopkins University), William Minicozzi II (Massachusetts Institute of Technology), Freydoon Shahidi (Purdue University), and Vyacheslav Shokurov (The Johns Hopkins University).
References
External links
Mathematics journals
Publications established in 1878
Bimonthly journals
English-language journals
Johns Hopkins University Press academic journals
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https://en.wikipedia.org/wiki/Conjugacy-closed%20subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be conjugacy-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgroup.
An alternative characterization of conjugacy-closed normal subgroups is that all class automorphisms of the whole group restrict to class automorphisms of the subgroup.
The following facts are true regarding conjugacy-closed subgroups:
Every central factor (a subgroup that may occur as a factor in some central product) is a conjugacy-closed subgroup.
Every conjugacy-closed normal subgroup is a transitively normal subgroup.
The property of being conjugacy-closed is transitive, that is, every conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed.
The property of being conjugacy-closed is sometimes also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of the base field is a conjugacy-closed subgroup of the general linear group over the extension field. This result is typically referred to as a stability theorem.
A subgroup is said to be strongly conjugacy-closed if all intermediate subgroups are also conjugacy-closed.
Examples and Non-Examples
Examples
Every subgroup of a commutative group is conjugacy closed.
Non-Examples
External links
Conjugacy-closed subgroup at the Group Properties Wiki
Central factor at the Group Properties Wiki
Subgroup properties
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https://en.wikipedia.org/wiki/Weakly%20normal%20subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be weakly normal if whenever , we have .
Every pronormal subgroup is weakly normal.
References
Subgroup properties
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https://en.wikipedia.org/wiki/Degen%27s%20eight-square%20identity
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In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.
Namely:
First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845). The latter two derived it while working on an extension of quaternions called octonions. In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms: . Similar statements are true for quaternions (Euler's four-square identity), complex numbers (the Brahmagupta–Fibonacci two-square identity) and real numbers. In 1898 Adolf Hurwitz proved that there is no similar bilinear identity for 16 squares (sedenions) or any other number of squares except for 1,2,4, and 8. However, in the 1960s, H. Zassenhaus, W. Eichhorn, and A. Pfister (independently) showed there can be a non-bilinear identity for 16 squares.
Note that each quadrant reduces to a version of Euler's four-square identity:
and similarly for the other three quadrants.
Comment: The proof of the eight-square identity is by algebraic evaluation. The eight-square identity can be written in the form of a product of two inner products of 8-dimensional vectors, yielding again an inner product of 8-dimensional vectors: . This defines the octonion multiplication rule , which reflects Degen's 8-square identity and the mathematics of octonions.
By Pfister's theorem, a different sort of eight-square identity can be given where the , introduced below, are non-bilinear and merely rational functions of the . Thus,
where,
and,
with,
Incidentally, the obey the identity,
See also
Pfister's sixteen-square identity
Cayley–Dickson construction
Hypercomplex number
Latin square
External links
Degen's eight-square identity on MathWorld
The Degen–Graves–Cayley Eight-Square Identity
Pfister's 16-Square Identity
Analytic number theory
Mathematical identities
Squares in number theory
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https://en.wikipedia.org/wiki/Fermat%20%28computer%20algebra%20system%29
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Fermat (named after Pierre de Fermat) is a program developed by Prof. Robert H. Lewis of Fordham University. It is a computer algebra system, in which items being computed can be integers (of arbitrary size), rational numbers, real numbers, complex numbers, modular numbers, finite field elements, multivariable polynomials, rational functions, or polynomials modulo other polynomials. The main areas of application are multivariate rational function arithmetic and matrix algebra over rings of multivariate polynomials or rational functions. Fermat does not do simplification of transcendental functions or symbolic integration.
A session with Fermat usually starts by choosing rational or modular "mode" to establish the ground field (or ground ring) as or . On top of this may be attached any number of symbolic variables thereby creating the polynomial ring and its quotient field. Further, some polynomials involving some of the can be chosen to mod out with, creating the quotient ring Finally, it is possible to allow Laurent polynomials, those with negative as well as positive exponents. Once the computational ring is established in this way, all computations are of elements of this ring. The computational ring can be changed later in the session.
The polynomial gcd procedures, which call each other in a highly recursive manner, are about 7000 lines of code.
Fermat has extensive built-in primitives for array and matrix manipulations, such as submatrix, sparse matrix, determinant, normalize, column reduce, row echelon, Smith normal form, and matrix inverse. It is consistently faster than some well known computer algebra systems, especially in multivariate polynomial gcd. It is also space efficient.
The basic data item in Fermat is a multivariate rational function or quolynomial. The numerator and denominator are polynomials with no common factor. Polynomials are implemented recursively as general linked lists, unlike some systems that implement polynomials as lists of monomials. To implement (most) finite fields, the user finds an irreducible monic polynomial in a symbolic variable, say and commands Fermat to mod out by it. This may be continued recursively, etc. Low level data structures are set up to facilitate arithmetic and gcd over this newly created ground field. Two special fields, and are more efficiently implemented at the bit level.
History
With Windows 10, and thanks to Bogdan Radu, it is now possible (May 2021) to run Fermat Linux natively on Windows. See the main web page http://home.bway.net/lewis
Fermat was last updated on 20 May 2020 (Mac and Linux; latest Windows version: 1 November 2011).
In an earlier version, called FFermat (Float Fermat), the basic number type is floating point numbers of 18 digits. That version allows for numerical computing techniques, has extensive graphics capabilities, no sophisticated polynomial gcd algorithms, and is available only for Mac OS 9.
Fermat was originally written in Pascal for
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https://en.wikipedia.org/wiki/J.%20Hyam%20Rubinstein
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Joachim Hyam Rubinstein FAA (born 7 March 1948, in Melbourne) an Australian top mathematician specialising in low-dimensional topology; he is currently serving as an honorary professor in the Department of Mathematics and Statistics at the University of Melbourne, having retired in 2019.
He has spoken and written widely on the state of the mathematical sciences in Australia, with particular focus on the impacts of reduced Government spending for university mathematics departments.
Education
In 1965, Rubinstein matriculated (i.e. graduated) from Melbourne High School in Melbourne, Australia winning the maximum of four exhibitions. In 1969, he graduated from Monash University in Melbourne, with a B.Sc.(Honours) degree in mathematics.
In 1974, Rubinstein received his Ph.D. from the University of California, Berkeley under the advisership of John Stallings. His dissertation was on the topic of Isotopies of Incompressible Surfaces in Three Dimensional Manifolds.
Research interests
His major contributions include results involving almost normal Heegaard splittings and the closely related joint work with Jon T. Pitts relating strongly irreducible Heegaard splittings to minimal surfaces, joint work with William Jaco on special triangulations of 3-manifolds (namely 0-efficient and 1-efficient triangulations), and joint work with Martin Scharlemann on the Rubinstein–Scharlemann graphic. He is a key figure in the algorithmic theory of 3-manifolds, and one of the initial developers of the Regina program, which implements his 3-sphere recognition algorithm.
His research interests also include: shortest networks applied to underground mine design, machine learning, learning theory, financial mathematics, and stock market trading systems.
Honours
Past President of the Australian Mathematical Society.
Chair of the Australian Committee for the Mathematical Sciences.
Elected Fellow of the Australian Academy of Science in 2003.
Recipient of the Hannan Medal in 2004.
Recipient of the George Szekeres Medal in 2008.
Fellow of the American Mathematical Society, 2012.
From July 11 to 22, 2011, a workshop and conference in his honour, jointly titled “Hyamfest: Geometry & Topology Down Under”, were held at the University of Melbourne.
References
External links
Interview
LinkedIn page
1948 births
20th-century Australian mathematicians
21st-century Australian mathematicians
Topologists
University of California, Berkeley alumni
Academic staff of the University of Melbourne
Mathematicians from Melbourne
Living people
People educated at Melbourne High School
Fellows of the Australian Academy of Science
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Genocchi%20number
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In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
The first few Genocchi numbers are 0, −1, −1, 0, 1, 0, −3, 0, 17 , see .
Properties
The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G2n+1 = 0 for n ≥ 1 and (−1)nG2n is an odd positive integer.
Genocchi numbers Gn are related to Bernoulli numbers Bn by the formula
Combinatorial interpretations
The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
They enumerate the following objects:
Permutations in S2n−1 with descents after the even numbers and ascents after the odd numbers.
Permutations π in S2n−2 with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.
Pairs (a1,…,an−1) and (b1,…,bn−1) such that ai and bi are between 1 and i and every k between 1 and n−1 occurs at least once among the ai's and bi's.
Reverse alternating permutations a1 < a2 > a3 < a4 >…>a2n−1 of [2n−1] whose inversion table has only even entries.
See also
Euler number
References
Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press.
Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminaire de Théorie des Nombres de Bordeaux, Volume 11 (1981-1982)
Serkan Araci, Mehmet Acikgoz, Erdoğan Şen, Some New Identities of Genocchi Numbers and Polynomials
Eponymous numbers in mathematics
Integer sequences
Factorial and binomial topics
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https://en.wikipedia.org/wiki/Moment%20matrix
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In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.)
Moment matrices play an important role in polynomial fitting, polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares) and econometrics.
Application in regression
A multiple linear regression model can be written as
where is the explained variable, are the explanatory variables, is the error, and are unknown coefficients to be estimated. Given observations , we have a system of linear equations that can be expressed in matrix notation.
or
where and are each a vector of dimension , is the design matrix of order , and is a vector of dimension . Under the Gauss–Markov assumptions, the best linear unbiased estimator of is the linear least squares estimator , involving the two moment matrices and defined as
and
where is a square normal matrix of dimension , and is a vector of dimension .
See also
Design matrix
Gramian matrix
Projection matrix
References
External links
Matrices
Least squares
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https://en.wikipedia.org/wiki/Ideal%20point
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In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line l and a point P not on l, right- and left-limiting parallels to l through P converge to l at ideal points.
Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself.
The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry.
For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model.
While the real line forms the Cayley absolute of the Poincaré half-plane model .
Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.
Properties
The hyperbolic distance between an ideal point and any other point or ideal point is infinite.
The centres of horocycles and horoballs are ideal points; two horocycles are concentric when they have the same centre.
Polygons with ideal vertices
Ideal triangles
if all vertices of a triangle are ideal points the triangle is an ideal triangle.
Some properties of ideal triangles include:
All ideal triangles are congruent.
The interior angles of an ideal triangle are all zero.
Any ideal triangle has an infinite perimeter.
Any ideal triangle has area where K is the (always negative) curvature of the plane.
Ideal quadrilaterals
if all vertices of a quadrilateral are ideal points, the quadrilateral is an ideal quadrilateral.
While all ideal triangles are congruent, not all quadrilaterals are; the diagonals can make different angles with each other resulting in noncongruent quadrilaterals. Having said this:
The interior angles of an ideal quadrilateral are all zero.
Any ideal quadrilateral has an infinite perimeter.
Any ideal (convex non intersecting) quadrilateral has area where K is the (always negative) curvature of the plane.
Ideal square
The ideal quadrilateral where the two diagonals are perpendicular to each other form an ideal square.
It was used by Ferdinand Karl Schweikart in his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility of hyperbolic geometry.
Ideal n-gons
An ideal n-gon can be subdivided into ideal triangles, with area times the area of an ideal triangle.
Representations in models of hyperbolic geometry
In the Klein disk model and the Poincaré disk model of the hyperbolic plane the ideal points are on the unit circle (hyperbolic plane) or unit sphere (higher dimensions) which is the unreachable boundary of the hyperbolic plane.
When projecting the same hyperbolic line to the Klein disk model and the Poincaré disk model both lines go through the same two ideal points (the ideal points in both models are on the same spot).
Klein disk model
Given two disti
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https://en.wikipedia.org/wiki/Cantor%20cube
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In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology).
If A is a countably infinite set, the corresponding Cantor cube is a Cantor space. Cantor cubes are special among compact groups because every compact group is a continuous image of one, although usually not a homomorphic image. (The literature can be unclear, so for safety, assume all spaces are Hausdorff.)
Topologically, any Cantor cube is:
homogeneous;
compact;
zero-dimensional;
AE(0), an absolute extensor for compact zero-dimensional spaces. (Every map from a closed subset of such a space into a Cantor cube extends to the whole space.)
By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties is homeomorphic to a Cantor cube.
In fact, every AE(0) space is the continuous image of a Cantor cube, and with some effort one can prove that every compact group is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube, and every compact group is a continuous image of a Cantor cube.
References
Topological groups
Georg Cantor
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https://en.wikipedia.org/wiki/Tijdeman%27s%20theorem
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In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation
for exponents n and m greater than one, is finite.
History
The theorem was proven by Dutch number theorist Robert Tijdeman in 1976, making use of Baker's method in transcendental number theory to give an effective upper bound for x,y,m,n. Michel Langevin computed a value of exp exp exp exp 730 for the bound.
Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu. Mihăilescu's theorem states that there is only one member to the set of consecutive power pairs, namely 9=8+1.
Generalized Tijdeman problem
That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of 1 by any other difference k and ask for the number of solutions
of
with n and m greater than one we have an unsolved problem, called the generalized Tijdeman problem. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture, stating that the equation only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture.
References
Theorems in number theory
Diophantine equations
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https://en.wikipedia.org/wiki/Israel%20Nathan%20Herstein
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Israel Nathan Herstein (March 28, 1923 – February 9, 1988) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and over a dozen books.
Education and career
Herstein was born in Lublin, Poland, in 1923. His family emigrated to Canada in 1926, and he grew up in a harsh and underprivileged environment where, according to him, "you either became a gangster or a college professor." During his school years he played football, ice hockey, golf, tennis, and pool. He also worked as a steeplejack and as a barker at a fair. He received his B.S. degree from the University of Manitoba and his M.A. from the University of Toronto. He received his Ph.D from Indiana University in 1948. His advisor was Max Zorn. He held positions at the University of Kansas, Ohio State University, University of Pennsylvania, and Cornell University before permanently settling at the University of Chicago in 1962. He was a Guggenheim Fellow for the academic year 1960–1961.
He is known for his lucid style of writing, as exemplified by his Topics in Algebra, an undergraduate introduction to abstract algebra that was first published in 1964, with a second edition in 1975. A more advanced text is his Noncommutative Rings in the Carus Mathematical Monographs series. His primary interest was in noncommutative ring theory, but he also wrote papers on finite groups, linear algebra, and mathematical economics.
He had 30 Ph.D. students, traveled and lectured widely, and spoke Italian, Hebrew, Polish, and Portuguese. He died from cancer in Chicago, Illinois, in 1988. His doctoral students include Miriam Cohen, Wallace S. Martindale, Susan Montgomery, Karen Parshall and Claudio Procesi.
Selected publications
Notes
References
External links
1923 births
1988 deaths
20th-century American mathematicians
Algebraists
Indiana University alumni
University of Chicago faculty
Ohio State University faculty
Polish emigrants to Canada
Canadian emigrants to the United States
University of Kansas faculty
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
Cornell University faculty
University of Manitoba alumni
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https://en.wikipedia.org/wiki/%27t%20Hooft%20symbol
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The t Hooft symbol is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft. It is used in the construction of the BPST instanton.
Definition
is the 't Hooft symbol:
Where and are instances of the Kronecker delta, and is the Levi-Civita symbol.
In other words, they are defined by
()
where the latter are the anti-self-dual 't Hooft symbols.
Matrix Form
In matrix form, the 't Hooft symbols are
and their anti-self-duals are the following:
Properties
They satisfy the self-duality and the anti-self-duality properties:
Some other properties are
The same holds for except for
and
Obviously due to different
duality properties.
Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.
See also
Instanton
't Hooft anomaly
't Hooft–Polyakov monopole
't Hooft loop
References
Gauge theories
Mathematical symbols
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https://en.wikipedia.org/wiki/Orthogonal%20coordinates
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In mathematics, orthogonal coordinates are defined as a set of coordinates in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, not exponents). A coordinate surface for a particular coordinate is the curve, surface, or hypersurface on which is a constant. For example, the three-dimensional Cartesian coordinates is an orthogonal coordinate system, since its coordinate surfaces constant, constant, and constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
Motivation
While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum mechanics, fluid flow, electrodynamics, plasma physics and the diffusion of chemical species or heat.
The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem. For example, the pressure wave due to an explosion far from the ground (or other barriers) depends on 3D space in Cartesian coordinates, however the pressure predominantly moves away from the center, so that in spherical coordinates the problem becomes very nearly one-dimensional (since the pressure wave dominantly depends only on time and the distance from the center). Another example is (slow) fluid in a straight circular pipe: in Cartesian coordinates, one has to solve a (difficult) two dimensional boundary value problem involving a partial differential equation, but in cylindrical coordinates the problem becomes one-dimensional with an ordinary differential equation instead of a partial differential equation.
The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may be solved by separation of variables. Separation of variables is a mathematical technique that converts a complex d-dimensional problem into d one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced to Laplace's equation or the Helmholtz equation. Laplace's equation is separable in 13 orthogonal coordinate systems (the 14 listed in the table below with the exception of toroidal), and the Helmholtz equation is separable in 11 orthogonal coordinate systems.
Orthogonal coordinates never have off-diagonal terms in their metric tensor. In other words, the infinitesimal squared distance ds2 can always be written as a scaled sum of the squared infinitesimal coordinate displacements
where d is the dimension and the scaling functions (or scale factors)
equal the square roots of the diagonal components of the metric tensor, or the lengths of the local b
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https://en.wikipedia.org/wiki/Star%20Flyer%20%28Tivoli%20Gardens%29
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Star Flyer () is a carousel-meets-watchtower style amusement ride in Tivoli Gardens, Copenhagen, Denmark. It was manufactured by Funtime and opened in May 2006.
Statistics
Height
Platform diameter
Chairs 12 (2 seats each)
Capacity circa 960 passengers/hour
Maximum rotation speed
Maximum vertical speed
References
External links
Tivoli.dk — The Star Flyer
Swing rides
Amusement rides introduced in 2006
Landmarks in Copenhagen
Amusement rides manufactured by Funtime
Towers in Denmark
2006 establishments in Denmark
Towers completed in 2006
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https://en.wikipedia.org/wiki/Elliptic%20coordinate%20system
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In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.
Basic definition
The most common definition of elliptic coordinates is
where is a nonnegative real number and
On the complex plane, an equivalent relationship is
These definitions correspond to ellipses and hyperbolae. The trigonometric identity
shows that curves of constant form ellipses, whereas the hyperbolic trigonometric identity
shows that curves of constant form hyperbolae.
Scale factors
In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates are equal to
Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as
Consequently, an infinitesimal element of area equals
and the Laplacian reads
Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
Alternative definition
An alternative and geometrically intuitive set of elliptic coordinates are sometimes used,
where and . Hence, the curves of constant are ellipses, whereas the curves of constant are hyperbolae. The coordinate must belong to the interval [-1, 1], whereas the
coordinate must be greater than or equal to one.
The coordinates have a simple relation to the distances to the foci and . For any point in the plane, the sum of its distances to the foci equals , whereas their difference equals .
Thus, the distance to is , whereas the distance to is . (Recall that and are located at and , respectively.)
A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates , so the conversion to Cartesian coordinates is not a function, but a multifunction.
Alternative scale factors
The scale factors for the alternative elliptic coordinates are
Hence, the infinitesimal area element becomes
and the Laplacian equals
Other differential operators such as
and can be expressed in the coordinates by substituting
the scale factors into the general formulae
found in orthogonal coordinates.
Extrapolation to higher dimensions
Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates:
The elliptic cylindrical coordinates are produced by projecting in the -direction.
The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis separating the foci.
Ellipsoidal coordinates are a formal extensi
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https://en.wikipedia.org/wiki/Hopf%20conjecture
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In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
Positively or negatively curved Riemannian manifolds
The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions of Heinz Hopf from 1931. A modern formulation is:
A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. A compact, (2d)-dimensional Riemannian manifold with negative sectional curvature has Euler characteristic of sign .
For surfaces, these statements follow from the Gauss–Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and Poincaré duality and Euler–Poincaré formula equating for 4-manifolds the Euler characteristic with and Synge's theorem, assuring that the orientation cover is simply connected, so that the Betti numbers vanish . For 4-manifolds, the statement also follows from the Chern–Gauss–Bonnet theorem as noticed by John Milnor in 1955 (written down by Shiing-Shen Chern in 1955.). For manifolds of dimension 6 or higher the conjecture is open. An example of Robert Geroch had shown that the Chern–Gauss–Bonnet integrand can become negative for . The positive curvature case is known to hold however for hypersurfaces in (Hopf) or codimension two surfaces embedded in . For sufficiently pinched positive curvature manifolds, the Hopf conjecture (in the positive curvature case) follows from the sphere theorem, a theorem which had also been conjectured first by Hopf. One of the lines of attacks is by looking for manifolds with more symmetry. It is particular for example that all known manifolds of positive sectional curvature allow for an isometric circle action. The corresponding vector field is called a killing vector field. The conjecture (for the positive curvature case) has also been proved for manifolds of dimension or admitting an isometric torus action of a k-dimensional torus and for manifolds M admitting an isometric action of a compact Lie group G with principal isotropy subgroup H and cohomogeneity k such that
Some references about manifolds with some symmetry are and
On the history of the problem:
the first written explicit appearance of the conjecture is in the proceedings of the German Mathematical Society, which is a paper based on talks, Heinz Hopf gave in the spring of 1931 in Fribourg, Switzerland and at Bad Elster in the fall of 1931. Marcel Berger discusses the conjecture in his book, and points to the work of Hopf from the 1920s which was influenced by such type of questions. The conjectures are listed as problem 8 (positive curvature case) and 10 (negative curvature case) in ``Yau's problems" of 1982.
Non-negatively or non-positively curved Riemannian manifolds
There are analogue conjectures if the curvature is allowed to become zero too. The statement should still be attributed to Hopf (for ex
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https://en.wikipedia.org/wiki/Parabolic%20cylindrical%20coordinates
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In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the
perpendicular -direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.
Basic definition
The parabolic cylindrical coordinates are defined in terms of the Cartesian coordinates by:
The surfaces of constant form confocal parabolic cylinders
that open towards , whereas the surfaces of constant form confocal parabolic cylinders
that open in the opposite direction, i.e., towards . The foci of all these parabolic cylinders are located along the line defined by . The radius has a simple formula as well
that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article.
Scale factors
The scale factors for the parabolic cylindrical coordinates and are:
Differential elements
The infinitesimal element of volume is
The differential displacement is given by:
The differential normal area is given by:
Del
Let be a scalar field. The gradient is given by
The Laplacian is given by
Let be a vector field of the form:
The divergence is given by
The curl is given by
Other differential operators can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
Relationship to other coordinate systems
Relationship to cylindrical coordinates :
Parabolic unit vectors expressed in terms of Cartesian unit vectors:
Parabolic cylinder harmonics
Since all of the surfaces of constant , and are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:
and Laplace's equation, divided by , is written:
Since the equation is separate from the rest, we may write
where is constant. has the solution:
Substituting for , Laplace's equation may now be written:
We may now separate the and functions and introduce another constant to obtain:
The solutions to these equations are the parabolic cylinder functions
The parabolic cylinder harmonics for are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:
Applications
The classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate.
See also
Parabolic coordinates
Orthogonal coordinate system
Curvilinear coor
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https://en.wikipedia.org/wiki/Tsuruichi%20Hayashi
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was a Japanese mathematician and historian of Japanese mathematics. He was born in Tokushima, Japan.
He was the founder of the Tohoku Mathematical Journal.
References
Further reading
External links
The Extremal Chords of an Oval, by TSURUICHI HAYASHI, Sendai.
A Remark on the integral Equation solved by Mr. Hirakawa, by TSURUICHI HAYASHI in Sendai.
1873 births
1935 deaths
19th-century Japanese mathematicians
20th-century Japanese mathematicians
Historians of mathematics
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https://en.wikipedia.org/wiki/Sine%20and%20cosine
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In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the and functions used in Indian astronomy during the Gupta period.
Notation
Sine and cosine are written using functional notation with the abbreviations sin and cos.
Often, if the argument is simple enough, the function value will be written without parentheses, as rather than as .
Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.
Definitions
Right-angled triangle definitions
To define the sine and cosine of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle α in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:
The opposite side is the side opposite to the angle of interest, in this case side a.
The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
The adjacent side is the remaining side, in this case side b. It forms a side of (and is adjacent to) both the angle of interest (angle A) and the right angle.
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side, divided by the length of the hypotenuse:
The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.
As stated, the values and appear to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratios are the same for each of them.
Unit circle definitions
In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate sy
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https://en.wikipedia.org/wiki/Weyl%20character%20formula
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In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.
By definition, the character of a representation of G is the trace of , as a function of a group element . The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character of gives a lot of information about itself.
Weyl's formula is a closed formula for the character , in terms of other objects constructed from G and its Lie algebra.
Statement of Weyl character formula
The character formula can be expressed in terms of representations of complex semisimple Lie algebras or in terms of the (essentially equivalent) representation theory of compact Lie groups.
Complex semisimple Lie algebras
Let be an irreducible, finite-dimensional representation of a complex semisimple Lie algebra . Suppose is a Cartan subalgebra of . The character of is then the function defined by
The value of the character at is the dimension of . By elementary considerations, the character may be computed as
,
where the sum ranges over all the weights of and where is the multiplicity of . (The preceding expression is sometimes taken as the definition of the character.)
The character formula states that may also be computed as
where
is the Weyl group;
is the set of the positive roots of the root system ;
is the half-sum of the positive roots, often called the Weyl vector;
is the highest weight of the irreducible representation ;
is the determinant of the action of on the Cartan subalgebra . This is equal to , where is the length of the Weyl group element, defined to be the minimal number of reflections with respect to simple roots such that equals the product of those reflections.
Discussion
Using the Weyl denominator formula (described below), the character formula may be rewritten as
,
or, equivalently,
The character is itself a large sum of exponentials. In this last expression, we then multiply the character by an alternating sum of exponentials—which seemingly will result in an even larger sum of exponentials. The surprising part of the character formula is that when we compute this product, only a small number of terms actually remain. Many more terms than this occur at least once in the product of the character and the We
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https://en.wikipedia.org/wiki/Symmetry%20set
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In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis, a subset of the symmetry set is a set of curves which roughly run along the middle of an object.
In 2 dimensions
Let be an open interval, and be a parametrisation of a smooth plane curve.
The symmetry set of is defined to be the closure of the set of centres of circles tangent to the curve at at least two distinct points (bitangent circles).
The symmetry set will have endpoints corresponding to vertices of the curve. Such points will lie at cusp of the evolute. At such points the curve will have 4-point contact with the circle.
In n dimensions
For a smooth manifold of dimension in (clearly we need ). The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.
As a bifurcation set
Let be an open simply connected domain and . Let be a parametrisation of a smooth piece of manifold.
We may define a parameter family of functions on the curve, namely
This family is called the family of distance squared functions. This is because for a fixed the value of is the square of the distance from to at
The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of such that has a repeated singularity for some
By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to .
The symmetry set is then the set of such that there exist with , and
together with the limiting points of this set.
References
J. W. Bruce, P. J. Giblin and C. G. Gibson, Symmetry Sets. Proc. of the Royal Soc.of Edinburgh 101A (1985), 163-186.
J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press (1993).
Differential geometry
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https://en.wikipedia.org/wiki/Normal%20measure
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In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f(α)<α for most α<κ, then there is a β<κ such that f(α)=β for most α<κ. (Here, "most" means that the set of elements of κ where the property holds is a member of the ultrafilter, i.e. has measure 1.) Also equivalent, the ultrafilter (set of sets of measure 1) is closed under diagonal intersection.
For a normal measure, any closed unbounded (club) subset of κ contains most ordinals less than κ. And any subset containing most ordinals less than κ is stationary in κ.
If an uncountable cardinal κ has a measure on it, then it has a normal measure on it.
See also
Measurable cardinal
Club set
References
pp 52–53
Large cardinals
Measures (set theory)
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https://en.wikipedia.org/wiki/Probabilistic%20logic
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Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.
Logical background
There are numerous proposals for probabilistic logics. Very roughly, they can be categorized into two different classes: those logics that attempt to make a probabilistic extension to logical entailment, such as Markov logic networks, and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics).
That the concept of probability can have different meanings may be understood by noting that, despite the mathematization of probability in the Enlightenment, mathematical probability theory remains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.
More precisely, in evidentiary logic, there is a need to distinguish the objective truth of a statement from our decision about the truth of that statement, which in turn must be distinguished from our confidence in its truth: thus, a suspect's real guilt is not necessarily the same as the judge's decision on guilt, which in turn is not the same as assigning a numerical probability to the commission of the crime, and deciding whether it is above a numerical threshold of guilt. The verdict on a single suspect may be guilty or not guilty with some uncertainty, just as the flipping of a coin may be predicted as heads or tails with some uncertainty. Given a large collection of suspects, a certain percentage may be guilty, just as the probability of flipping "heads" is one-half. However, it is incorrect to take this law of averages with regard to a single criminal (or single coin-flip): the criminal is no more "a little bit guilty" than predicting a single coin flip to be "a little bit heads and a little bit tails": we are merely uncertain as to which it is. Expressing uncertainty as a numerical probability may be acceptable when making scientific measurements of physical quantities, but it is merely a mathematical model of the uncertainty we perceive in the context of "common sense" reasoning and logic. Just as in courtroom reasoning, the goal of employing uncertain inference is to gather evidence to strengthen the confidence of a proposition, as opposed to performing some sort of probabilistic entailment.
Histor
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https://en.wikipedia.org/wiki/Critical%20pair
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In mathematics, a critical pair may refer to:
Critical pair (term rewriting), terms resulting from two overlapping rules in a term rewriting system
Critical pair (order theory), two incomparable elements of a partial order that could be made comparable without changing any other relation in the partial order
The pair of polynomials associated with an S-polynomial in Buchberger's algorithm for computing a Gröbner basis
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https://en.wikipedia.org/wiki/Dependency%20graph
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In mathematics, computer science and digital electronics, a dependency graph is a directed graph representing dependencies of several objects towards each other. It is possible to derive an evaluation order or the absence of an evaluation order that respects the given dependencies from the dependency graph.
Definition
Given a set of objects and a transitive relation with modeling a dependency "a depends on b" ("a needs b evaluated first"), the dependency graph is a graph with the transitive reduction of R.
For example, assume a simple calculator. This calculator supports assignment of constant values to variables and assigning the sum of exactly two variables to a third variable. Given several equations like "A = B+C; B = 5+D; C=4; D=2;", then and . You can derive this relation directly: A depends on B and C, because you can add two variables if and only if you know the values of both variables. Thus, B must be calculated before A can be calculated. However, the values of C and D are known immediately, because they are number literals.
Recognizing impossible evaluations
In a dependency graph, the cycles of dependencies (also called circular dependencies) lead to a situation in which no valid evaluation order exists, because none of the objects in the cycle may be evaluated first. If a dependency graph does not have any circular dependencies, it forms a directed acyclic graph, and an evaluation order may be found by topological sorting. Most topological sorting algorithms are also capable of detecting cycles in their inputs; however, it may be desirable to perform cycle detection separately from topological sorting in order to provide appropriate handling for the detected cycles.
Assume the simple calculator from before. The equation system "A=B; B=D+C; C=D+A; D=12;" contains a circular dependency formed by A, B and C, as B must be evaluated before A, C must be evaluated before B, and A must be evaluated before C.
Deriving an evaluation order
A correct evaluation order is a numbering of the objects that form the nodes of the dependency graph so that the following equation holds: with . This means, if the numbering orders two elements and so that will be evaluated before , then must not depend on .
There can be more than one correct evaluation order. In fact, a correct numbering is a topological order, and any topological order is a correct numbering. Thus, any algorithm that derives a correct topological order derives a correct evaluation order.
Assume the simple calculator from above once more. Given the equation system "A = B+C; B = 5+D; C=4; D=2;", a correct evaluation order would be (D, C, B, A). However, (C, D, B, A) is a correct evaluation order as well.
Monoid structure
An acyclic dependency graph corresponds to a trace of a trace monoid as follows:
A function labels each vertex with a symbol from the alphabet
There is an edge or if and only if is in the dependency relation .
Two graphs are considered to be
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https://en.wikipedia.org/wiki/Conway%20polyhedron%20notation
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In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, represents a truncated cube, and , parsed as , is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.g., a dual cube is an octahedron: . Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators (ambo), (bevel), (dual), (expand), (gyro), (join), (kis), (meta), (ortho), (snub), and (truncate), while Hart added (reflect) and (propellor). Later implementations named further operators, sometimes referred to as "extended" operators. Conway's basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation (), while a truncation after ambo produces bevel ().
Polyhedra can be studied topologically, in terms of how their vertices, edges, and faces connect together, or geometrically, in terms of the placement of those elements in space. Different implementations of these operators may create polyhedra that are geometrically different but topologically equivalent. These topologically equivalent polyhedra can be thought of as one of many embeddings of a polyhedral graph on the sphere. Unless otherwise specified, in this article (and in the literature on Conway operators in general) topology is the primary concern. Polyhedra with genus 0 (i.e. topologically equivalent to a sphere) are often put into canonical form to avoid ambiguity.
Operators
In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a cuboctahedron is an ambo cube, i.e. , and a truncated cuboctahedron is . Repeated application of an operator can be denoted with an exponent: j2 = o. In general, Conway operators are not commutative.
Individual operators can be visualized in terms of fundamental domains (or chambers), as below. Each right triangle is a fundamental domain. Each white chamber is a rotated version of the others, and so is each colored chamber. For achiral operators, the colored chambers are a reflection of the white chambers, and all are transitive. In group terms, achiral operators correspond to dihedral groups where n is the number of sides of a face, while chiral operators correspond to cyclic groups lacking the reflective symmetry of the dihedral groups. Achiral and chiral operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively.
LSPs should be understood as local operations that preserve symmetr
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https://en.wikipedia.org/wiki/Colorado%20Model%20Content%20Standards
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The Colorado Model Content Standards were a set of curriculum standards for teaching civics, dance, economics, foreign language, geography, history, mathematics, music, physical education, reading and writing, science, theatre, and visual arts.
Of the 13 standards only three (mathematics, reading and writing, and science) were testing subjects included in the CSAP.
The standards were replaced by the Colorado Academic Standards in 2011.
External links
Colorado Department of Education: Colorado K-12 Academic Standards
Education in Colorado
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https://en.wikipedia.org/wiki/Apollonian%20circles
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In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer.
Definition
The Apollonian circles are defined in two different ways by a line segment denoted .
Each circle in the first family (the blue circles in the figure) is associated with a positive real number , and is defined as the locus of points such that the ratio of distances from to and to equals ,
For values of close to zero, the corresponding circle is close to , while for values of close to , the corresponding circle is close to ; for the intermediate value , the circle degenerates to a line, the perpendicular bisector of . The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points.
Each circle in the second family (the red circles in the figure) is associated with an angle , and is defined as the locus of points such that the inscribed angle equals ,
Scanning from 0 to π generates the set of all circles passing through the two points and .
The two points where all the red circles cross are the limiting points of pairs of circles in the blue family.
Bipolar coordinates
A given blue circle and a given red circle intersect in two points. In order to obtain bipolar coordinates, a method is required to specify which point is the right one. An isoptic arc is the locus of points that sees points under a given oriented angle of vectors i.e.
Such an arc is contained into a red circle and is bounded by points . The remaining part of the corresponding red circle is . When we really want the whole red circle, a description using oriented angles of straight lines has to be used:
Pencils of circles
Both of the families of Apollonian circles are pencils of circles. Each is determined by any two of its members, called generators of the pencil. Specifically, one is an elliptic pencil (red family of circles in the figure) that is defined by two generators that pass through each other in exactly two points (). The other is a hyperbolic pencil (blue family of circles in the figure) that is defined by two generators that do not intersect each other at any point.
Radical axis and central line
Any two of these circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxial circles or coaxal circles.
The elliptic pencil of circles passing through the two points (the set of red circles, in the figure) has the line as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of .
The hyperbolic pencil defined by points (the blue circles) has its radical axis on the perpendicular bisector of line , and all its
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https://en.wikipedia.org/wiki/257-gon
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In Geometry, 257-gon, also known broadly as the Dihectapentacontakaiheptagon, is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
Regular 257-gon
The area of a regular 257-gon is (with )
A whole regular 257-gon is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 24 parts per million.
Construction
The regular 257-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime, being of the form 22n + 1 (in this case n = 3). Thus, the values and are 128-degree algebraic numbers, and like all constructible numbers they can be written using square roots and no higher-order roots.
Although it was known to Gauss by 1801 that the regular 257-gon was constructible, the first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822) and Friedrich Julius Richelot (1832). Another method involves the use of 150 circles, 24 being Carlyle circles: this method is pictured below. One of these Carlyle circles solves the quadratic equation x2 + x − 64 = 0.
Symmetry
The regular 257-gon has Dih257 symmetry, order 514. Since 257 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z257, and Z1.
257-gram
A 257-gram is a 257-sided star polygon. As 257 is prime, there are 127 regular forms generated by Schläfli symbols {257/n} for all integers 2 ≤ n ≤ 128 as .
Below is a view of {257/128}, with 257 nearly radial edges, with its star vertex internal angles 180°/257 (~0.7°).
See also
17-gon
List of polygons
List of self-intersecting polygons
References
External links
Robert Dixon Mathographics. New York: Dover, p. 53, 1991.
Benjamin Bold, Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982.
H. S. M. Coxeter Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons
Leonard Eugene Dickson Constructions with Ruler and Compasses; Regular Polygons. Ch. 8 in Monographs on Topics of Modern Mathematics *Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352–386, 1955.
257-gon, exact construction the 1st side using the quadratrix according of Hippias as an additional aid (German)
Constructible polygons
Polygons by the number of sides
Euclidean plane geometry
Carl Friedrich Gauss
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https://en.wikipedia.org/wiki/65537-gon
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In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non–self-intersecting is 11796300°.
Regular 65537-gon
The area of a regular is (with )
A whole regular is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 15 parts per billion.
Construction
The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime, being of the form 22n + 1 (in this case n = 4).
Thus, the values and are 32768-degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots.
Although it was known to Gauss by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. Another method involves the use of at most 1332 Carlyle circles, and the first stages of this method are pictured below. This method faces practical problems, as one of these Carlyle circles solves the quadratic equation x2 + x − 16384 = 0 (16384 being 214).
Symmetry
The regular 65537-gon has Dih65537 symmetry, order 131074. Since 65,537 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z65537, and Z1.
65537-gram
A 65537-gram is a 65,537-sided star polygon. As 65,537 is prime, there are 32,767 regular forms generated by Schläfli symbols {65537/n} for all integers 2 ≤ n ≤ 32768 as .
See also
Circle
Equilateral triangle
Pentagon
Heptadecagon (17-sides)
257-gon
References
Bibliography
Robert Dixon Mathographics. New York: Dover, p. 53, 1991.
Benjamin Bold, Famous Problems of Geometry and How to Solve Them New York: Dover, p. 70, 1982.
H. S. M. Coxeter Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons
Leonard Eugene Dickson Constructions with Ruler and Compasses; Regular Polygons Ch. 8 in Monographs on Topics of Modern Mathematics
Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352–386, 1955.
External links
65537-gon mathematik-olympiaden.de (German), with images of the documentation HERMES; retrieved on July 9, 2018
Wikibooks 65573-Eck (German) Approximate construction of the first side in two main steps
65537-gon, exact construction for the 1st side, using the Quadratrix of Hippias and GeoGebra as additional aids, with brief description (German)
Constructible polygons
Polygons by the number of sides
Euclidean plane geometry
Carl Friedrich Gauss
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https://en.wikipedia.org/wiki/Vampirium
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Vampirium is the twenty-seventh book of the award-winning Lone Wolf series of gamebooks created by Joe Dever.
Gameplay
Lone Wolf books rely on a combination of thought and luck. Certain statistics such as combat skill and endurance attributes are determined randomly before play (reading). The player is then allowed to choose which Kai disciplines or skills he or she possess. This number depends directly on how many books in the series have been completed ("Kai rank"). In his first book, the player starts with five disciplines. With each additional book completed, the player chooses one additional Kai discipline.
Plot
References
External links
Gamebooks - Lone Wolf
Origins of Lone Wolf
Book entry
Lone Wolf (gamebooks)
1998 fiction books
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https://en.wikipedia.org/wiki/Macdonald%20identities
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In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by , and a 10-fold product identity found by .
and pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebras and superalgebras.
References
Lie algebras
Mathematical identities
Infinite products
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https://en.wikipedia.org/wiki/Peter%20Ozsv%C3%A1th
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Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds.
Education
Ozsváth received his Ph.D. from Princeton in 1994 under the supervision of John Morgan; his dissertation was entitled On Blowup Formulas For SU(2) Donaldson Polynomials.
Awards
In 2007, Ozsváth was one of the recipients of the Oswald Veblen Prize in Geometry. In 2008 he was named a Guggenheim Fellow. In July 2017, he was a plenary lecturer in the Mathematical Congress of the Americas. He was elected a member of the National Academy of Sciences in 2018.
Selected publications
Grid Homology for Knots and Links, American Math Society, (2015)
References
External links
Personal homepage
Living people
1967 births
20th-century American mathematicians
20th-century Hungarian mathematicians
21st-century American mathematicians
21st-century Hungarian mathematicians
Princeton University faculty
Columbia University faculty
Topologists
Mathematicians from Texas
People from Dallas
Princeton University alumni
Members of the United States National Academy of Sciences
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https://en.wikipedia.org/wiki/Suslin%20tree
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In mathematics, a Suslin tree is a tree of height ω1 such that
every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin.
Every Suslin tree is an Aronszajn tree.
The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by ) or a Suslin algebra. The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees.
More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin tree is the same as a ω1-Suslin tree. showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesis implies the existence of an ℵ2-Suslin tree, is a longstanding open problem.
See also
Glossary of set theory
Kurepa tree
List of statements independent of ZFC
List of unsolved problems in set theory
Suslin's problem
References
Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics,Springer,
erratum, ibid. 4 (1972), 443.
Trees (set theory)
Independence results
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https://en.wikipedia.org/wiki/Trail%20of%20the%20Wolf
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Trail of the Wolf is the twenty-fifth book of the Lone Wolf book series created by Joe Dever.
Gameplay
Lone Wolf books rely on a combination of thought and luck. Certain statistics such as combat skill and endurance attributes are determined randomly before play (reading). The player is then allowed to choose which Kai disciplines or skills he or she possess. This number depends directly on how many books in the series have been completed ("Kai rank"). With each additional book completed, the player chooses one additional Kai discipline. In this first book, the player starts with five disciplines.
References
External links
Gamebooks - Lone Wolf
Origins of Lone Wolf
Book entry
Lone Wolf (gamebooks)
1997 fiction books
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https://en.wikipedia.org/wiki/Rune%20War
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Rune War is the twenty-fourth book of the award-winning Lone Wolf book series created by Joe Dever.
Gameplay
Lone Wolf books rely on a combination of thought and luck. Certain statistics such as combat skill and endurance attributes are determined randomly before play (reading). The player is then allowed to choose which Kai disciplines or skills he or she possess. This number depends directly on how many books in the series have been completed ("Kai rank"). With each additional book completed, the player chooses one additional Kai discipline. In this first book, the player starts with five disciplines.
Reception
Chris Read reviewed Rune War for Arcane magazine, rating it a 6 out of 10 overall. Read comments that "the book is nicely illustrated and the narrative is generally good with plenty of decision-making and 'interaction' with NPCs. Naturally, Lady Luck plays her fickle part, but your skills and common sense are much more relevant to your success. This, above all, gives the book a good feel."
References
External links
Gamebooks - Lone Wolf
Origins of Lone Wolf
Book entry
Lone Wolf (gamebooks)
1995 fiction books
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https://en.wikipedia.org/wiki/Mydnight%27s%20Hero
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Mydnight's Hero is the twenty-third book of the award-winning Lone Wolf book series created by Joe Dever.
Gameplay
Lone Wolf books rely on a combination of thought and luck. Certain statistics such as combat skill and endurance attributes are determined randomly before play (reading). The player is then allowed to choose which Kai disciplines or skills he or she possess. This number depends directly on how many books in the series have been completed ("Kai rank"). With each additional book completed, the player chooses one additional Kai discipline. In this first book, the player starts with five disciplines.
Plot
References
External links
Gamebooks - Lone Wolf
Origins of Lone Wolf
Book entry
Lone Wolf (gamebooks)
1995 fiction books
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https://en.wikipedia.org/wiki/Bernhard%20Neumann
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Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory.
Early life and education
After gaining a D.Phil. from Friedrich-Wilhelms Universität in Berlin in 1932 he earned a Ph.D. at the University of Cambridge in 1935 and a Doctor of Science at the University of Manchester in 1954. His doctoral students included Gilbert Baumslag, László Kovács, Michael Newman, and James Wiegold. After war service with the British Army, he became a lecturer at University College, Hull, before moving in 1948 to the University of Manchester, where he spent the next 14 years. In 1954 he received a DSc from the University of Cambridge.
In 1962 he migrated to Australia to take up the Foundation Chair of the Department of Mathematics within the Institute of Advanced Studies of the Australian National University (ANU), where he served as head of the department until retiring in 1975. In addition he was a senior research fellow at the CSIRO Division of Mathematics and Statistics from 1975 to 1977 and then honorary research fellow from 1978 until his death in 2002.
His wife, Hanna Neumann, and sons, Peter M. Neumann and Walter Neumann, are also notable for their contributions to group theory.
He was an invited speaker of the International Congress of Mathematicians in 1936 at Oslo and in 1970 at Nice. He was elected a Fellow of the Royal Society in 1959. In 1994, he was appointed a Companion of the Order of Australia (AC).
The Australian Mathematical Society awards a student prize named in his honour. The group-theoretic notion of HNN (Higman-Neumann-Neumann) extension bears the names of Bernard and his wife Hanna, from their joint paper with Graham Higman (who later supervised the PhD of their son Peter).
Career
Assistant lecturer, University College, Cardiff, 1937–40.
Army Service, 1940–45.
Lecturer, University College, Hull, (now University of Hull), 1946–48
Lecturer, senior lecturer, reader, Manchester, 1948–61
Professor and head of Department of Mathematics, Institute of Advanced Studies, ANU, Canberra, 1962–74; Emeritus Professor, 1975–2002.
Senior research fellow, CSIRO Division of Mathematics and Statistics, 1975–77; honorary research fellow, 1978–99.
Founding member of the World Cultural Council, 1981.
Awards
1984 Matthew Flinders Medal and Lecture
1952 Adams Prize, University of Cambridge
References
External links
1909 births
2002 deaths
20th-century British mathematicians
20th-century German mathematicians
Academics of Cardiff University
Academics of the University of Manchester
Australian mathematicians
Companions of the Order of Australia
Fellows of the Australian Academy of Science
Fellows of the Royal Society
Founding members of the World Cultural Council
German emigrants to Australia
Group theorists
Jewish emigrants from Nazi Germany to the United Kingdom
Academic staff of the Australian National University
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https://en.wikipedia.org/wiki/Monster%20vertex%20algebra
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The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by applying the Goddard–Thorn theorem of string theory to construct the monster Lie algebra, an infinite-dimensional generalized Kac–Moody algebra acted on by the monster.
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products. It can be constructed as conformal field theory describing 24 free bosons compactified on the torus induced by the Leech lattice and orbifolded by the two-element reflection group.
References
Non-associative algebra
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https://en.wikipedia.org/wiki/James%20Lepowsky
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James Lepowsky (born July 5, 1944) is a professor of mathematics at Rutgers University, New Jersey. Previously he taught at Yale University. He received his Ph.D. from Massachusetts Institute of Technology in 1970 where his advisors were Bertram Kostant and Sigurdur Helgason. Lepowsky graduated from Stuyvesant High School in 1961, 16 years after Kostant. His current research is in the areas of infinite-dimensional Lie algebras and vertex algebras. He has written several books on vertex algebras and related topics. In 1988, in a joint work with Igor Frenkel and Arne Meurman, he constructed the monster vertex algebra (also known as the Moonshine module). His PhD students include Stefano Capparelli, Yi-Zhi Huang, Haisheng Li, Arne Meurman, and Antun Milas.
In 2012, he became a fellow of the American Mathematical Society.
Notes
References
External links
Stuyvesant High School alumni
Massachusetts Institute of Technology alumni
Rutgers University faculty
Fellows of the American Mathematical Society
20th-century American mathematicians
21st-century American mathematicians
1944 births
Living people
Mathematicians from New York (state)
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https://en.wikipedia.org/wiki/Arne%20Meurman
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Arne Meurman (born 6 April 1956) is a Swedish mathematician working on finite groups and vertex operator algebras. Currently, he is a professor at Lund University.
He is best known for constructing the monster vertex algebra together with Igor Frenkel and James Lepowsky.
He is interested in chess.
Publications
Igor Frenkel, James Lepowsky, Arne Meurman, "Vertex operator algebras and the Monster". Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988. liv+508 pp.
Arne Meurman, Mirko Primc, "Annihilating fields of standard modules of sl (2,C) and combinatorial identities", Memoirs AMS 1999
References
External links
Homepage
1956 births
Living people
People connected to Lund University
20th-century Swedish mathematicians
21st-century Swedish mathematicians
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https://en.wikipedia.org/wiki/Osem
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Osem may refer to:
Osem (mathematics) – algorithm for image reconstruction in nuclear medical imaging
Osem (company) – Israeli food corporation
Orquesta Sinfonica del Estado de Mexico, an official State symphony orchestra in Mexico.
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https://en.wikipedia.org/wiki/Robert%20Griess
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Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan.
Education
Griess developed a keen interest in mathematics prior to entering undergraduate studies at the University of Chicago in the fall of 1963. There, he eventually earned a Ph.D. in 1971 after defending a dissertation on the Schur multipliers of the then-known finite simple groups.
Career
Griess' work has focused on group extensions, cohomology and Schur multipliers, as well as on vertex operator algebras and the classification of finite simple groups. In 1982, he published the first construction of the monster group using the Griess algebra, and in 1983 he was an invited speaker at the International Congress of Mathematicians in Warsaw to give a lecture on the sporadic groups and his construction of the monster group. In the same landmark 1982 paper where he published his construction, Griess detailed an organization of the twenty-six sporadic groups into two general families of groups: the Happy Family and the pariahs.
He became a member of the American Academy of Arts and Sciences in 2007, and a fellow of the American Mathematical Society in 2012. In 2020 he became a member of the National Academy of Sciences. Since 2006, Robert Griess has been an editor for Electronic Research Announcements of the AIMS (ERA-AIMS), a peer-review journal.
In 2010, he was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research for his construction of the monster group, which he named the Friendly Giant.
Selected publications
Books
Journal articles
References
External links
Homepage at the Department of Mathematics at the University of Michigan
for the Mathematical Science Literature lecture series, Harvard University (2020)
Living people
20th-century American mathematicians
21st-century American mathematicians
Group theorists
University of Michigan faculty
University of Chicago alumni
Fellows of the American Mathematical Society
1945 births
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https://en.wikipedia.org/wiki/O%27Leary%2C%20Prince%20Edward%20Island
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O'Leary is a town located in Prince County, Prince Edward Island. Its population in the 2016 Census was 815 people.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, O'Leary 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.
Economy
The community's economy is tied to the potato farming industry. O'Leary is home to the Canadian Potato Museum.
Climate
References
External links
Communities in Prince County, Prince Edward Island
Towns in Prince Edward Island
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https://en.wikipedia.org/wiki/Sofya%20Yanovskaya
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Sofya Aleksandrovna Yanovskaya (also Janovskaja; ; 31 January 1896 – 24 October 1966) was a Soviet mathematician and historian, specializing in the history of mathematics, mathematical logic, and philosophy of mathematics. She is best known for her efforts in restoring the research of mathematical logic in the Soviet Union and publishing and editing the mathematical works of Karl Marx.
Biography
Yanovskaya was born in Pruzhany, a town near Brest, to a Jewish family of accountant Alexander Neimark. From 1915 to 1918, she studied in a woman's college in Odessa, when she became a communist. She worked as a party official until 1924, when she started teaching at the Institute of Red Professors. With exception of the war years (1941–1945), she worked at Moscow State University until retirement.
Engels had noted in his writings that Karl Marx had written some mathematics. Yanonskaya found Marx's ''Mathematical Manuscripts'' and she arranged for their first publication in 1933 in Russian. She received her doctoral degree in 1935.
Her work on Karl Marx's mathematical manuscripts began in 1930s and may have had some influence on the study of non-standard analysis in China. In the academia she is most remembered now for her work on history and philosophy of mathematics, as well as for her influence on young generation of researchers. She persuaded Ludwig Wittgenstein when he was visiting Soviet Union in 1935 to give up his idea to relocate to the Soviet Union. In 1968 Yanovskaya arranged for a better publication of Marx's work.
She died from diabetes in Moscow.
Awards and honours
For her work, Yanovskaya received the Order of Lenin and other medals.
References
Sources
Irving Anellis (1987) "The heritage of S.A. Janovskaja". History and Philosophy of Logic 8: 45-56.
B.A. Kushner (1996) "Sof'ja Aleksandrovna Janovskaja: a few reminiscences", Modern Logic 6: 67-72.
V.A. Bazhanov (2002) Essays on the Social History of Logic in Russia. Simbirsk-Ulyanovsk. Chapter 5 (bibliography of S.A. Yanovskaya's works is presented here). (in Russian).
B.V. Biryukov and L.G. Biryukova (2004) "Ludwig Wittgenstein and Sof'ya Aleksandrovna Yanovskaya. The 'Cambridge Genius' becomes acquainted with Soviet mathematicians in the 1930s" (in Russian). Logical Investigations. No. 11 (Russian), 46-94, Nauka, Moscow.
Further reading
"Sof'ya Aleksandrovna Janovskaja", Biographies of Women Mathematicians, Agnes Scott College
Remembrances and more remembrances of S.A. Yanovskaya, by Boris A. Kushner (in Russian).
a review of Yanovskaya's Methodological problems in science monograph – an article by B.V. Biryukov and O.A. Borisova (in Russian).
1965 Moscow Interview with Sofya Yanovskaya, Eugene Dynkin Collection of Mathematics Interviews, Cornell University Library (in Russian).
Vadim Valilyev on the meeting between Ludwig Wittgenstein and Sophia Yanovskaya (in Russian).
1896 births
1966 deaths
Burials at Novodevichy Cemetery
People from Pruzhany
People from Pruzhan
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https://en.wikipedia.org/wiki/Josip%20Belu%C5%A1i%C4%87
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Josip Belušić (March 12, 1847 – January 8, 1905) was a Croatian inventor and professor of physics and mathematics. He was born in the small settlement of Županići, in the region of Labin, Istria, and schooled in Pazin and Koper. Belušić continued his studies in Vienna, later resettling in Trieste before coming back to Istria, where he built his best known invention, the speedometer. After completing his studies, Belušić was employed as a professor of physics and mathematics at the Royal School of Koper. Later, he became director of the Maritime School of Castelnuovo, and was employed as an assistant professor in that institution.
In 1887 Belušić publicly experimented for the first time with his new invention, an electric speedometer. The invention was patented in Austria-Hungary under the name of "Velocimeter."
Belušić exhibited his invention at the 1889 Exposition Universell in Paris, renaming it Controllore automatico per vetture. In the same year, the Municipality of Paris announced a public competition, and over 120 patents were registered to compete. His design won as the most precise and reliable and was accepted in June 1890. Within a year, a hundred devices were installed on Parisian carriages. In 1889, the Croatian newspaper Naša sloga predicted that "[Belušić's invention] will spread all over the world, and with it the name of our virtuous Istrian, friend and patriot."
Belušić's invention was also the first monitoring device in history, a forerunner of measuring monitoring devices used today in trucks, buses and taxis. Thus, Belušić is also credited as the father of monitoring and surveillance devices.
Birth date, death date and nationality
Belušić was born on March 12, 1847, in the small village of Županići, on the outskirts of Labin, Istria, then part of the Austro-Hungarian Empire (now in Croatia). Županići is closer to Sveta Nedelja than Labin, and in fact today it is part of the former's municipality. There is some debate surrounding Belušić's place of death, with the Croatian Technical Encyclopedia leaning toward Trieste. He died on January 8, 1905. Belušić was 57 years old at the time of his death. He was an ethnic Croat born in Austria-Hungary.
Early life and education
Little is known about Belušić's early life. He was born to Marin Belušić and Katarina Ružić. Belušić spent his childhood in Županići, and was educated in nearby Pazin (5 miles northeast of Županići) and Koper (today Slovenia). In Pazin, the school's priests who tended to him were the first to notice his talent for the natural sciences. Belušić later enrolled in the Higher State Gymnasium of Koper.
The Koper high school had eight classes. The official languages of the institution were Italian and German. In the upper grades, there were also elective courses in the Slavic language. As a state grammar school, the institution was loyal to the Monarchy and the emperor. According to its curriculum, religious education, Italian, German, Latin and mathematics were s
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https://en.wikipedia.org/wiki/Constructive%20set%20theory
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Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory.
The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach.
On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.
In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be set bounded, motivated by results tied to impredicativity.
Introduction
Constructive outlook
Preliminary on the use of intuitionistic logic
The logic of the set theories discussed here is constructive in that it rejects the principle of excluded middle , i.e. that the disjunction automatically holds for all propositions . As a rule, to prove the excluded middle for a proposition , i.e. to prove the particular disjunction , either or needs to be explicitly proven. When either such proof is established, one says the proposition is decidable, and this then logically implies the disjunction holds. Similarly, a predicate for in a domain is said to be decidable when the more intricate statement is provable. Non-constructive axioms may enable proofs that formally claim decidability of such (and/or ) in the sense that they prove excluded middle for (resp. the statement using the quantifier above) without demonstrating the truth of either side of the disjunction(s). This is often the case in classical logic. In contrast, axiomatic theories deemed constructive tend to not permit many classical proofs of statements involving properties that are provenly computationally undecidable.
The law of noncontradiction is a special case of the propositional form of modus ponens. Using the former with any negated statement , one valid De Morgan's law thus implies already in the more conservative minimal logic. In words, intuitionistic logic still posits: It is impossible to rule out a proposition and rule out its negation both at once, and thus the rejection of any instantiated excluded middle statement for an individual proposition is inconsistent. Here the double-negation captures that the disjunction statement now provenly can never be ruled out or rejected, even in cases where the disjunction may not be provable (for example, by demonstrating one of the disjuncts, thus deciding ) from the assumed axioms.
More generally, constructive mathematical theories tend to prove classically equivalent reformulations of classical theorems. For example, in constructive analysis, one cannot prove the intermediate value theorem in its textbook formulation, but one can prove theorems with algorithmic content that, as soon as double negation elimination and its consequences are assumed legal, are at once classically equivalent to the classical statement. The difference is that the constructive proofs are harder to find.
The intuitioni
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https://en.wikipedia.org/wiki/Absolute%20value%20%28algebra%29
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In algebra, an absolute value (also called a valuation, magnitude, or norm, although "norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping |x| from D to the real numbers R satisfying:
It follows from these axioms that |1| = 1 and |-1| = 1. Furthermore, for every positive integer n,
|n| = |1 + 1 + ... + 1 (n times)| = |−1 − 1 − ... − 1 (n times)| ≤ n.
The classical "absolute value" is one in which, for example, |2|=2, but many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value (but not the square thereof).
An absolute value induces a metric (and thus a topology) by
Examples
The standard absolute value on the integers.
The standard absolute value on the complex numbers.
The p-adic absolute value on the rational numbers.
If R is the field of rational functions over a field F and is a fixed irreducible element of R, then the following defines an absolute value on R: for in R define to be , where and
Types of absolute value
The trivial absolute value is the absolute value with |x|=0 when x=0 and |x|=1 otherwise. Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a finite field because any non-zero element can be raised to some power to yield 1.
If an absolute value satisfies the stronger property |x + y| ≤ max(|x|, |y|) for all x and y, then |x| is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value.
Places
If |x|1 and |x|2 are two absolute values on the same integral domain D, then the two absolute values are equivalent if |x|1 < 1 if and only if |x|2 < 1 for all x. If two nontrivial absolute values are equivalent, then for some exponent e we have |x|1e = |x|2 for all x. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it violates the rule |x+y| ≤ |x|+|y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place.
Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p. For a given prime p, any rational number q can be written as pn(a/b), where a and b are integers not divisible by p and n is an integer. The p-adic absolute value of q is
Since the ordinary absolute value and the p-adic absolute values are absolute values according to the definition above, these define places.
Valuations
If for some ultrametric absolute value and any base b > 1, we define ν(x) = −lo
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https://en.wikipedia.org/wiki/Paul%20Hoffert
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Paul Matthew Hoffert, LLD, CM (born 22 September 1943, in Brooklyn, New York) is a recording artist, performer, media music composer, author, academic, and corporate executive. He studied mathematics and physics at the University of Toronto. He later studied music composition with Gordon Delamont. In 1969, the 26-year-old Hoffert co-founded Lighthouse, a rock group that sold millions of records and earned three Juno Awards as one of Canada's leading pop bands. His film music earned him a San Francisco Film Festival and three SOCAN Film Composer of the Year awards and included films such as The Proud Rider (1971), The Groundstar Conspiracy (1972), Outrageous! (1977), High-Ballin' (1978), The Shape of Things to Come (1979), Wild Horse Hank (1979), Mr. Patman (1980), Deadly Companion (1981), Paradise (1982), Fanny Hill (1983), Bedroom Eyes (1984), and Mr. Nice Guy (1987).
In 2001, Hoffert received the Pixel Award as the New Media industry's "Visionary of the Year".
Hoffert has parallel achievements in science and technology. He was a researcher at the National Research Council of Canada in the early 1970s and returned to research in 1988 as Vice President of DHJ Research, where he invented precursor algorithms to MP3 audio compression, as well as microchips for Newbridge Microsystems and products for Mattel, Akai, and Yamaha.
In 1992, Hoffert founded the CulTech Research Centre at York University, where he developed advanced media such as digital videophones and networked distribution of CD-ROMs. From 1994 to 1999, he directed Intercom Ontario, a $100 million trial of the world's first completely connected broadband community that landed him on the cover of the Financial Post and in the Wall Street Journal. He is an expert in online content distribution and usage consumption.
Hoffert was awarded the Order of Canada [CM] in 2004 for his contributions to music and the arts. The Canadian Government citation reads: "[Mr. Hoffert] is multitalented, determined, and a visionary. Paul Hoffert is a founding member of the rock group Lighthouse and an award-winning composer who has scored countless feature films and television productions. He received an honorary PhD from the University of Toronto in June 2012."
"Formerly a teacher at the Faculty of Fine Arts at York University, Hoffert founded the University's CulTech Research Centre and is an expert on new media and technology. A founding director of the Canadian Independent Record Producers Association and the Academy of Canadian Cinema and Television, he was instrumental in bringing about the Gemini and Prix Gémeaux awards. He was the first artist to chair the Ontario Arts Council, and he continues to be involved in multiple arts organizations and the Bell Broadcast and New Media Fund."
Current positions
Chair of the Bell Broadcast and New Media Fund, 1997–
Chair of the Screen Composers Guild of Canada, 1999–
President of the Glenn Gould Foundation US
Director, Ontario Cultural Attractions Fund, 1999
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https://en.wikipedia.org/wiki/Belt%20transect
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Belt transects are used in biology, more specifically in biostatistics, to estimate the distribution of organisms in relation to a certain area, such as the seashore or a meadow.
The belt transect method is similar to the line transect method but gives information on abundance as well as presence, or absence of species.
Method
The method involves laying out a transect line and then placing quadrats over the line, starting the quadrat at the first marked point of the line. Any consistent measurement size for the quadrat and length of the line can be chosen, depending on the species. With the quadrats applied, all the individuals of a species can be counted, and the species abundance can be estimated. The method is also suitable for long-term observations with a permanent installation.
References
Ecological techniques
Sampling techniques
Environmental statistics
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https://en.wikipedia.org/wiki/Pauli%20group
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In physics and mathematics, the Pauli group on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix and all of the Pauli matrices
,
together with the products of these matrices with the factors and :
.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.
The Pauli group on qubits, , is the group generated by the operators described above applied to each of qubits in the tensor product Hilbert space .
As an abstract group, is the central product of a cyclic group of order 4 and the dihedral group of order 8.
The Pauli group is a representation of the gamma group in three-dimensional Euclidean space. It is not isomorphic to the gamma group; it is less free, in that its chiral element is whereas there is no such relationship for the gamma group.
References
External links
Finite groups
Quantum information science
2. https://arxiv.org/abs/quant-ph/9807006
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https://en.wikipedia.org/wiki/National%20Science%20%26%20Mathematics%20Access%20to%20Retain%20Talent%20Grant
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The National Science and Mathematics Access to Retain Talent (SMART) Grant was a need based federal grant that was awarded to undergraduate students in their third and fourth year of undergraduate studies. The National SMART grant was introduced to help maintain the edge that United States has in the fields of Science and Technology. Only specific majors were eligible for the SMART grant, the complete list is given below.
History
The National Science and Mathematics Access to Retain Talent (SMART) Grant was introduced by Senator Bill Frist, R-Tennessee and approved by the Senate on 21 December 2005 as part of the Higher Education Reconciliation Act. President Bush signed the bill into law on Feb 8, 2006. This program ended June 30, 2011.
Application
Applying for the National SMART grant requires the student and the student's family to complete a Free Application For Federal Student Aid (FAFSA) form. Eligible students based on GPA, major and Pell Grant eligibility will be identified by the educational institute. The applicant does not need to file a separate application for being considered for a SMART grant.
Eligibility
The student must be a U.S Citizen, must be enrolled in a qualifying four year degree program, must be in the third or fourth year of the program, must be eligible to receive a Pell Grant in the same year, must maintain a minimum GPA of 3.0
Qualifying Degree Programs
Eligible degree programs or majors for the SMART grant are Science (including Computer Science, Physical and Life Sciences), Engineering, Technology, Mathematics, Liberal Arts and Sciences, Critical foreign language studies and certain natural resource conservation and multidisciplinary programs.
In August 2006, The Chronicle of Higher Education noted that evolutionary biology had been removed from the list of qualifying majors. This caused concern among some scientists and educators, who feared that the omission was deliberate and politically motivated. The Department of Education has denied this, stating that the major was removed from the list inadvertently, and that it would correct the omission.
In September 2006, evolutionary biology, along with exercise physiology, were added to the list of eligible majors.
Award Amount
The grant awards a maximum of $4,000 a year. The amount of the SMART Grant, when combined with a Pell Grant, may not exceed the student's cost of attendance. In addition, if the number of eligible students is large enough that payment of the full grant amounts would exceed the program appropriation in any fiscal year, then the amount of the grant to each eligible student may be ratably reduced.
According to the Binghamton University website, "The SMART Grant program is only funded through the 2010-11 academic year. The grant will not be available for the 2011-2012 year and beyond." The SMART Grant was dissolved in April 2011.
External links
SMART on studentaid.ed.gov
References
Student financial aid in the United States
Federal as
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https://en.wikipedia.org/wiki/Evidential%20reasoning
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Evidential reason or evidential reasoning may refer to:
Probabilistic logic, a combination of the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure
"Evidential reason", a type of reason (argument) in contrast to an "explanatory reason"
Evidential reasoning approach, in decision theory, an approach for multiple criteria decision analysis (MCDA) under uncertainty
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https://en.wikipedia.org/wiki/SMART%20Defense%20Scholarship%20Program
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The Science, Mathematics, And Research For Transformation (SMART) Defense Scholarship Program was tested as a program in 2005 under the Air
Force Office of Scientific Research. SMART was fully established by the National Defense Authorization Act for fiscal year 2006, and was assigned to the Navy Postgraduate School (NPS) as the managing agency in late 2005-early 2006.
The SMART Scholarship-for-Service Program is a Department of Defense (DoD) workforce development program created to address the growing gap between America and the rest of the world in the Science, Technology, Engineering and Mathematics (STEM) disciplines. SMART facilitates this goal by recruiting and retaining some of the best and brightest STEM candidates in the nation.
SMART is a DoD civilian scholarship-for-service program which is a part of the National Defense Education Program (NDEP). Like other NDEP programs, SMART is funded through the Office of the Secretary of Defense.
Requirements
The program is open to current and prospective students, including current DoD employees who meet the following requirements:
U.S. citizen (exceptions include: UK, New Zealand, Australia and Canada)
Minimum cumulative grade point average (GPA) of 3.0 on a 4.0 scale
Pursuing a degree in one of the 21 STEM disciplines
Able to participate in summer internships
Able to accept post-graduation employment within the DoD
Able to obtain and maintain a SECRET clearance
While these are the minimum requirements, the average GPA is well over 3.0, with the current 2017 cohort year averaging 3.7 on a 4.0 scale.
Benefits
Students who are accepted receive the following benefits:
Full tuition at any accredited college or university within the U.S.
Cash awards paid at a rate of $25,000 – $38,000 per year depending on prior educational experience
Health insurance allowance of $1,200 per academic year
Book allowance of $1,000 per academic year
Mentoring by a DoD sponsoring facility
Post-graduation employment placement within the DoD
Students are required to pay back the scholarship by working with the Department of Defense in a 1-1 year ratio. This work is a paid position within a specific DoD agency. Any changes to the original contract agreement made between the DoD and the award recipient may result in a longer duration of required employment. Multi-year recipients are also required to complete internships during the summers at their sponsoring facility at no additional compensation.
In the event that a recipient chooses to leave the program or fails to meet the program requirements, the recipient may be required to pay back in full all award funding that has been paid on their behalf to the DoD. At $25000-$38000 per year, plus orientation expenses, plus book and health insurance allowances, plus all tuition and fees, this amount can be very high. This payback amount is prorated depending on how long the recipient decides to stay in their post-graduation employment. As an example, i
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https://en.wikipedia.org/wiki/Singapore%20Mathematical%20Olympiad
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The Singapore Mathematical Olympiad (SMO) is a mathematics competition organised by the Singapore Mathematical Society. It comprises three sections, Junior, Senior and Open, each of which is open to all pre-university students studying in Singapore who meet the age requirements for the particular section. The competition is held annually, and the first round of each section is usually held in late May or early June. The second round is usually held in late June or early July.
History
The Singapore Mathematical Society (SMS) has been organising mathematical competitions since the 1950's, launching the first inter-school Mathematical Competition in 1956. The Mathematical Competition was renamed to Singapore Mathematical Olympiad in 1995.
In 2016, the SMS attempted to make the SMO more inviting to students by aligning questions more closely with school curriculum, although solutions still require considerable insight and creativity in addition to sound mathematical knowledge.
In 2020 and 2021, the written round (Round 1) in all sections were postponed to September due to the COVID-19 pandemic, while the invitational round (Round 2) in all sections were cancelled. The normal competition timeline was resumed in 2022.
Junior Section
There are two rounds in the Junior Section: a written round (Round 1) and an invitational round (Round 2).
The paper in Round 1 comprises 5 multiple-choice questions, each with five options, and 20 short answer questions. The Junior section is geared towards Lower Secondary students, and topics tested include number theory, combinatorics, geometry, algebra, and probability.
Beginning in 2006, a second round was added, based on the Senior Invitational Round, in the form of a 5-question, 3-hour long paper requiring full-length solutions. Only the top 10% of students from Round 1 are eligible to take Round 2.
Senior Section
There are two rounds in the Senior Section: a written round (Round 1) and an invitational round (Round 2).
The paper in Round 1 comprises 5 multiple-choice questions, each with five options, and 20 short answer questions. The Senior section is geared towards Upper Secondary students, and topics tested include number theory, combinatorics, geometry, algebra, and probability.
The second round, the Senior Invitational Round consists of a 5-question, 4-hour long paper requiring full-length solutions. Only the top 10% of students from Round 1 are eligible to take Round 2.
Open Section
Similar to the Senior Section, there are also two rounds, a written round (Round 1) and an invitational round (Round 2).
The paper in Round 1 comprises 25 short answer questions, and is geared towards pre-university students. Topics tested include number theory, combinatorics, geometry, algebra, calculus (occasionally), probability, but of a higher difficulty level than the Senior Section.
The Open Invitational Round consists of a 5-question, 4-hour long paper requiring full-length solutions, in which only the to
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https://en.wikipedia.org/wiki/Jonathan%20Partington
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Jonathan Richard Partington (born 4 February 1955) is an English mathematician who is Emeritus Professor of pure mathematics at the University of Leeds.
Education
Professor Partington was educated at Gresham's School, Holt, and Trinity College, Cambridge, where he completed his PhD thesis entitled "Numerical ranges and the Geometry of Banach Spaces" under the supervision of Béla Bollobás.
Career
Partington works in the area of functional analysis, sometimes applied to control theory, and is the author of several books in this area. He was formerly editor-in-chief of the Journal of the London Mathematical Society, a position he held jointly with his Leeds colleague John Truss.
Partington's extra-mathematical activities include the invention of the March March march, an annual walk starting at March, Cambridgeshire. He is also known as a writer or co-writer of some of the earliest British text-based computer games, including Acheton, Hamil, Murdac, Avon, Fyleet, Crobe, Sangraal, and SpySnatcher, which started life on the Phoenix computer system at the University of Cambridge Computer Laboratory. These are still available on the IF Archive.
Books
External links
Professor Jonathan R. Partington at the University of Leeds
1955 births
Living people
People from Holt, Norfolk
20th-century English mathematicians
21st-century English mathematicians
Mathematical analysts
People educated at Gresham's School
Alumni of Trinity College, Cambridge
Fellows of Pembroke College, Cambridge
Fellows of Fitzwilliam College, Cambridge
Academics of the University of Leeds
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https://en.wikipedia.org/wiki/Metacompact%20space
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In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.
A space is countably metacompact if every countable open cover has a point-finite open refinement.
Properties
The following can be said about metacompactness in relation to other properties of topological spaces:
Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.
Every metacompact space is orthocompact.
Every metacompact normal space is a shrinking space
The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.
An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane.
In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact and pseudocompact (see Watson).
Covering dimension
A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for which this is true. If no such minimal n exists, the space is said to be of infinite covering dimension.
See also
Compact space
Paracompact space
Normal space
Realcompact space
Pseudocompact space
Mesocompact space
Tychonoff space
Glossary of topology
References
.
P.23.
Properties of topological spaces
Compactness (mathematics)
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https://en.wikipedia.org/wiki/Orthocompact%20space
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In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement.
That is, given an open cover of the topological space, there is a refinement that is also an open cover, with the further property that at any point, the intersection of all open sets in the refinement containing that point is also open.
If the number of open sets containing the point is finite, then their intersection is definitionally open. That is, every point-finite open cover is interior preserving.
Hence, we have the following: every metacompact space, and in particular, every paracompact space, is orthocompact.
Useful theorems:
Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms.
Every closed subspace of an orthocompact space is orthocompact.
A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has an interior-preserving refinement that is an open cover of X.
The product X × [0,1] of the closed unit interval with an orthocompact space X is orthocompact if and only if X is countably metacompact. (B.M. Scott)
Every orthocompact space is countably orthocompact.
Every countably orthocompact Lindelöf space is orthocompact.
See also
References
P. Fletcher, W.F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, 1982, . Chap.V.
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Supercompact%20space
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In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements. Supercompactness and the related notion of superextension was introduced by J. de Groot in 1967.
Examples
By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not all) compact spaces are supercompact. The following are examples of supercompact spaces:
Compact linearly ordered spaces with the order topology and all continuous images of such spaces
Compact metrizable spaces (due originally to , see also )
A product of supercompact spaces is supercompact (like a similar statement about compactness, Tychonoff's theorem, it is equivalent to the axiom of choice.)
Properties
Some compact Hausdorff spaces are not supercompact; such an example is given by the Stone–Čech compactification of the natural numbers (with the discrete topology).
A continuous image of a supercompact space need not be supercompact.
In a supercompact space (or any continuous image of one), the cluster point of any countable subset is the limit of a nontrivial convergent sequence.
Notes
References
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Siegel%E2%80%93Walfisz%20theorem
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In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.
Statement
Define
where denotes the von Mangoldt function, and let φ denote Euler's totient function.
Then the theorem states that given any real number N there exists a positive constant CN depending only on N such that
whenever (a, q) = 1 and
Remarks
The constant CN is not effectively computable because Siegel's theorem is ineffective.
From the theorem we can deduce the following bound regarding the prime number theorem for arithmetic progressions: If, for (a, q) = 1, by we denote the number of primes less than or equal to x which are congruent to a mod q, then
where N, a, q, CN and φ are as in the theorem, and Li denotes the logarithmic integral.
References
Theorems in analytic number theory
Theorems about prime numbers
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https://en.wikipedia.org/wiki/Pseudonormal%20space
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In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Note the following:
Every normal space is pseudonormal.
Every pseudonormal space is regular.
An example of a pseudonormal Moore space that is not metrizable was given by , in connection with the conjecture that all normal Moore spaces are metrizable.
References
Topology
Properties of topological spaces
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https://en.wikipedia.org/wiki/Collectionwise%20Hausdorff%20space
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In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.
Here a subset being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of are isolated in ).
Properties
Every T1 space that is collectionwise Hausdorff is also Hausdorff.
Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset of , every singleton is closed in and the family of such singletons is a discrete family in .)
Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.
Remarks
References
Topology
Properties of topological spaces
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https://en.wikipedia.org/wiki/Volterra%20space
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In mathematics, in the field of topology, a topological space is said to be a Volterra space if any finite intersection of dense Gδ subsets is dense. Every Baire space is Volterra, but the converse is not true. In fact, any metrizable Volterra space is Baire.
The name refers to a paper of Vito Volterra in which he uses the fact that (in modern notation) the intersection of two dense G-delta sets in the real numbers is again dense.
References
Cao, Jiling and Gauld, D, "Volterra spaces revisited", J. Aust. Math. Soc. 79 (2005), 61–76.
Cao, Jiling and Junnila, Heikki, "When is a Volterra space Baire?", Topology Appl. 154 (2007), 527–532.
Gauld, D. and Piotrowski, Z., "On Volterra spaces", Far East J. Math. Sci. 1 (1993), 209–214.
Gruenhage, G. and Lutzer, D., "Baire and Volterra spaces", Proc. Amer. Math. Soc. 128 (2000), 3115–3124.
Volterra, V., "Alcune osservasioni sulle funzioni punteggiate discontinue", Giornale di Matematiche 19 (1881), 76–86.
Properties of topological spaces
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https://en.wikipedia.org/wiki/A-paracompact%20space
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In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not required to be open.
Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.
References
Compactness (mathematics)
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https://en.wikipedia.org/wiki/Perfect%20set
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In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of .
In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of and any neighborhood of the point, there is another point of that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of belongs to .
Note that the term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a Gδ space. As another possible source of confusion, also note that having the perfect set property is not the same as being a perfect set.
Examples
Examples of perfect subsets of the real line are the empty set, all closed intervals, the real line itself, and the Cantor set. The latter is noteworthy in that it is totally disconnected.
Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space. For instance, the set is perfect as a subset of the space but not perfect as a subset of the space .
Connection with other topological properties
Every topological space can be written in a unique way as the disjoint union of a perfect set and a scattered set.
Cantor proved that every closed subset of the real line can be uniquely written as the disjoint union of a perfect set and a countable set. This is also true more generally for all closed subsets of Polish spaces, in which case the theorem is known as the Cantor–Bendixson theorem.
Cantor also showed that every non-empty perfect subset of the real line has cardinality , the cardinality of the continuum. These results are extended in descriptive set theory as follows:
If X is a complete metric space with no isolated points, then the Cantor space 2ω can be continuously embedded into X. Thus X has cardinality at least . If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly .
If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X, and so X has cardinality at least .
See also
Dense-in-itself
Finite intersection property
Subspace topology
Notes
References
Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989.
Topology
Properties of topological spaces
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https://en.wikipedia.org/wiki/Bombieri%27s%20theorem
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Bombieri's theorem may refer to:
Bombieri–Vinogradov theorem, a result in analytic number theory
Schneider–Lang theorem for Bombieri's theorem on transcendental numbers
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https://en.wikipedia.org/wiki/Door%20space
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In mathematics, specifically in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
Properties and examples
Every door space is T0 (because if and are two topologically indistinguishable points, the singleton is neither open nor closed).
Every subspace of a door space is a door space. So is every quotient of a door space.
Every topology finer than a door topology on a set is also a door topology.
Every discrete space is a door space. These are the spaces without accumulation point, that is, whose every point is an isolated point.
Every space with exactly one accumulation point (and all the other point isolated) is a door space (since subsets consisting only of isolated points are open, and subsets containing the accumulation point are closed). Some examples are: (1) the one-point compactification of a discrete space (also called Fort space), where the point at infinity is the accumulation point; (2) a space with the excluded point topology, where the "excluded point" is the accumulation point.
Every Hausdorff door space is either discrete or has exactly one accumulation point. (To see this, if is a space with distinct accumulations points and having respective disjoint neighbourhoods and the set is neither closed nor open in )
An example of door space with more than one accumulation point is given by the particular point topology on a set with at least three points. The open sets are the subsets containing a particular point together with the empty set. The point is an isolated point and all the other points are accumulation points. (This is a door space since every set containing is open and every set not containing is closed.) Another example would be the topological sum of a space with the particular point topology and a discrete space.
Door spaces with no isolated point are exactly those with a topology of the form for some free ultrafilter on Such spaces are necessarily infinite.
There are exactly three types of connected door spaces :
a space with the excluded point topology;
a space with the included point topology;
a space with topology such that is a free ultrafilter on
See also
Notes
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/Richard%20S.%20Kayne
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Richard Stanley Kayne is Professor of Linguistics in the Linguistics Department at New York University.
Born in 1944, after receiving an A.B. in mathematics from Columbia College, New York City in 1964, he studied linguistics at the Massachusetts Institute of Technology, receiving his Ph.D. in 1969. He then taught at the University of Paris VIII (1969–1986), MIT (1986–1988) and the City University of New York (1988–1997), becoming Professor at New York University in 1997.
He has made prominent contributions to the study of the syntax of English and the Romance languages within the framework of transformational grammar. His theory of Antisymmetry has become part of the canon of the Minimalist syntax literature.
Publications
Movement and Silence, Oxford University Press, New York, 2005
(with Thomas Leu & Raffaella Zanuttini) Lasting Insights and Questions: An Annotated Syntax Reader, Wiley/Blackwell, Malde, Mass., 2014
References
External links
Homepage
Linguists from the United States
Generative linguistics
Syntacticians
Living people
Academic staff of Paris 8 University Vincennes-Saint-Denis
New York University faculty
Year of birth missing (living people)
Columbia College (New York) alumni
Fellows of the Linguistic Society of America
Silver professors
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https://en.wikipedia.org/wiki/Pseudocompact%20space
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In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.
Properties related to pseudocompactness
For a Tychonoff space X to be pseudocompact requires that every locally finite collection of non-empty open sets of X be finite. There are many equivalent conditions for pseudocompactness (sometimes some separation axiom should be assumed); a large number of them are quoted in Stephenson 2003. Some historical remarks about earlier results can be found in Engelking 1989, p. 211.
Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true.
As a consequence of the above result, every sequentially compact space is pseudocompact. The converse is true for metric spaces. As sequential compactness is an equivalent condition to compactness for metric spaces this implies that compactness is an equivalent condition to pseudocompactness for metric spaces also.
The weaker result that every compact space is pseudocompact is easily proved: the image of a compact space under any continuous function is compact, and every compact set in a metric space is bounded.
If Y is the continuous image of pseudocompact X, then Y is pseudocompact. Note that for continuous functions g : X → Y and h : Y → R, the composition of g and h, called f, is a continuous function from X to the real numbers. Therefore, f is bounded, and Y is pseudocompact.
Let X be an infinite set given the particular point topology. Then X is neither compact, sequentially compact, countably compact, paracompact nor metacompact (although it is orthocompact). However, since X is hyperconnected, it is pseudocompact. This shows that pseudocompactness doesn't imply any of these other forms of compactness.
For a Hausdorff space X to be compact requires that X be pseudocompact and realcompact (see Engelking 1968, p. 153).
For a Tychonoff space X to be compact requires that X be pseudocompact and metacompact (see Watson).
Pseudocompact topological groups
A relatively refined theory is available for pseudocompact topological groups. In particular, W. W. Comfort and Kenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).
Notes
See also
Compact space
Paracompact space
Normal space
Realcompact space
Metacompact space
Orthocompact space
Tychonoff space
References
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External links
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Properties of topological spaces
Compactness (mathematics)
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https://en.wikipedia.org/wiki/Realcompact%20space
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In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and it contains every point of its Stone–Čech compactification which is real (meaning that the quotient field at that point of the ring of real functions is the reals). Realcompact spaces have also been called Q-spaces, saturated spaces, functionally complete spaces, real-complete spaces, replete spaces and Hewitt–Nachbin spaces (named after Edwin Hewitt and Leopoldo Nachbin). Realcompact spaces were introduced by .
Properties
A space is realcompact if and only if it can be embedded homeomorphically as a closed subset in some (not necessarily finite) Cartesian power of the reals, with the product topology. Moreover, a (Hausdorff) space is realcompact if and only if it has the uniform topology and is complete for the uniform structure generated by the continuous real-valued functions (Gillman, Jerison, p. 226).
For example Lindelöf spaces are realcompact; in particular all subsets of are realcompact.
The (Hewitt) realcompactification υX of a topological space X consists of the real points of its Stone–Čech compactification βX. A topological space X is realcompact if and only if it coincides with its Hewitt realcompactification.
Write C(X) for the ring of continuous real-valued functions on a topological space X. If Y is a real compact space, then ring homomorphisms from C(Y) to C(X) correspond to continuous maps from X to Y. In particular the category of realcompact spaces is dual to the category of rings of the form C(X).
In order that a Hausdorff space X is compact it is necessary and sufficient that X is realcompact and pseudocompact (see Engelking, p. 153).
See also
Compact space
Paracompact space
Normal space
Pseudocompact space
Tychonoff space
References
Gillman, Leonard; Jerison, Meyer, "Rings of continuous functions". Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp.
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Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Locally%20Hausdorff%20space
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In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology.
Examples and sufficient conditions
Every Hausdorff space is locally Hausdorff.
There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
The line with two origins is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff.
The etale space for the sheaf of differentiable functions on a differential manifold is not Hausdorff, but it is locally Hausdorff.
Let be a set given the particular point topology with particular point The space is locally Hausdorff at since is an isolated point in and the singleton is a Hausdorff neighbourhood of For any other point any neighbourhood of it contains and therefore the space is not locally Hausdorff at
Properties
A space is locally Hausdorff exactly if it can be written as a union of Hausdorff open subspaces. And in a locally Hausdorff space each point belongs to some Hausdorff dense open subspace.
Every locally Hausdorff space is T1. The converse is not true in general. For example, an infinite set with the cofinite topology is a T1 space that is not locally Hausdorff.
Every locally Hausdorff space is sober.
If is a topological group that is locally Hausdorff at some point then is Hausdorff. This follows from the fact that if there exists a homeomorphism from to itself carrying to so is locally Hausdorff at every point, and is therefore T1 (and T1 topological groups are Hausdorff).
References
Separation axioms
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https://en.wikipedia.org/wiki/Mesocompact%20space
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In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many members of the refinement.
The following facts are true about mesocompactness:
Every compact space, and more generally every paracompact space is mesocompact. This follows from the fact that any locally finite cover is automatically compact-finite.
Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that points are compact, and hence any compact-finite cover is automatically point finite.
Notes
References
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Shrinking%20space
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In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding original open set.
Properties
The following facts are known about shrinking spaces:
Every shrinking space is normal.
Every shrinking space is countably paracompact.
In a normal space, every locally finite, and in fact, every point-finite open cover admits a shrinking.
Thus, every normal metacompact space is a shrinking space. In particular, every paracompact space is a shrinking space.
These facts are particularly important because shrinking of open covers is a common technique in the theory of differential manifolds and while constructing functions using a partition of unity.
See also
References
General topology, Stephen Willard, definition 15.9 p. 104
Topology
Properties of topological spaces
Topological spaces
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https://en.wikipedia.org/wiki/Hemicompact%20space
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In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.
Examples
Every compact space is hemicompact.
The real line is hemicompact.
Every locally compact Lindelöf space is hemicompact.
Properties
Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets.
Applications
If is a hemicompact space, then the space of all continuous functions to a metric space with the compact-open topology is metrizable. To see this, take a sequence of compact subsets of such that every compact subset of lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of ). Define pseudometrics
Then
defines a metric on which induces the compact-open topology.
See also
Compact space
Exhaustible by compact sets
Locally compact space
Lindelöf space
Notes
References
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Dice%20notation
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Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles) is a system to represent different combinations of dice in wargames and tabletop role-playing games using simple algebra-like notation such as d8+2.
Standard notation
In most tabletop role-playing games, die rolls required by the system are given in the form AdX. A and X are variables, separated by the letter d, which stands for die or dice. The letter d is most commonly lower-case, but some forms of notation use upper-case D (non-English texts can use the equivalent form of the first letter of the given language's word for "dice", but also often use the English "d").
A is the number of dice to be rolled (usually omitted if 1).
X is the number of faces of each dice.
For example, if a game calls for a roll of d4 or 1d4, it means "roll one 4-sided die."
If the final number is omitted, it is typically assumed to be a six, but in some contexts, other defaults are used.
3d6 would mean "roll three six-sided dice." Commonly, these dice are added together, but some systems could direct the player use them in some other way, such as choosing the best die rolled.
To this basic notation, an additive modifier can be appended, yielding expressions of the form AdX+B. The plus sign is sometimes replaced by a minus sign ("−") to indicate subtraction. B is a number to be added to the sum of the rolls. So, 1d20−10 would indicate a roll of a single 20-sided die with 10 being subtracted from the result. These expressions can also be chained (e.g. 2d6+1d8), though this usage is less common. Additionally, notation such as AdX−L is not uncommon, the L (or H, less commonly) being used to represent "the lowest result" (or "the highest result"). For instance, 4d6−L means a roll of 4 six-sided dice, dropping the lowest result. This application skews the probability curve towards the higher numbers, as a result a roll of 3 can only occur when all four dice come up 1 (probability ), while a roll of 18 results if any three dice are 6 (probability = ).
Rolling three or more dice gives a probability distribution that is approximately Gaussian, in accordance with the central limit theorem.
History
Miniatures wargamers began using dice in the shape of Platonic solids in the late 1960s and early ’70s, to obtain results that could not easily be produced on a conventional six-sided die. Dungeons & Dragons emerged in this milieu, and was the first game with widespread commercial availability to use such dice. In its earliest edition (1974), D&D had no standardized way to call for polyhedral die rolls or to refer to the results of such rolls. In some places the text gives a verbal instruction; in others, it only implies the roll to be made by describing the range of its results. For example, the spell sticks to snakes says, "From 2–16 snakes can be conjured (roll two eight-sided dice)." When only a range is listed, the exact method of rolling can be ambiguous. For
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https://en.wikipedia.org/wiki/Cochrane%20Lake%2C%20Alberta
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Cochrane Lake is a hamlet in southern Alberta under the jurisdiction of Rocky View County. Statistics Canada also recognizes a smaller portion of the hamlet as a designated place under the name of Cochrane Lake Subdivision.
Cochrane Lake is located approximately 45 km (23 mi) northwest of the City of Calgary and 1.6 km (1.0 mi) north of the Town of Cochrane on the west side of Highway 22. Cochrane Lake gets its name from Senator Matthew Henry Cochrane who in 1881 founded the Cochrane Ranche (later known as the British-American Ranche) which was a major producer of beef.
Cochrane Lake is also currently the site of a housing development, managed by property developer Monterra.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Cochrane Lake had a population of 767 living in 240 of its 252 total private dwellings, a change of from its 2016 population of 799. With a land area of , it had a population density of in 2021.
The population of Cochrane Lake according to the 2018 municipal census conducted by Rocky View County is 769. Rocky View County's 2013 municipal census counted a population of 792 in the Hamlet of Cochrane Lake, a 226% change from its 2006 municipal census population of 243.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Karamitsanis, Aphrodite (1992). Place Names of Alberta – Volume II, Southern Alberta, University of Calgary Press, Calgary, Alberta.
Read, Tracey (1983). Acres and Empires – A History of the Municipal District of Rocky View, Calgary, Alberta.
Rocky View County
Hamlets in Alberta
Calgary Region
Designated places in Alberta
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https://en.wikipedia.org/wiki/List%20of%20NHL%20statistical%20leaders%20by%20country%20of%20birth
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This is a list of National Hockey League statistical leaders by country of birth, sorted by total points. The top ten players from each country are included. Statistics are current through the end of the 2022–23 NHL season and players currently playing in the National Hockey League are marked in boldface.
All players are listed by the current country of the players' birth location, regardless of their citizenship, where they were trained in hockey or what country they represented internationally.
Country
Canada
Czech Republic
Russia
Slovakia
Finland
United States
Sweden
Slovenia
United Kingdom
Ukraine
Serbia
Germany
Austria
France
Switzerland
Lithuania
Norway
Latvia
Denmark
Kazakhstan
Paraguay
Poland
Republic of China (Taiwan)
South Korea
Belarus
Netherlands
Brazil
Estonia
Brunei
Italy
Venezuela
Uzbekistan
Haiti
South Africa
Tanzania
Jamaica
Australia
Lebanon
Japan
Nigeria
Bulgaria
Indonesia
Belgium
Croatia
Bahamas
See also
List of NHL statistical leaders
List of countries with their first National Hockey League player
Notes
Almost all players on this list from Russia, Ukraine, Latvia, Lithuania, Kazakhstan, and Belarus were born in the Soviet Union – in the Russian SFSR, Ukrainian SSR, Latvian SSR, Lithuanian SSR, Kazakh SSR, and Byelorussian SSR respectively. The Soviet Union officially dissolved at the end of 1991. Many of these players have represented both the Soviet Union and their respective nation in international competitions.
Almost all players on this list from the Czech Republic or Slovakia were born in Czechoslovakia. Czechoslovakia officially dissolved at the end of 1992. Many of these players have represented both Czechoslovakia and their respective nation in international competitions.
Almost every player on this list from Germany was born in West Germany. The exceptions are Mikhail Grabovski, born in East Germany, and Walt Tkaczuk, born shortly after World War II in the portion of Allied-occupied Germany that became West Germany in 1949. West Germany and East Germany reunited in 1990. Some of these players have represented both West Germany and Germany in international competitions.
External links
Career stats from NHL.com
The Internet Hockey Database
Legends of Hockey
NHL Finland
Birth Countries of NHL Players, databasehockey.com
Statistical leaders by country
Statistical leaders by country
Ice hockey
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https://en.wikipedia.org/wiki/Cuzick%E2%80%93Edwards%20test
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In statistics, the Cuzick–Edwards test is a significance test whose aim is to detect the possible clustering of sub-populations within a clustered or non-uniformly-spread overall population. Possible applications of the test include examining the spatial clustering of childhood leukemia and lymphoma within the general population, given that the general population is spatially clustered.
The test is based on:
using control locations within the general population as the basis of a second or "control" sub-population in addition to the original "case" sub-population;
using "nearest-neighbour" analyses to form statistics based on either:
the number of other "cases" among the neighbours of each case;
the number "cases" which are nearer to each given case than the k-th nearest "control" for that case.
An example application of this test was to spatial clustering of leukaemias and lymphomas among young people in New Zealand.
See also
Clustering (demographics)
References
Further reading
Epidemiology
Medical statistics
Statistical tests
Spatial analysis
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https://en.wikipedia.org/wiki/Rearrangement
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Rearrangement may refer to:
Chemistry
Rearrangement reaction
Mathematics
Rearrangement inequality
The Riemann rearrangement theorem, also called the Riemann series theorem
see also Lévy–Steinitz theorem
A permutation of the terms of a conditionally convergent series
Genetics
Chromosomal rearrangements, such as:
Translocations
Ring chromosomes
Chromosomal inversions
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https://en.wikipedia.org/wiki/Bitopological%20space
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In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is and the topologies are and then the bitopological space is referred to as . The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.
Continuity
A map from a bitopological space to another bitopological space is called continuous or sometimes pairwise continuous if is continuous both as a map from to and as map from to .
Bitopological variants of topological properties
Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.
A bitopological space is pairwise compact if each cover of with , contains a finite subcover. In this case, must contain at least one member from and at least one member from
A bitopological space is pairwise Hausdorff if for any two distinct points there exist disjoint and with and .
A bitopological space is pairwise zero-dimensional if opens in which are closed in form a basis for , and opens in which are closed in form a basis for .
A bitopological space is called binormal if for every -closed and -closed sets there are -open and -open sets such that , and
Notes
References
Kelly, J. C. (1963). Bitopological spaces. Proc. London Math. Soc., 13(3) 71–89.
Reilly, I. L. (1972). On bitopological separation properties. Nanta Math., (2) 14–25.
Reilly, I. L. (1973). Zero dimensional bitopological spaces. Indag. Math., (35) 127–131.
Salbany, S. (1974). Bitopological spaces, compactifications and completions. Department of Mathematics, University of Cape Town, Cape Town.
Kopperman, R. (1995). Asymmetry and duality in topology. Topology Appl., 66(1) 1--39.
Fletcher. P, Hoyle H.B. III, and Patty C.W. (1969). The comparison of topologies. Duke Math. J.,36(2) 325–331.
Dochviri, I., Noiri T. (2015). On some properties of stable bitopological spaces. Topol. Proc., 45 111–119.
Topology
Topological spaces
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https://en.wikipedia.org/wiki/Stinespring%20dilation%20theorem
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In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra A as a composition of two completely positive maps each of which has a special form:
A *-representation of A on some auxiliary Hilbert space K followed by
An operator map of the form T ↦ V*TV.
Moreover, Stinespring's theorem is a structure theorem from a C*-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms.
Formulation
In the case of a unital C*-algebra, the result is as follows:
Theorem. Let A be a unital C*-algebra, H be a Hilbert space, and B(H) be the bounded operators on H. For every completely positive
there exists a Hilbert space K and a unital *-homomorphism
such that
where is a bounded operator. Furthermore, we have
Informally, one can say that every completely positive map can be "lifted" up to a map of the form .
The converse of the theorem is true trivially. So Stinespring's result classifies completely positive maps.
Sketch of proof
We now briefly sketch the proof. Let . For , define
and extend by semi-linearity to all of K. This is a Hermitian sesquilinear form because is compatible with the * operation. Complete positivity of is then used to show that this sesquilinear form is in fact positive semidefinite. Since positive semidefinite Hermitian sesquilinear forms satisfy the Cauchy–Schwarz inequality, the subset
is a subspace. We can remove degeneracy by considering the quotient space . The completion of this quotient space is then a Hilbert space, also denoted by . Next define and . One can check that and have the desired properties.
Notice that is just the natural algebraic embedding of H into K. One can verify that holds. In particular holds so that is an isometry if and only if . In this case H can be embedded, in the Hilbert space sense, into K and , acting on K, becomes the projection onto H. Symbolically, we can write
In the language of dilation theory, this is to say that is a compression of . It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some *-homomorphism.
Minimality
The triple (, V, K) is called a Stinespring representation of Φ. A natural question is now whether one can reduce a given Stinespring representation in some sense.
Let K1 be the closed linear span of (A) VH. By property of *-representations in general, K1 is an invariant subspace of (a) for all a. Also, K1 contains VH. Define
We can compute directly
and if k and ℓ lie in K1
So (1, V, K1) is also a Stinespring representation of Φ and has the additional property that K1 is the closed linear span of (A) V H. Such a representation is called a minimal Stinespring representation.
Uniqueness
Let (1, V
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https://en.wikipedia.org/wiki/Free-standing%20Mathematics%20Qualifications
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Free-standing Mathematics Qualifications (FSMQ) are a suite of mathematical qualifications available at levels 1 to 3 in the National Qualifications Framework – Foundation, Intermediate and Advanced.
Educational standard
They bridge a gap between GCSE and A-Level Mathematics. The advanced course is especially ideal for pupils who do not find GCSE maths particularly challenging and who often have extra time in their second year of GCSEs, having taken their Maths GCSE a year early. The qualification is commonly offered in private schools and is useful in allowing pupils to determine whether or not to pursue maths in subsequent stages of their schooling.
The highest grade achievable is an A. An FSMQ Unit at Advanced level is roughly equivalent to a single AS module with candidates receiving 10 UCAS points for an A grade. Intermediate level is equivalent to a GCSE in Mathematics. Coursework is often a key part of the FSMQ, but is sometimes omitted depending on the examining board.
Exam boards
The only examining board currently offering FSMQs is OCR.
Edexcel withdrew the qualification, the last exam being held in June 2004. AQA also withdrew the pilot advanced level FSMQ, the last exam being in June 2018, and a final re-sit opportunity in June 2019.
Examples
Additional Mathematics/AdMaths (OCR) (No coursework)
References
External links
Edexcel
Oxford, Cambridge and RSA (OCR)
Assessment and Qualifications Alliance (AQA)
Qualifications and Curriculum Authority (QCA)
Educational qualifications in the United Kingdom
Mathematics education in the United Kingdom
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https://en.wikipedia.org/wiki/Contact%20process%20%28mathematics%29
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The contact process is a stochastic process used to model population growth on the set of sites of a graph in which occupied sites become vacant at a constant rate, while vacant sites become occupied at a rate proportional to the number of occupied neighboring sites. Therefore, if we denote by the proportionality constant, each site remains occupied for a random time period which is exponentially distributed parameter 1 and places descendants at every vacant neighboring site at times of events of a Poisson process parameter during this period. All processes are independent of one another and of the random period of time sites remains occupied.
The contact process can also be interpreted as a model for the spread of an infection by
thinking of particles as a bacterium spreading over individuals that are positioned at the sites of , occupied sites correspond to infected individuals, whereas vacant correspond to healthy ones.
The main quantity of interest is the number of particles in the process, say , in the first interpretation, which corresponds to the number of infected sites in the second one. Therefore, the process survives whenever the number of particles is positive for all times, which corresponds to the case that there are always infected individuals in the second one. For any infinite graph there exists a positive and finite critical value so that if then survival of the process starting from a finite number of particles occurs with positive probability, while if their extinction is almost certain. Note that by and the infinite monkey theorem, survival of the process is equivalent to , as , whereas extinction is equivalent to , as , and therefore, it is natural to ask about the rate at which when the process survives.
Mathematical definition
If the state of the process at time is , then a site in is occupied, say by a particle, if and vacant if .
The contact process is a continuous-time Markov process with state space , where is a finite or countable graph, usually , and a special case of an interacting particle system.
More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site ,
where the sum is over all the neighbors of in . This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and vice versa).
Connection to percolation
The contact process is a stochastic process that is closely connected to percolation theory. Ted Harris (1974) noted that the contact process on when infections and recoveries can occur only in discrete times corresponds to one-step-at-a-time bond percolation on the graph obtained by orienting each edge of in the direction of increasing coordinate-value.
The law of large numbers on the integers
A law of large numbers for the number of particles in the process on the integers informally means that for all large , is approximately equal to for some positive constant . Harris
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https://en.wikipedia.org/wiki/James%20Stewart%20%28mathematician%29
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James Drewry Stewart, (March 29, 1941December 3, 2014) was a Canadian mathematician, violinist, and professor emeritus of mathematics at McMaster University. Stewart is best known for his series of calculus textbooks used for high school, college, and university level courses.
Career
Stewart received his master of science at Stanford University and his doctor of philosophy from the University of Toronto in 1967. He worked for two years as a postdoctoral fellow at the University of London, where his research focused on harmonic and functional analysis. His books are standard textbooks in universities in many countries. One of his most well-known textbooks is Calculus: Early Transcendentals (1995), a set of textbooks which is accompanied by a website for students.
Stewart was also a violinist, and a former member of the Hamilton Philharmonic Orchestra.
Integral House
From 2003 to 2009 a house designed by Brigitte Shim and Howard Sutcliffe was constructed for Stewart in the Rosedale neighbourhood of Toronto at a cost of $32 million. He paid an additional $5.4 million for the existing house and lot which was torn down to make room for his new home. Called Integral House (a reference to its curved walls, and their similarity to the mathematical integral symbol), the house includes a concert hall that seats 150. Stewart has said, "My books and my house are my twin legacies. If I hadn't commissioned the house I'm not sure what I would have spent the money on." Glenn Lowry, director of the Museum of Modern Art, called the house "one of the most important private houses built in North America in a long time."
Personal life and political activism
Stewart was gay and involved in LGBT activism. According to Joseph Clement, a documentary filmmaker who is working on a film about Stewart and Integral House, Stewart brought gay rights activist George Hislop to speak at McMaster in the early 1970s, when the LGBT liberation movement was in its infancy, and was involved in protests and demonstrations.
Death
In the summer of 2013, Stewart was diagnosed with multiple myeloma, a blood cancer. He died on December 3, 2014, aged 73.
Honours
In 2015, he was posthumously awarded the Meritorious Service Cross.
References
Further reading
Article about Stewart's "Integral House".
External links
Stewart Calculus Official Biography
The house that math built by Katie Daubs at the Toronto Star
Integral Man , a documentary about Stewart and Integral House
1941 births
2014 deaths
Deaths from cancer in Ontario
Deaths from multiple myeloma
Canadian mathematicians
Canadian textbook writers
Canadian classical violinists
Canadian male violinists and fiddlers
Academic staff of McMaster University
Stanford University alumni
Academic staff of the University of Toronto
University of Toronto alumni
Canadian LGBT rights activists
Musicians from Toronto
Scientists from Toronto
Writers from Toronto
20th-century Canadian violinists and fiddlers
Canadian gay writers
Canadian
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https://en.wikipedia.org/wiki/Manjul%20Bhargava
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Manjul Bhargava (born 8 August 1974) is a Canadian-American mathematician. He is the Brandon Fradd, Class of 1983, Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds Adjunct Professorships at the Tata Institute of Fundamental Research, the Indian Institute of Technology Bombay, and the University of Hyderabad. He is known primarily for his contributions to number theory.
Bhargava was awarded the Fields Medal in 2014. According to the International Mathematical Union citation, he was awarded the prize "for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves".
Education and career
Bhargava was born to an Indian family in Hamilton, Ontario, Canada, but grew up and attended school primarily in Long Island, New York. His mother Mira Bhargava, a mathematician at Hofstra University, was his first mathematics teacher. He completed all of his high school math and computer science courses by age 14. He attended Plainedge High School in North Massapequa, and graduated in 1992 as the class valedictorian. He obtained his AB from Harvard University in 1996. For his research as an undergraduate, he was awarded the 1996 Morgan Prize. Bhargava went on to pursue graduate studies at Princeton University, where he completed a doctoral dissertation titled "Higher composition laws" under the supervision of Andrew Wiles and received his PhD in 2001, with the support of a Hertz Fellowship. He was a visiting scholar at the Institute for Advanced Study in 2001–02, and at Harvard University in 2002–03. Princeton appointed him as a tenured Full Professor in 2003. He was appointed to the Stieltjes Chair in Leiden University in 2010.
Bhargava is also an accomplished tabla player, having studied under gurus such as Zakir Hussain. He also studied Sanskrit from his grandfather Purushottam Lal Bhargava, a well-known scholar of Sanskrit and ancient Indian history. He is an admirer of Sanskrit poetry.
Career and research
Bhargava’s PhD thesis generalized Gauss's classical law for composition of binary quadratic forms to many other situations. One major use of his results is the parametrization of quartic and quintic orders in number fields, thus allowing the study of asymptotic behavior of arithmetic properties of these orders and fields.
His research also includes fundamental contributions to the representation theory of quadratic forms, to interpolation problems and p-adic analysis, to the study of ideal class groups of algebraic number fields, and to the arithmetic theory of elliptic curves. A short list of his specific mathematical contributions are:
Fourteen new Gauss-style composition laws.
Determination of the asymptotic density of discriminants of quartic and quintic number fields.
Proofs of the first-known cases of the Cohen-Lenstra-Martinet heuristics for class groups.
Proof of the 15 the
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https://en.wikipedia.org/wiki/Isodynamic%20point
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In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid(as well as its orthocenter, its incenter, and its circumcenter, which are concurrent); every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by .
Distance ratios
The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If and are the isodynamic points of a triangle then the three products of distances are equal. The analogous equalities also hold for Equivalently to the product formula, the distances and are inversely proportional to the corresponding triangle side lengths and
and are the common intersection points of the three circles of Apollonius associated with triangle of a triangle the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices. Hence, line is the common radical axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment is the Lemoine line, which contains the three centers of the circles of Apollonius.
Transformations
The isodynamic points and of a triangle may also be defined by their properties with respect to transformations of the plane, and particularly with respect to inversions and Möbius transformations (products of multiple inversions).
Inversion of the triangle with respect to an isodynamic point transforms the original triangle into an equilateral triangle.
Inversion with respect to the circumcircle of triangle leaves the triangle invariant but transforms one isodynamic point into the other one.
More generally, the isodynamic points are equivariant under Möbius transformations: the unordered pair of isodynamic points of a transformation of is equal to the same transformation applied to the pair The individual isodynamic points are fixed by Möbius transformations that map the interior of the circumcircle of to the interior of the circumcircle of the transformed triangle, and swapped by transformations that exchange the interior and exterior of the circumcircle.
Angles
As well as being the intersections of the circles of Apollonius, each isodynamic point is the intersection points of another triple of circles. The first isodynam
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https://en.wikipedia.org/wiki/Overlap%20%28term%20rewriting%29
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In mathematics, computer science and logic, overlap, as a property of the reduction rules in term rewriting system, describes a situation where a number of different reduction rules specify potentially contradictory ways of reducing a reducible expression, also known as a redex, within a term.
More precisely, if a number of different reduction rules share function symbols on the left-hand side, overlap can occur. Often we do not consider trivial overlap with a redex and itself.
Examples
Consider the term rewriting system defined by the following reduction rules:
The term can be reduced via ρ1 to yield , but it can also be reduced via ρ2 to yield . Note how the redex is contained in the redex . The result of reducing different redexes is described in a what is known as a critical pair; the critical pair arising out of this term rewriting system is .
Overlap may occur with fewer than two reduction rules.
Consider the term rewriting system defined by the following reduction rule:
The term has overlapping redexes, which can be either applied to the innermost occurrence or to the outermost occurrence of the term.
References
Rewriting systems
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https://en.wikipedia.org/wiki/Weber%27s%20theorem%20%28Algebraic%20curves%29
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In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following.
Consider two non-singular curves C and having the same genus g > 1. If there is a rational correspondence φ between C and , then φ is a birational transformation.
References
Further reading
External links
Algebraic curves
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Heteroclinic%20cycle
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In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching trajectories spend longer and longer periods of time in a neighbourhood of successive equilibria.
In generic dynamical systems heteroclinic connections are of high co-dimension, that is, they will not persist if parameters are varied.
Robust heteroclinic cycles
A robust heteroclinic cycle is one which persists under small changes in the underlying dynamical system. Robust cycles often arise in the presence of symmetry or other constraints which force the existence of invariant hyperplanes. A prototypical example of a robust heteroclinic cycle is the Guckenheimer–Holmes cycle. This cycle has also been studied in the context of rotating convection, and as three competing species in population dynamics.
See also
Heteroclinic bifurcation
Heteroclinic network
References
Guckenheimer J and Holmes, P, 1988, Structurally Stable Heteroclinic Cycles, Math. Proc. Cam. Phil. Soc. 103: 189-192.
F. M. Busse and K. E. Heikes (1980), Convection in a rotating layer: A simple case of turbulence, Science, 208, 173–175.
R. May and W. Leonard (1975), Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29, 243–253.
External links
Dynamical systems
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https://en.wikipedia.org/wiki/Eric%20Lengyel
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Eric Lengyel is a computer scientist specializing in game engine development, computer graphics, and geometric algebra. He holds a Ph.D. in computer science from the University of California, Davis and a master's degree in mathematics from Virginia Tech.
Lengyel is an expert in font rendering technology for 3D applications and is the inventor of the Slug font rendering algorithm, which allows glyphs to be rendered directly from outline data on the GPU with full resolution independence. Lengyel is also the inventor of the Transvoxel algorithm, which is used to seamlessly join multiresolution voxel data at boundaries between different levels of detail that have been triangulated with the Marching cubes algorithm.
Among his many written contributions to the field of game development, Lengyel is the author of the four-volume book series Foundations of Game Engine Development. The first volume, covering the mathematics of game engines, was published in 2016 and is now known for its unique treatment of Grassmann algebra. The second volume, covering a wide range of rendering topics, was published in 2019. Lengyel is also the author of the textbook Mathematics for 3D Game Programming and Computer Graphics and the editor for the three-volume Game Engine Gems book series.
Lengyel founded Terathon Software in 2000 and is currently President and Chief Technology Officer at the company, where he leads development of the C4 Engine. He has previously worked in the advanced technology group at Naughty Dog, and before that was the lead programmer for the fifth installment of Sierra's popular RPG adventure series Quest for Glory. In addition to the C4 Engine, Lengyel is the creator of the Open Data Description Language (OpenDDL) and the Open Game Engine Exchange (OpenGEX) file format.
Lengyel is originally from Reynoldsburg, Ohio, but now lives in Lincoln, California. He is a cousin of current Ohioan and "Evolution of Dance" creator Judson Laipply.
Games
Eric Lengyel is credited on the following games:
Heavenly Sword (2007), Sony Computer Entertainment America, Inc.
Ratchet & Clank Future: Tools of Destruction (2007), Sony Computer Entertainment America, Inc.
Warhawk (2007), Sony Computer Entertainment America, Inc.
Formula One Championship Edition (2006), Sony Computer Entertainment America, Inc.
MotorStorm (2006), Sony Computer Entertainment Incorporated
Resistance: Fall of Man (2006), Sony Computer Entertainment Incorporated
Jak 3 (2004), Sony Computer Entertainment America, Inc.
Quest for Glory V: Dragon Fire (1998), Sierra On-Line, Inc.
Patents
Eric Lengyel is the primary inventor on the following patents:
Method for rendering resolution-independent shapes directly from outline control points
Graphics processing apparatus, graphics library module and graphics processing method
References
External links
List of publications by Eric Lengyel
Moby Games rap sheet
Year of birth missing (living people)
Living people
University of California,
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