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https://en.wikipedia.org/wiki/Jonathan%20Borwein
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Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics.
Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. He was not associated with MathResources at the time of his death.
Borwein was also an expert on the number pi and especially its computation.
Early life and education
Borwein was born in St. Andrews, Scotland in 1951 into a Jewish family. His father was mathematician David Borwein, with whom he collaborated. His brother Peter Borwein was also a mathematician.
Borwein was married to Judith, and had three daughters.
He received his B.A. (Honours Math) from University of Western Ontario in 1971, and his D.Phil. from Oxford University in 1974 as a Rhodes Scholar at Jesus College.
Career
Prior to joining Simon Fraser University in 1993, he worked at Dalhousie University (1974–91), Carnegie-Mellon (1980–82) and the University of Waterloo (1991–93).
He was Shrum Professor of Science (1993–2003) and a Canada Research Chair in Information Technology (2001–08) at Simon Fraser University, where he was founding Director of the Centre for Experimental and Constructive Mathematics and developed the Inverse Symbolic Calculator together with his brother and Simon Plouffe. In 2004, he (re-)joined the Faculty of Computer Science at Dalhousie University as a Canada Research Chair in Distributed and Collaborative Research, cross-appointed in Mathematics, while preserving an adjunct appointment at Simon Fraser.
Borwein was Governor at large of the Mathematical Association of America (2004–07), was president of the Canadian Mathematical Society (2000–02) and chair of (the Canadian National Science Library) NRC-CISTI Advisory Board (2000–2003). He served as chair of various NATO scientific programs. He was also Chair of the Scientific Advisory Committee of the Australian Mathematical Sciences Institute (AMSI). He chaired the Canadian HPC consortium, later Compute Canada, and the International Mathematical Union's Committee on Electronic Information and Communications (2002–2008).
Awards
Borwein received various awards including the Chauvenet Prize (1993), Fellowship in the Royal Society of Canada (1994), Fellowship in the American Association for the Advancement of Science (2002), an honorary degree from Limoges (1999), and foreign membership in the Bulgarian Academy of Sciences (2
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https://en.wikipedia.org/wiki/Climbing%20Mount%20Improbable
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Climbing Mount Improbable is a 1996 popular science book by Richard Dawkins. The book is about probability and how it applies to the theory of evolution. It is designed to debunk claims by creationists about the probability of naturalistic mechanisms like natural selection.
The main metaphorical treatment is of a geographical landscape upon which evolution can ascend only gradually and cannot climb cliffs (that is known as an adaptive landscape). In the book, Dawkins gives ideas about a seemingly complex mechanism coming about from many gradual steps that were previously unseen.
The book grew out of the annual Royal Institution Christmas Lectures, which Dawkins delivered in 1991 (see Growing Up in the Universe). It is illustrated by Dawkins' wife at the time, Lalla Ward, and is dedicated to Robert Winston, "a good doctor and a good man".
References
1996 non-fiction books
Books about evolution
Books by Richard Dawkins
Criticism of creationism
English-language books
English non-fiction books
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https://en.wikipedia.org/wiki/Pairing%20function
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In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.
Definition
A pairing function is a bijection
More generally, a pairing function on a set A is a function that maps each pair of elements from A into an element of A, such that any two pairs of elements of A are associated with different elements of A, or a bijection from to A.
Hopcroft and Ullman pairing function
Hopcroft and Ullman (1979) define the following pairing function: , where . This is the same as the Cantor pairing function below, shifted to exclude 0 (i.e., , , and ).
Cantor pairing function
The Cantor pairing function is a primitive recursive pairing function
defined by
where .
It can also be expressed as .
It is also strictly monotonic w.r.t. each argument, that is, for all , if , then ; similarly, if , then .
The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to and we often denote the resulting number as .
This definition can be inductively generalized to the
for as
with the base case defined above for a pair:
Inverting the Cantor pairing function
Let be an arbitrary natural number. We will show that there exist unique values such that
and hence that the function is invertible. It is helpful to define some intermediate values in the calculation:
where is the triangle number of . If we solve the quadratic equation
for as a function of , we get
which is a strictly increasing and continuous function when is non-negative real. Since
we get that
and thus
where is the floor function.
So to calculate and from , we do:
Since the Cantor pairing function is invertible, it must be one-to-one and onto.
Examples
To calculate :
,
,
,
,
,
so .
To find and such that :
,
,
,
,
,
,
so ;
,
,
,
so ;
,
so ;
,
so ; thus .
Derivation
The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability. The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane.
A pairing function can usually be defined inductively – that is, given the th pair, what is the th pair? The way Cantor's function progresses diagonally across the plane can be expressed as
.
The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step
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https://en.wikipedia.org/wiki/Lucas%E2%80%93Carmichael%20number
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In mathematics, a Lucas–Carmichael number is a positive composite integer n such that
if p is a prime factor of n, then p + 1 is a factor of n + 1;
n is odd and square-free.
The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1).
They are named after Édouard Lucas and Robert Carmichael.
Properties
The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.
The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.
The smallest Lucas–Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.
It is not known whether any Lucas–Carmichael number is also a Carmichael number.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers. If we let denote the number of Lucas–Carmichael numbers up to , Wright showed that there exists a positive constant such that
.
List of Lucas–Carmichael numbers
The first few Lucas–Carmichael numbers and their prime factors are listed below.
References
External links
Eponymous numbers in mathematics
Integer sequences
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https://en.wikipedia.org/wiki/GGP
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GGP may refer to:
Gan–Gross–Prasad conjecture, a conjecture in number theory
Garden Grove Playhouse, a former theater group in Orange County, California
Gateway-to-Gateway Protocol
General game playing, in artificial intelligence
General Growth Properties, since 2018 part of Brookfield Properties
Generations and Gender Programme of the United Nations Economic Commission for Europe
Georgia Green Party, a state-level political party in the U.S.
Global Goods Partners, a fair-trade nonprofit organization
Golden Gate Park in San Francisco
Gondwana Ganatantra Party, an Indian political party
Good guidance practice
Guernsey pound, the currency of Guernsey
Logansport/Cass County Airport, in Indiana
nProtect GameGuard Personal 2007, security software
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https://en.wikipedia.org/wiki/Kac%E2%80%93Moody%20algebra
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In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting.
A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated that Rogers–Ramanujan identities can be derived in a similar fashion.
History of Kac–Moody algebras
The initial construction by Élie Cartan and Wilhelm Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent. In 1966 Jean-Pierre Serre showed that relations of Claude Chevalley and Harish-Chandra, with simplifications by Nathan Jacobson, give a defining presentation for the Lie algebra. One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally positive definite.
"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A. J. Coleman
In his 1967 thesis, Robert Moody considered Lie algebras whose Cartan matrix is no longer positive definite. This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, Z-graded Lie algebras were being studied in Moscow where I. L. Kantor introduced and studied a general class of Lie algebras including what eventually became known as Kac–Moody algebras. Victor Kac was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras evolved. An account of the subject, which also includes works of many others is given in (Kac 1990). See also (Seligman 1987).
Introduction
Given an n×n generalized Cartan matrix , one can construct a Lie algebra defined by generators , , and and relations given by:
for all ;
;
;
, where is the Kronecker delta;
If (so ) then and , where is the adjoint representation of .
Under a "symmetrizability" assumption, identifies with the derived subalgebra of the affine Kac-Moody algebra defined below.
Definition
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https://en.wikipedia.org/wiki/Current%20algebra
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Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra.
History
The original current algebra, proposed in 1964 by Murray Gell-Mann, described weak and electromagnetic currents of the strongly interacting particles, hadrons, leading to the Adler–Weisberger formula and other important physical results. The basic concept, in the era just preceding quantum chromodynamics, was that even without knowing the Lagrangian governing hadron dynamics in detail, exact kinematical information – the local symmetry – could still be encoded in an algebra of
currents.
The commutators involved in current algebra amount to an infinite-dimensional extension of the Jordan map, where the quantum fields represent infinite arrays of oscillators.
Current algebraic techniques are still part of the shared background of particle physics when analyzing symmetries and indispensable in discussions of the Goldstone theorem.
Example
In a non-Abelian Yang–Mills symmetry, where and are flavor-current and axial-current 0th components (charge densities), respectively, the paradigm of a current algebra is
and
where are the structure constants of the Lie algebra. To get meaningful expressions, these must be normal ordered.
The algebra resolves to a direct sum of two algebras, and , upon defining
whereupon
Conformal field theory
For the case where space is a one-dimensional circle, current algebras arise naturally as a central extension of the loop algebra, known as Kac–Moody algebras or, more specifically, affine Lie algebras. In this case, the commutator and normal ordering can be given a very precise mathematical definition in terms of integration contours on the complex plane, thus avoiding some of the formal divergence difficulties commonly encountered in quantum field theory.
When the Killing form of the Lie algebra is contracted with the current commutator, one obtains the energy–momentum tensor of a two-dimensional conformal field theory. When this tensor is expanded as a Laurent series, the resulting algebra is called the Virasoro algebra. This calculation is known as the Sugawara construction.
The general case is formalized as the vertex operator algebra.
See also
Affine Lie algebra
Chiral model
Jordan map
Virasoro algebra
Vertex operator algebra
Kac–Moody algebra
Notes
References
Sample.
Quantum field theory
Lie algebras
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https://en.wikipedia.org/wiki/Cartan%20matrix
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In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.
Lie algebras
A (symmetrizable) generalized Cartan matrix is a square matrix with integer entries such that
For diagonal entries, .
For non-diagonal entries, .
if and only if
can be written as , where is a diagonal matrix, and is a symmetric matrix.
For example, the Cartan matrix for G2 can be decomposed as such:
The third condition is not independent but is really a consequence of the first and fourth conditions.
We can always choose a D with positive diagonal entries. In that case, if S in the above decomposition is positive definite, then A is said to be a Cartan matrix.
The Cartan matrix of a simple Lie algebra is the matrix whose elements are the scalar products
(sometimes called the Cartan integers) where ri are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for is a root which is a linear combination of the simple roots ri and rj with a positive coefficient for rj and so, the coefficient for ri has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let and . Because the simple roots span a Euclidean space, S is positive definite.
Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See Kac–Moody algebra for more details).
Classification
An matrix A is decomposable if there exists a nonempty proper subset such that whenever and . A is indecomposable if it is not decomposable.
Let A be an indecomposable generalized Cartan matrix. We say that A is of finite type if all of its principal minors are positive, that A is of affine type if its proper principal minors are positive and A has determinant 0, and that A is of indefinite type otherwise.
Finite type indecomposable matrices classify the finite dimensional simple Lie algebras (of types ), while affine type indecomposable matrices classify the affine Lie algebras (say over some algebraically closed field of characteristic 0).
Determinants of the Cartan matrices of the simple Lie algebras
The determinants of the Cartan matrices of the simple Lie algebras are given in the following table (along with A1=B1=C1, B2=C2, D3=A3, D2=A1A1, E5=D5, E4=A4, and E3=A2A1).
Another property of this determinant is that it is equal to the index of the associated root system, i.e. it is equal to where denote the weight lattice and root lattice, respectively.
Representations of finite-dimensional algebras
In modular representation theory, and more generally in the theory of representations of finite-dimensional associative algebras A that are not semisimple, a Cartan matrix is defined by consi
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https://en.wikipedia.org/wiki/Whitney%20immersion%20theorem
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In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for , any smooth -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean -space, and a (not necessarily one-to-one) immersion in -space. Similarly, every smooth -dimensional manifold can be immersed in the -dimensional sphere (this removes the constraint).
The weak version, for , is due to transversality (general position, dimension counting): two m-dimensional manifolds in intersect generically in a 0-dimensional space.
Further results
William S. Massey went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in where is the number of 1's that appear in the binary expansion of . In the same paper, Massey proved that for every n there is manifold (which happens to be a product of real projective spaces) that does not immerse in .
The conjecture that every n-manifold immerses in became known as the immersion conjecture. This conjecture was eventually solved in the affirmative by .
See also
Whitney embedding theorem
References
External links
(Exposition of Cohen's work)
Theorems in differential topology
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https://en.wikipedia.org/wiki/Buddhabrot
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The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka), a traditional oval crown (ushnisha), and ringlet of hair.
Discovery
The Buddhabrot rendering technique was discovered by Melinda Green, who later described it in a 1993 Usenet post to sci.fractals.
Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988, Linas Vepstas relayed similar images to Cliff Pickover for inclusion in Pickover's then-forthcoming book Computers, Pattern, Chaos, and Beauty. This led directly to the discovery of Pickover stalks. Noel Griffin also implemented this idea in the 1993 "Mandelcloud" option in the Fractint renderer. However, these researchers did not filter out non-escaping trajectories required to produce the ghostly forms reminiscent of Hindu art. The inverse, "Anti-Buddhabrot" filter produces images similar to no filtering.
Green first named this pattern Ganesh, since an Indian co-worker "instantly recognized it as the god 'Ganesha' which is the one with the head of an elephant." The name Buddhabrot was coined later by Lori Gardi.
Rendering method
Mathematically, the Mandelbrot set consists of the set of points in the complex plane for which the iteratively defined sequence
does tend to infinity as goes to infinity for .
The Buddhabrot image can be constructed by first creating a 2-dimensional array of boxes, each corresponding to a final pixel in the image. Each box for and has size in complex coordinates of and , where and for an image of width and height . For each box, a corresponding counter is initialized to zero. Next, a random sampling of points are iterated through the Mandelbrot function. For points which escape within a chosen maximum number of iterations, and therefore are not in the Mandelbrot set, the counter for each box entered during the escape to infinity is incremented by 1. In other words, for each sequence corresponding to that escapes, for each point during the escape, the box that lies within is incremented by 1. Points which do not escape within the maximum number of iterations (and considered to be in the Mandelbrot set) are discarded. After a large number of values have been iterated, grayscale shades are then chosen based on the distribution of values recorded in the array. The result is a density plot highlighting regions where values spend the most time on their way to infinity.
Nuances
Rendering Buddhabrot images is typically more computationally intensive than standard Mandelbrot rendering techniques. This is partly due to requiring more random points to be iterated than pixels in the image in order to build up a sharp image. Rendering highly zoomed areas requires even more computation than for standard Mandelbrot images in which a given pixel can be compu
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https://en.wikipedia.org/wiki/Net%20%28polyhedron%29
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In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.
An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book A Course in the Art of Measurement with Compass and Ruler (Unterweysung der Messung mit dem Zyrkel und Rychtscheyd ) included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel.
Existence and uniqueness
Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net. Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together. If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that the sum of these defects is exactly 4, then there necessarily exists exactly one polyhedron that can be folded from it; this is Alexandrov's uniqueness theorem. However, the polyhedron formed in this way may have different faces than the ones specified as part of the net: some of the net polygons may have folds across them, and some of the edges between net polygons may remain unfolded. Additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra.
In 1975, G. C. Shephard asked whether every convex polyhedron has at least one net, or simple edge-unfolding. This question, which is also known as Dürer's conjecture, or Dürer's unfolding problem, remains unanswered. There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron (for instance along a cut locus) so that the set of subdivided faces has a net. In 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation. Furthermore, in 2019 Barvinok and Ghomi showed that a generalization of Dürer's conjecture fails for pseudo edges, i.e., a network of geodesics which connect vertices of the polyhedron and form a graph with convex faces.
A related open question asks whether every net of a convex polyhedron has a blooming, a continuous non-self-intersecting motion from its flat to its folded state that keeps each face flat throughout the motion.
Shortest path
The shortest path over the surface between two points on the surface
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https://en.wikipedia.org/wiki/John%20Britton%20%28mathematician%29
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John Leslie Britton (18 November 1927 – 13 June 1994) was an English mathematician from Yorkshire who worked in combinatorial group theory and was an expert on the word problem for groups. Britton was a member of the London Mathematical Society and was Secretary of Meetings and Membership with that organization from 1973-1976.
Britton died in a climbing accident on the Isle of Skye.
References
External links
MacTutor biography
1927 births
1994 deaths
20th-century English mathematicians
Group theorists
Mountaineering deaths
Sport deaths in Scotland
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https://en.wikipedia.org/wiki/Ranked%20list%20of%20Paraguayan%20departments
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Population figures from the 2021 statistics by the INE, the National Statistics Institute.
By population
By area
By density
This is a list of regions of Paraguay by Human Development Index as of 2017.
References
Paraguay
Human Development Index
List
departments
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https://en.wikipedia.org/wiki/1591%20in%20science
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The year 1591 in science and technology included many events, some of which are listed here.
Mathematics
François Viète publishes In Artem Analyticien Isagoge, introducing the new algebra with innovative use of letters as parameters in equations.
Giordano Bruno publishes and in Francfort.
Technology
The Rialto Bridge in Venice, designed by Antonio da Ponte, is completed.
Publications
Prospero Alpini publishes De Medicina Egyptiorum in Venice, including accounts of coffee, bananas and the baobab.
Publication of the first of the Conimbricenses commentaries on Aristotle by the Jesuits of the University of Coimbra, Commentarii Collegii Conimbricensis Societatis Jesu in octo libros physicorum Aristotelis Stagyritæ, on Aristotle's Physics.
Births
February 21 – Gérard Desargues, French geometer (died 1661)
Deaths
July 2 – Vincenzo Galilei, Italian scientist and musician (born 1520)
References
16th century in science
1590s in science
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https://en.wikipedia.org/wiki/ALGO
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ALGO is an algebraic programming language developed for the Bendix G-15 computer.
ALGO was one of several programming languages inspired by the Preliminary Report on the International Algorithmic Language written in Zürich in 1958. This report underwent several modifications before becoming the Revised Report on which most ALGOL implementations are based. As a result, ALGO and other early ALGOL-related languages have a very different syntax from ALGOL 60.
Example
Here is the Trabb Pardo – Knuth algorithm in ALGO:
TITLE TRABB PARDO-KNUTH ALGORITHM
SUBSCript I,J
DATA A(11)
FORMAt FI(2DT), FLARGE(3D)
PROCEDURE F(T=Z)
BEGIN
Z=SQRT(ABS(T))+5*T^3
END
FOR I=0(1)10
A[I]=KEYBD
FOR J=0(1)10 BEGIN
I=J-10
F(A[I]=Y)
PRINT(FI)=I
IF Y > 400
GO TO LARGE
PRINT(FL)=Y
GO TO NEXT
LARGE: PRINT(FLARGE)=999
NEXT: CARR(1) END
2END
Remarks
See also
ALGOL 58
ALGOL 60
References
External links
ALGO manual (PDF)
ALGOL 58 dialect
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https://en.wikipedia.org/wiki/Antonio%20Abetti
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Antonio Abetti (19 June 1846 – 20 February 1928) was an Italian astronomer.
Born in San Pietro di Gorizia (Šempeter-Vrtojba), he earned a degree in mathematics and engineering at the University of Padua. He was married to Giovanna Colbachini in 1879 and they had two sons. He died in Arcetri.
Work
Abetti mainly worked in positional astronomy and made many observations of minor planets, comets, and star occultations. In 1874 he was part of an expedition led by Pietro Tacchini to observe a transit of Venus with a spectroscope. Later he became director of the Osservatorio Astrofisico di Arcetri and a professor at the University of Florence. He refurbished the observatory at Arcetri by installing a new telescope.
Honors
Member of the Accademia dei Lincei.
Member of the Royal Astronomical Society.
The crater Abetti on the Moon is named after both Antonio and his son Giorgio Abetti.
The minor planet 2646 Abetti is also named after Antonio and his son.
References
External links
Biography of Abetti
1846 births
1928 deaths
People from Šempeter pri Gorici
19th-century Italian astronomers
20th-century Italian astronomers
University of Padua alumni
Academic staff of the University of Florence
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https://en.wikipedia.org/wiki/157%20%28number%29
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157 (one hundred [and] fifty-seven) is the number following 156 and preceding 158.
In mathematics
157 is:
the 37th prime number. The next prime is 163 and the previous prime is 151.
a balanced prime, because the arithmetic mean of those primes yields 157.
an emirp.
a Chen prime.
the largest known prime p which is also prime. (see ).
the least irregular prime with index 2.
a palindromic number in bases 7 (3137) and 12 (11112).
a repunit in base 12, so it is a unique prime in the same base.
a prime whose digits sum to a prime. (see ).
a prime index prime.
In base 10, 1572 is 24649, and 1582 is 24964, which uses the same digits. Numbers having this property are listed in . The previous entry is 13, and the next entry after 157 is 913.
The simplest right angle triangle with rational sides that has area 157 has the longest side with a denominator of 45 digits.
In the military
was a United States Coast Guard cutter built in 1926
was a United States Navy Type T2 tanker during World War II
was a United States Navy Alamosa-class cargo ship during World War II
was a United States Navy Admirable-class minesweeper during World War II
was a United States Navy Wickes-class destroyer during World War II
was a United States Navy Buckley-class destroyer escort during World War II
was a United States Navy General G. O. Squier-class transport ship during World War II
was a United States Navy LST-542-class tank landing ship during World War II
was a United States Navy ship during World War II
was a United States Navy transport military ship during World War II
was a United States Navy yacht during World War I
ZIL-157 is a 2.5-ton truck produced in post-World War II Russia
In music
"157 Riverside Avenue" is a song by REO Speedwagon from their debut album, REO Speedwagon in 1971. Its title refers to a Westport, Connecticut address where the band stayed while recording it.
Piano Sonata No. 1 in E major, D. 157 is a piano sonata in three movements by Franz Schubert.
"157" is a song by Tom Rosenthal where the lyrics merely consist of the numbers from 1 to 157. The song was released on April Fools' Day, 2020.
In sports
Ken Carpenter held the US record in discus, and won the NCAA national title with a toss of 157 feet in 1936.
Steph Curry of the Golden State Warriors holds the NBA record for 157 consecutive games with a 3-point field goal made (from November 13, 2014 to November 4, 2016).
In transportation
The British Rail Class 157 was the designation for a range of Diesel multiple unit trains of the Sprinter family
London Buses route 157
American Airlines Flight 157 from New York City bound for Mexico City crashed on November 29, 1949
157th Street (IRT Broadway – Seventh Avenue Line), a New York City Subway station at Broadway in Manhattan served by the
157th Street (Manhattan), a street in New York City
In other fields
157 is also:
The year AD 157 or 157 BC
157 AH is a year in the Islamic calendar that corresponds
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https://en.wikipedia.org/wiki/Power%20series%20solution%20of%20differential%20equations
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In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
Method
Consider the second-order linear differential equation
Suppose is nonzero for all . Then we can divide throughout to obtain
Suppose further that and are analytic functions.
The power series method calls for the construction of a power series solution
If is zero for some , then the Frobenius method, a variation on this method, is suited to deal with so called "singular points". The method works analogously for higher order equations as well as for systems.
Example usage
Let us look at the Hermite differential equation,
We can try to construct a series solution
Substituting these in the differential equation
Making a shift on the first sum
If this series is a solution, then all these coefficients must be zero, so for both k=0 and k>0:
We can rearrange this to get a recurrence relation for .
Now, we have
We can determine A0 and A1 if there are initial conditions, i.e. if we have an initial value problem.
So we have
and the series solution is
which we can break up into the sum of two linearly independent series solutions:
which can be further simplified by the use of hypergeometric series.
A simpler way using Taylor series
A much simpler way of solving this equation (and power series solution in general) using the Taylor series form of the expansion.
Here we assume the answer is of the form
If we do this, the general rule for obtaining the recurrence relationship for the coefficients is
and
In this case we can solve the Hermite equation in fewer steps:
becomes
or
in the series
Nonlinear equations
The power series method can be applied to certain nonlinear differential equations, though with less flexibility. A very large class of nonlinear equations can be solved analytically by using the Parker–Sochacki method. Since the Parker–Sochacki method involves an expansion of the original system of ordinary differential equations through auxiliary equations, it is not simply referred to as the power series method. The Parker–Sochacki method is done before the power series method to make the power series method possible on many nonlinear problems. An ODE problem can be expanded with the auxiliary variables which make the power series method trivial for an equivalent, larger system. Expanding the ODE problem with auxiliary variables produces the same coefficients (since the power series for a function is unique) at the cost of also calculating the coefficients of auxiliary equations. Many times, without using auxiliary variables, there is no known way to get the power series for the solution to a system, hence the power series method alone is difficult to apply to most nonlinear equations.
The power series method will
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https://en.wikipedia.org/wiki/Thue%20equation
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In mathematics, a Thue equation is a Diophantine equation of the form
ƒ(x,y) = r,
where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers x and y, a result known as Thue's theorem,
The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form where constants C1 and C2 depend only on the form ƒ. A stronger result holds: if K is the field generated by the roots of ƒ, then the equation has only finitely many solutions with x and y integers of K, and again these may be effectively determined.
Finiteness of solutions and diophantine approximation
Thue's original proof that the equation named in his honour has finitely many solutions is through the proof of what is now known as Thue's theorem: it asserts that for any algebraic number having degree and for any there exists only finitely many co-prime integers with such that . Applying this theorem allows one to almost immediately deduce the finiteness of solutions. However, Thue's proof, as well as subsequent improvements by Siegel, Dyson, and Roth were all ineffective.
Solution algorithm
Finding all solutions to a Thue equation can be achieved by a practical algorithm, which has been implemented in the following computer algebra systems:
in PARI/GP as functions thueinit() and thue().
in Magma computer algebra system as functions ThueObject() and ThueSolve().
in Mathematica through Reduce
Bounding the number of solutions
While there are several effective methods to solve Thue equations (including using Baker's method and Skolem's -adic method), these are not able to give the best theoretical bounds on the number of solutions. One may qualify an effective bound of the Thue equation by the parameters it depends on, and how "good" the dependence is.
The best result known today, essentially building on pioneering work of Bombieri and Schmidt, gives a bound of the shape , where is an absolute constant (that is, independent of both and ) and is the number of distinct prime divisors of . The most significant qualitative improvement to the theorem of Bombieri and Schmidt is due to Stewart, who obtained a bound of the form where is a divisor of exceeding in absolute value. It is conjectured that one may take the bound ; that is, depending only on the degree of but not its coefficients, and completely independent of the integer on the right hand side of the equation.
This is a weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness conjecture for rational points. This conjecture has been proven for "small" integers , where smallness is measured in terms of the discriminant of the form , by various authors, including Evertse, Stewart, and Akhtari. Stewart and Xiao demonstrated a strong form of this conjecture, asserting tha
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https://en.wikipedia.org/wiki/BRL-CAD
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BRL-CAD is a constructive solid geometry (CSG) solid modeling computer-aided design (CAD) system. It includes an interactive geometry editor, ray tracing support for graphics rendering and geometric analysis, computer network distributed framebuffer support, scripting, image-processing and signal-processing tools. The entire package is distributed in source code and binary form.
Although BRL-CAD can be used for a variety of engineering and graphics applications, the package's primary purpose continues to be the support of ballistic and electromagnetic analyses. In keeping with the Unix philosophy of developing independent tools to perform single, specific tasks and then linking the tools together in a package, BRL-CAD is basically a collection of libraries, tools, and utilities that work together to create, raytrace, and interrogate geometry and manipulate files and data. In contrast to many other 3D modelling applications, BRL-CAD primarily uses CSG rather than boundary representation. This means BRL-CAD can "study physical phenomena such as ballistic penetration and thermal, radiative, neutron, and other types of transport". It does also support boundary representation.
The BRL-CAD libraries are designed primarily for the geometric modeler who also wants to tinker with software and design custom tools. Each library is designed for a specific purpose: creating, editing, and ray tracing geometry, and image handling. The application side of BRL-CAD also offers a number of tools and utilities that are primarily concerned with geometric conversion, interrogation, image format conversion, and command-line-oriented image manipulation.
History
In 1979, the U.S. Army Ballistic Research Laboratory (BRL) expressed a need for tools that could assist with the computer simulation and engineering analysis of combat vehicle systems and environments. When no CAD package was found to be adequate for this purpose, BRL software developers – led by Mike Muuss – began assembling a suite of utilities capable of interactively displaying, editing, and interrogating geometric models. This suite became known as BRL-CAD. Development on BRL-CAD as a package subsequently began in 1983; the first public release was made in 1984. BRL-CAD became an open-source project in December 2004.
The BRL-CAD source code repository is the oldest known public version-controlled codebase in the world that's still under active development, dating back to 1983-12-16 00:10:31 UTC.
See also
PLaSM - Programming Language of Solid Modeling
Comparison of CAD editors
References
External links
Computer-aided design software for Linux
Computer-aided design software
Engineering software that uses Qt
Free computer-aided design software
Free software programmed in C
Free software programmed in Tcl
MacOS computer-aided design software
Software that uses Tk (software)
Software using the BSD license
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https://en.wikipedia.org/wiki/Steiner%20point
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A Steiner point (named after Jakob Steiner) may refer to:
Steiner point (computational geometry), a point added in solving a geometric optimization problem to make its solution better
Steiner point (triangle), a certain point on the circumcircle of a given triangle
One of 20 points associated with a given set of six points on a conic; see
See also
Steiner tree problem, an algorithmic problem of finding extra Steiner points to add to a point set to reduce the cost of connecting the points
The median of three vertices in a median graph, the solution to the Steiner tree problem for those three vertices
The Fermat point of a triangle, the solution to the Steiner tree problem for the three vertices of the triangle
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https://en.wikipedia.org/wiki/Mathematics%20Genealogy%20Project
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The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians. it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a typical mathematician, the project entry includes graduation year, thesis title (in its Mathematics Subject Classification), alma mater, doctoral advisor, and doctoral students.
Origin of the database
The project grew out of founder Harry Coonce's desire to know the name of his advisor's advisor. Coonce was Professor of Mathematics at Minnesota State University, Mankato, at the time of the project's founding, and the project went online there in fall 1997. Coonce retired from Mankato in 1999, and in fall 2002 the university decided that it would no longer support the project. The project relocated at that time to North Dakota State University. Since 2003, the project has also operated under the auspices of the American Mathematical Society and in 2005 it received a grant from the Clay Mathematics Institute. Harry Coonce has been assisted by Mitchel T. Keller, Assistant Professor at Morningside College. Keller is currently the Managing Director of the project.
Mission and scope
The Mathematics Genealogy Mission statement: "Throughout this project when we use the word 'mathematics' or 'mathematician' we mean that word in a very inclusive sense. Thus, all relevant data from statistics, computer science, philosophy or operations research is welcome."
Scope
The genealogy information is obtained from sources such as Dissertation Abstracts International and Notices of the American Mathematical Society, but may be supplied by anyone via the project's website. The searchable database contains the name of the mathematician, university which awarded the degree, year when the degree was awarded, title of the dissertation, names of the advisor and second advisor, a flag of the country where the degree was awarded, a listing of doctoral students, and a count of academic descendants. Some historically significant figures who lacked a doctoral degree are listed, notably Joseph-Louis Lagrange.
Reliability and completeness
It has been noted that "the data collected by the mathematics genealogy project are self-reported, so there is no guarantee that the observed genealogy network is a complete description of the mentorship network. In fact, 16,147 mathematicians do not have a recorded mentor, and of these, 8,336 do not have any recorded proteges." Maimgren, Ottino and Amaral (2010) stated that "for [mathematicians who graduated between 1900 and 1960] we believe that the graduation and mentorship record is the most reliable."
See also
Neurotree, Academic Family Tree
References
External links
Projects established in 1997
Internet properties established in 1997
Mathematical projects
Historiography of mathematics
Mathematical databases
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https://en.wikipedia.org/wiki/Signature%20%28topology%29
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In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem.
Definition
Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group
.
The basic identity for the cup product
shows that with p = q = 2k the product is symmetric. It takes values in
.
If we assume also that M is compact, Poincaré duality identifies this with
which can be identified with . Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature of M is by definition the signature of Q, that is, where any diagonal matrix defining Q has positive entries and negative entries. If M is not connected, its signature is defined to be the sum of the signatures of its connected components.
Other dimensions
If M has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in L-theory: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group or as the 4k-dimensional quadratic L-group and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of ) for framed manifolds of dimension 4k+2 (the quadratic L-group ), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group ); the other dimensional L-groups vanish.
Kervaire invariant
When is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.
Properties
Compact oriented manifolds M and N satisfy by definition, and satisfy by a Künneth formula.
If M is an oriented boundary, then .
René Thom (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers. For example, in four dimensions, it is given by . Friedrich Hirzebruch (1954) found an explicit expression for this linear combination as the L genus of the manifold.
William Browder (1962) proved that a simply connected compact polyhedron with 4n-di
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https://en.wikipedia.org/wiki/Capitulation
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Capitulation may have the following special meanings.
Capitulation (surrender)
Stock market capitulation
Capitulation (treaty)
Capitulations of the Ottoman Empire
Capitulation (algebra)
Conclave capitulation
Electoral capitulation
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https://en.wikipedia.org/wiki/William%20Smith%20%28teacher%29
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William Macdonald Smith (born 25 June 1939) is a South African science and mathematics teacher who is best known for his maths and science lessons on television. Born in Makhanda (Grahamstown), he is the son of the ichthyologist Margaret Mary Smith and Professor J. L. B. Smith, the renowned chemist and ichthyologist who identified the coelacanth.
Early life and education
He attended St. Andrew's Prep before matriculating at Union High School in Graaff-Reinet. He then went on to study at Rhodes University, where he obtained a Bachelor of Science degree in physics and chemistry, followed by an honours degree (cum laude) in chemistry at the same institution. Following that, he obtained a master's degree from the University of Natal (Pietermaritzburg campus) in only seven months.
During his time at school and university, Smith showed an interest in film and camerawork, scripting, shooting, and producing the 50-minute feature documentary, ‘The Garden Route,’ in 1960. The film was digitised and relaunched in 2010.
He started working at African Explosives and Chemical Industries (AECI). Deciding that he would rather pursue a teaching career, Smith left the industry and moved to the education sector, where he started 'Star Schools,' named for the mass-circulation Johannesburg newspaper, The Star, which published material that Smith prepared to support his lessons. The aim of these schools is to provide value-for-money supplementary education with top-class teachers to prepare learners for their matriculation exams. During the next 25 years, Smith became famous throughout South Africa, winning a 'Teacher of the Year' award in 1991.
Smith ran his first multi-racial school in the 1970s, despite problems with the Apartheid authorities. At that time, education facilities were segregated under legislation such as the Bantu Education Act (1953), and black children were prohibited from attending classes on campuses that had been reserved for whites. Smith, however, never turned any black student away from any class, and made Star Schools more accessible by offering instruction in subjects that weren’t adequately covered by the Bantu Education system - such as mathematics and physical science. (Bantu Education was reserved for black learners, while Christian National Education was reserved for whites. Under Apartheid legislation, South Africa had as many as nineteen different education departments).
In 1990, Smith began producing The Learning Channel educational television programmes with the financial backing of Hylton Appelbaum, then executive director of the Liberty Life Foundation. As a result of his work on this programme, Smith was voted as one of the top three presenters on South African television in 1998.
Other achievements
Smith is also a renowned conservationist and owned the Featherbed Nature Reserve in Knysna, where he lived until the sale of the land and company in 2008. He was also the owner of 'Rivercat Ferries', which has several craft th
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https://en.wikipedia.org/wiki/Karl%20Adams%20%28mathematician%29
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Karl Adams (1811 in Merscheid – 14 November 1849, in Winterthur) was a Swiss mathematician and teacher who specialised in synthetic geometry.
Publications
Lehre von den Transversalen, 1843
Die harmonischen Verhältnisse, 1845
Die merkwürdigen Eigenschaften des geradlinigen Dreiecks, 1846
Das Malfattische Problem, 1846 and 1848, on the Malfatti circles
Geometrische Aufgaben mit besonderer Rücksicht auf geometrische ConstruCtion, 1847 and 1849
Sources
Allgemeine Deutsche Biographie – online version at Wikisource
19th-century Swiss mathematicians
Swiss schoolteachers
Swiss Calvinist and Reformed Christians
1811 births
1849 deaths
19th-century Swiss educators
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https://en.wikipedia.org/wiki/Darboux%20vector
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In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. It is also called angular momentum vector, because it is directly proportional to angular momentum.
In terms of the Frenet-Serret apparatus, the Darboux vector ω can be expressed as
and it has the following symmetrical properties:
which can be derived from Equation (1) by means of the Frenet-Serret theorem (or vice versa).
Let a rigid object move along a regular curve described parametrically by β(t). This object has its own intrinsic coordinate system. As the object moves along the curve, let its intrinsic coordinate system keep itself aligned with the curve's Frenet frame. As it does so, the object's motion will be described by two vectors: a translation vector, and a rotation vector ω, which is an areal velocity vector: the Darboux vector.
Note that this rotation is kinematic, rather than physical, because usually when a rigid object moves freely in space its rotation is independent of its translation. The exception would be if the object's rotation is physically constrained to align itself with the object's translation, as is the case with the cart of a roller coaster.
Consider the rigid object moving smoothly along the regular curve. Once the translation is "factored out", the object is seen to rotate the same way as its Frenet frame. The total rotation of the Frenet frame is the combination of the rotations of each of the three Frenet vectors:
Each Frenet vector moves about an "origin" which is the centre of the rigid object (pick some point within the object and call it its centre). The areal velocity of the tangent vector is:
Likewise,
Now apply the Frenet-Serret theorem to find the areal velocity components:
so that
as claimed.
The Darboux vector provides a concise way of interpreting curvature κ and torsion τ geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector.
References
Differential geometry
Vectors (mathematics and physics)
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https://en.wikipedia.org/wiki/Graded%20vector%20space
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In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.
For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures.
Integer gradation
Let be the set of non-negative integers. An -graded vector space, often called simply a graded vector space without the prefix , is a vector space together with a decomposition into a direct sum of the form
where each is a vector space. For a given n the elements of are then called homogeneous elements of degree n.
Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.
General gradation
The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elements of any set I. An I-graded vector space V is a vector space together with a decomposition into a direct sum of subspaces indexed by elements i of the set I:
Therefore, an -graded vector space, as defined above, is just an I-graded vector space where the set I is (the set of natural numbers).
The case where I is the ring (the elements 0 and 1) is particularly important in physics. A -graded vector space is also known as a supervector space.
Homomorphisms
For general index sets I, a linear map between two I-graded vector spaces is called a graded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called a homomorphism (or morphism) of graded vector spaces, or homogeneous linear map:
for all i in I.
For a fixed field and a fixed index set, the graded vector spaces form a category whose morphisms are the graded linear maps.
When I is a commutative monoid (such as the natural numbers), then one may more generally define linear maps that are homogeneous of any degree i in I by the property
for all j in I,
where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into an abelian group A that it generates (for instance the integers if I is the natural numbers), then one may also define linear maps that are homogeneous of degree i in A by the same property (but now "+" denotes the group operation in A). Specifically, for i in I a linear map will be homogeneous of degree −i if
for all j in I, while
if is not in I.
Just as the set of linear maps from a vector space to itself forms an associative algebra (the algebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees to I or allowing any degrees in the group A – form associative graded al
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https://en.wikipedia.org/wiki/Doubling
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Doubling may refer to:
Mathematics
Arithmetical doubling of a count or a measure, expressed as:
Multiplication by 2
Increase by 100%, i.e. one-hundred percent
Doubling the cube (i. e., hypothetical geometric construction of a cube with twice the volume of a given cube)
Doubling time, the length of time required for a quantity to double in size or value
Doubling map, a particular infinite two-dimensional geometrical construction
see also: Period-doubling bifurcation
Music
The composition or performance of a melody with itself or itself transposed at a constant interval such as the octave, third, or sixth, Voicing (music)#Doubling
The assignment of a melody to two instruments in an arrangement
The playing of two (or more) instruments alternately by a single player, e.g. Flute, doubling piccolo
Musicians who play more than one woodwind instrument are called woodwind doublers or reed players
Doubletracking, a recording technique in which a musical part (or vocal) is recorded twice and mixed together, to strengthen or "fatten" the tone.
Other
Doubling (psychodrama) is a technique of provoking a protagonist by a participant, for effect.
Doubling in the theatre is where one actor plays more than one part in the same performance.
Doubling (textiles) is the process where six slivers of cotton are fed into a draw frame, stretched and drawn together to improve the uniformity of the roving before it is spun
Doubling (naval tactic) was a means of focusing gunfire in formations of sailing warships maneuvering as a line of battle.
Double knitting is the process of combining two or more lengths of yarn into a single thread.
Doubling in two-way radio, where two or more transmitters transmit at once on the same frequency, interfering with one another and garbling all messages.
Syntactic doubling is a phenomenon consisting in the lengthening (gemination) of the initial consonant of certain words
When more than one round is fired in a semiautomatic gas powered rifle with only one pull of the trigger, also known as a slam fire.
Doubling trains tracks has two tracks for two direction traffic.
See also
Double (disambiguation)
Dublin
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https://en.wikipedia.org/wiki/Helmholtz%20equation
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In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation
where is the Laplace operator, is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.
Motivation and uses
The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.
For example, consider the wave equation
Separation of variables begins by assuming that the wave function is in fact separable:
Substituting this form into the wave equation and then simplifying, we obtain the following equation:
Notice that the expression on the left side depends only on , whereas the right expression depends only on . As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for , the other for
where we have chosen, without loss of generality, the expression for the value of the constant. (It is equally valid to use any constant as the separation constant; is chosen only for convenience in the resulting solutions.)
Rearranging the first equation, we obtain the Helmholtz equation:
Likewise, after making the substitution , where is the wave number, and is the angular frequency (assuming a monochromatic field), the second equation becomes
We now have Helmholtz's equation for the spatial variable and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.
Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.
Solving the Helmholtz equation using separation of variables
The solution to the spatial Helmholtz equation:
can be obtained for simple geometries using separation of variables.
Vibrating membrane
The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equatio
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https://en.wikipedia.org/wiki/Heinrich%20Heesch
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Heinrich Heesch (June 25, 1906 – July 26, 1995) was a German mathematician. He was born in Kiel and died in Hanover.
In Göttingen he worked on Group theory. In 1933 Heesch witnessed the National Socialist purges of university staff. Not willing to become a member of the National Socialist organization of university teachers as required, he resigned from his university position in 1935 and worked privately at his parents' home in Kiel until 1948.
During this time he did research on tilings. In 1955 Heesch began teaching at Leibniz University Hannover and worked on graph theory. In this period Heesch did pioneering work in developing methods for a computer-aided proof of the then unproved four color theorem. In particular, he was the first to investigate the notion of "discharging", which turned out to be a fundamental ingredient of the eventual computer-aided proof by Kenneth Appel and Wolfgang Haken.
Between 1967 and 1971, Heesch made several visits to the United States, where bigger and faster computers were available, working with Haken at University of Illinois at Urbana-Champaign and with Karl Durre and Yoshio Shimamoto at Brookhaven National Laboratory.
During the crucial phase of his project, the German national research fund DFG cancelled financial support. After the 1977 success of Appel and Haken, Heesch worked on refining and shortening their proof, even after his retirement.
Works
Heinrich Heesch, Otto Kienzle: Flächenschluß. Berlin / Göttingen / Heidelberg: Springer-Verlag 1963
Heinrich Heesch, Untersuchungen zum Vierfarbenproblem, Bibliographisches Institut, Mannheim 1969
Bigalke, Hans-Günther (Hrsg.). Heinrich Heesch, Gesammelte Abhandlungen, Bad Salzdetfurth 1986.
Literature on Heinrich Heesch's work
Bigalke, Hans-Günther. Heinrich Heesch, Kristallgeometrie, Parkettierungen, Vierfarbenforschung, Basel 1988.
See also
Heesch's problem
External links
http://www.ifg.uni-kiel.de/eckenundkanten/hk-02_de.html Biography (in German)
1906 births
1995 deaths
20th-century German mathematicians
Scientists from Kiel
People from the Province of Schleswig-Holstein
Academic staff of the University of Göttingen
Academic staff of the University of Hanover
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https://en.wikipedia.org/wiki/Nilakantha%20Somayaji
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Keļallur Nilakantha Somayaji (14 June 1444 – 1545), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapariksakrama is a manual on making observations in astronomy based on instruments of the time. Known popularly as Kelallur Chomaathiri, he is considered an equal to Vatasseri Parameshwaran Nambudiri.
Early life
Nilakantha was born into a Brahmin family which came from South Malabar in Kerala.
Biographical details
Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times.
In one of his works titled Siddhanta-star and also in his own commentary on Siddhanta-darpana, Nilakantha Somayaji has stated that he was born on Kali-day 1,660,181 which works out to 14 June 1444 CE. A contemporary reference to Nilakantha Somayaji in a Malayalam work on astrology implies that Somayaji lived to a ripe old age even to become a centenarian. Sankara Variar, a pupil of Nilakantha Somayaji, in his commentary on Tantrasamgraha titled Tantrasamgraha-vyakhya, points out that the first and last verses of Tantrasamgraha contain chronograms specifying the Kali-days of the commencement (1,680,548) and of completion (1,680,553) of Somayaji's magnum opus Tantrasamgraha. Both these days occur in 1500 CE.
In Aryabhatiya-bhashya, Nilakantha Somayaji has stated that he was the son of Jatavedas and he had a brother named Sankara. Somayaji has further stated that he was a Bhatta belonging to the Gargya gotra and was a follower of Asvalayana-sutra of Rigveda. References in his own Laghuramayana indicate that Nilakantha Somayaji was a member of the Kelallur family (Sanskritised as Kerala-sad-grama) residing at Kundagrama, now known as Trikkandiyur in modern Tirur, Kerala. His wife was named Arya and he had two sons Rama and Dakshinamurti.
Nilakantha Somayaji studied vedanta and some aspects of astronomy under one Ravi. However, it was Damodara, son of Kerala-drgganita author Paramesvara, who initiated him into the science of astronomy and instructed him in the basic principles of mathematical computations. The great Malayalam poet Thunchaththu Ramanujan Ezhuthachan is said to have been a student of Nilakantha Somayaji.
The epithet Somayaji is a title assigned to or assumed by a Namputiri who has performed the vedic ritual of Somayajna. So it could be surmised that Nilakantha Somayaji had also performed a Somayajna ritual and assumed the title of a Somayaji in later life. In colloquial Malayalam usage the word Somayaji has been corrupted to Comatiri.
Nilakant
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https://en.wikipedia.org/wiki/Narayana%20Pandita%20%28mathematician%29
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Nārāyaṇa Paṇḍita () (1340–1400) was an Indian mathematician. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala school. He wrote the Ganita Kaumudi (lit "Moonlight of mathematics") in 1356 about mathematical operations. The work anticipated many developments in combinatorics. About his life, the most that is known is that:
Narayana Pandit wrote two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijaganita Vatamsa. Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati). Although the Karmapradipika contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares.
Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order indeterminate equation nq2 + 1 = p2 (Pell's equation), solutions of indeterminate higher-order equations, mathematical operations with zero, several geometrical rules, methods of integer factorization, and a discussion of magic squares and similar figures. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.
Narayana is also credited with developing a method for systematic generation of all permutations of a given sequence.
Narayana's cows is an integer sequence which Narayana described as the number of cows present each year, starting from one cow in the first year, where every cow has one baby cow each year starting in its fourth year of life. The first few terms of the sequence are as follows: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, … Narayana's cows is sequence in OEIS. The ratio of consecutive terms approaches the supergolden ratio.
References
1340 births
1400 deaths
Indian Hindus
14th-century Indian mathematicians
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https://en.wikipedia.org/wiki/Transcendental%20number%20theory
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Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendence
The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial with rational coefficients there will be a complex number such that . Transcendence theory is concerned with the converse question: given a complex number , is there a polynomial with rational coefficients such that If no such polynomial exists then the number is called transcendental.
More generally the theory deals with algebraic independence of numbers. A set of numbers {α1, α2, …, αn} is called algebraically independent over a field K if there is no non-zero polynomial P in n variables with coefficients in K such that P(α1, α2, …, αn) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where n = 1 and the field K is the field of rational numbers.
A related notion is whether there is a closed-form expression for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.
History
Approximation by rational numbers: Liouville to Roth
Use of the term transcendental to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function. The question of whether certain classes of numbers could be transcendental dates back to 1748 when Euler asserted that the number logab was not algebraic for rational numbers a and b provided b is not of the form b = ac for some rational c.
Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim Joseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure. His original papers on the matter in the 1840s sketched out arguments using continued fractions to construct transcendental numbers. Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental. This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number e is transcendental. But his work did provide a larger class of transcendental numbers, now known as Liouville numbers in his honour.
Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers. So if a number can be very well approximated by rationa
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https://en.wikipedia.org/wiki/Hasse%E2%80%93Weil%20zeta%20function
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In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number p. It is a global L-function defined as an Euler product of local zeta functions.
Hasse–Weil L-functions form one of the two major classes of global L-functions, alongside the L-functions associated to automorphic representations. Conjecturally, these two types of global L-functions are actually two descriptions of the same type of global L-function; this would be a vast generalisation of the Taniyama-Weil conjecture, itself an important result in number theory.
For an elliptic curve over a number field K, the Hasse–Weil zeta function is conjecturally related to the group of rational points of the elliptic curve over K by the Birch and Swinnerton-Dyer conjecture.
Definition
The description of the Hasse–Weil zeta function up to finitely many factors of its Euler product is relatively simple. This follows the initial suggestions of Helmut Hasse and André Weil, motivated by the case in which V is a single point, and the Riemann zeta function results.
Taking the case of K the rational number field Q, and V a non-singular projective variety, we can for almost all prime numbers p consider the reduction of V modulo p, an algebraic variety Vp over the finite field Fp with p elements, just by reducing equations for V. Scheme-theoretically, this reduction is just the pullback of V along the canonical map Spec Fp → Spec Z. Again for almost all p it will be non-singular. We define
to be the Dirichlet series of the complex variable s, which is the infinite product of the local zeta functions
.
Then , according to our definition, is well-defined only up to multiplication by rational functions in a finite number of .
Since the indeterminacy is relatively harmless, and has meromorphic continuation everywhere, there is a sense in which the properties of Z(s) do not essentially depend on it. In particular, while the exact form of the functional equation for Z(s), reflecting in a vertical line in the complex plane, will definitely depend on the 'missing' factors, the existence of some such functional equation does not.
A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in ramification theory, 'bad' primes carry good information (theory of the conductor). This manifests itself in the étale theory in the Ogg–Néron–Shafarevich criterion for good reduction; namely that there is good reduction, in a definite sense, at all primes p for which the Galois representation ρ on the étale cohomology groups of V is unramified. For those, the definition of local zeta function can be recovered in terms of the characteristic polynomial of
Frob(p) being a Frobenius
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https://en.wikipedia.org/wiki/List%20of%20Indian%20mathematicians
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The chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India.
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. Hindu-Arabic numerals predominantly used today and likely into the future.
Ancient (Before 320 CE)
Baudhayana sutras (fl. c. 900 BCE)
Yajnavalkya (700 BCE)
Manava (fl. 750–650 BCE)
Apastamba Dharmasutra (c. 600 BCE)
Pāṇini (c. 520–460 BCE)
Kātyāyana (fl. c. 300 BCE)
Akṣapada Gautama(c. 600 BCE–200 CE)
Bharata Muni (200 BCE-200 CE)
Pingala (c. 3rd/2nd century BCE)
Bhadrabahu (367 – 298 BCE)
Umasvati (c. 200 CE)
Yavaneśvara (2nd century)
Vasishtha Siddhanta, 4th century CE
Classical (320 CE–520 CE)
Vasishtha Siddhanta, 4th century CE
Aryabhata (476–550 CE)
Yativrsabha (500–570)
Varahamihira (505–587 CE)
Yativṛṣabha, (6th-century CE)
Virahanka (6th century CE)
Early Medieval Period (521 CE–1206 CE)
Brahmagupta (598–670 CE)
Bhaskara I (600–680 CE)
Shridhara (between 650–850 CE)
Lalla (c. 720–790 CE)
Virasena (792–853 CE)
Govindasvāmi (c. 800 – c. 860 CE)
Prithudaka (c. 830 – c. 900CE)
Śaṅkaranārāyaṇa, (c. 840 – c. 900 CE)
Vaṭeśvara (born 880 CE)
Mahavira (9th century CE)
Jayadeva 9th century CE
Aryabhata II (920 – c. 1000)
Vijayanandi (c. 940–1010)
Halayudha 10th Century
Śrīpati (1019–1066)
Abhayadeva Suri (1050 CE)
Brahmadeva (1060–1130)
Pavuluri Mallana (11th century CE)
Hemachandra (1087–1172 CE)
Bhaskara II (1114–1185 CE)
Someshvara III (1127–1138 CE)
Śārṅgadeva (1175-1247)
Late Medieval Period (1206–1526)
13th Century
Thakkar Pheru( 1291– 1347)
14th century
Mahendra Suri (1340 – 1400)
Keshava of Nandigrama (fl. 1496–1507)
Narayana Pandita (1325–1400)
Navya-Nyāya (Neo-Logical) School
Gangesha Upadhyaya (first half of the 14th century)
Kerala School of Mathematics and Astronomy
Madhava of Sangamagrama (c. 1340 – c. 1425)
Parameshvara (1360–1455), discovered drek-ganita, a mode of astronomy based on observations
Puthumana Somayaji (c. 1380–1460)
15th century
Kerala School of Mathematics and Astronomy
Chennas Narayanan Namboodiripad (born 1428)
Nilakantha Somayaji (1444–1545), mathematician and astronomer
Damodara (15th century)
Navya-Nyāya (Neo-Logical) School
Raghunatha Siromani (1475–1550)
Early Modern Period (1527– 1800)
16th Century
Gaṇeśa Daivajna (born 1507, fl. 1520-1554)
Kerala School of Mathematics and Astronomy
Chitrabhanu (16th Century)
Shankara Variyar (c. 1530)
Jyeshtadeva (1500–1610), author of Yuktibhāṣā
Paarangot Jyeshtadevan Namboodiri (AD 1500–1610)
Achyuta Pisharati (1550–1621), mathematician and astronomer
Melpathur Narayana Bhattathiri (1560–1646/1666)
Golagrama school of astronomy
Nṛsiṃha (born 1586)
Mallari (fl.1575)
17th Century
Muhammad Saleh Thattvi ( fl. 1663–64)
Ali Kashmiri ibn Luqman (fl. 1589-90)
Ataullah Rashidi (17th century)
Munishvara (born 1603)
Mulla
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https://en.wikipedia.org/wiki/Virahanka
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Virahanka (Devanagari: विरहाङ्क) was an Indian prosodist who is also known for his work on mathematics. He may have lived in the 6th century, but it is also possible that he worked as late as the 8th century.
His work on prosody builds on the Chhanda-sutras of Pingala (4th century BCE), and was the basis for a 12th-century commentary by Gopala.
He was the first to propose the so-called Fibonacci Sequence.
See also
Indian mathematicians
References
External links
The So-called Fibonacci Numbers in Ancient and Medieval India by Parmanand Singh
8th-century Indian mathematicians
Fibonacci numbers
Medieval Sanskrit grammarians
Ancient Indian mathematical works
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https://en.wikipedia.org/wiki/Parameshvara%20Nambudiri
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Vatasseri Parameshvara Nambudiri ( 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita or Drig system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.
Biographical details
Parameshvara was a Hindu of Bhrgugotra following the Ashvalayanasutra of the Rigveda. Parameshvara's family name (Illam) was Vatasseri and his family resided in the village of Alathiyur (Sanskritised as Asvatthagrama) in Tirur, Kerala. Alathiyur is situated on the northern bank of the river Nila (river Bharathappuzha) at its mouth in Kerala. He was a grandson of a disciple of Govinda Bhattathiri (1237–1295 CE), a legendary figure in the astrological traditions of Kerala.
Parameshvara studied under teachers Rudra and Narayana, and also under Madhava of Sangamagrama (c. 1350 – c. 1425) the founder of the Kerala school of astronomy and mathematics. Damodara, another prominent member of the Kerala school, was his son and also his pupil. Parameshvara was also a teacher of Nilakantha Somayaji (1444–1544) the author of the celebrated Tantrasamgraha.
Work
Parameshvara wrote commentaries on many mathematical and astronomical works such as those by Bhāskara I and Aryabhata. He made a series of eclipse observations over a 55-year period. Constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.
One of Parameshvara's more significant contributions was his mean value type formula for the inverse interpolation of the sine.
He was the first mathematician to give a formula for the radius of the circle circumscribing a cyclic quadrilateral. The expression is sometimes attributed to Lhuilier [1782], 350 years later. With the sides of the cyclic quadrilateral being a, b, c, and d, the radius R of the circumscribed circle is:
Works by Parameshvara
The following works of Parameshvara are well-known. A complete list of all manuscripts attributed to Parameshvara is available in Pingree.
Bhatadipika – Commentary on Āryabhaṭīya of Āryabhaṭa I
Karmadipika – Commentary on Mahabhaskariya of Bhaskara I
Paramesvari – Commentary on Laghubhaskariya of Bhaskara I
Sidhantadipika – Commentary on Mahabhaskariyabhashya of Govindasvāmi
Vivarana – Commentary on Surya Siddhanta and Lilāvati
Drgganita – Description of the Drig system (compose
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https://en.wikipedia.org/wiki/Baudhayana%20sutras
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The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. They belong to the Taittiriya branch of the Krishna Yajurveda school and are among the earliest texts of the genre.
The Baudhayana sūtras consist of six texts:
the , probably in 19 (questions),
the in 20 (chapters),
the in 4 ,
the Grihyasutra in 4 ,
the in 4 and
the in 3 .
The is noted for containing several early mathematical results, including an approximation of the square root of 2 and the statement of the Pythagorean theorem.
Baudhāyana Shrautasūtra
His Śrauta sūtras related to performing Vedic sacrifices have followers in some Smārta brāhmaṇas (Iyers) and some Iyengars of Tamil Nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal Brahmins (Aadi Saivas) and Kongu Vellalars. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord Krishna had done tarpaṇa on the day before amāvāsyā; they call themselves Baudhāyana Amavasya.
Baudhāyana Dharmasūtra
The Dharmasūtra of Baudhāyana like that of Apastamba also forms a part of the larger Kalpasutra. Likewise, it is composed of praśnas which literally means 'questions' or books. The structure of this Dharmasūtra is not very clear because it came down in an incomplete manner. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The praśnas consist of the Srautasutra and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the Grhyasutra which deals with domestic rituals.
There are no commentaries on this Dharmasūtra with the exception of Govindasvāmin's Vivaraṇa. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Also the commentary is inferior in comparison to that of Haradatta on Āpastamba and Gautama.
This Dharmasūtra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the 'Proto-Baudhayana' even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmasūtra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.
The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation. Book two refers to penances, inheritance, women, householder, orders of life, ancestral offerings. Book three refers to holy householders, forest hermit and penances. Book four primarily refers to the yogic practices and penances along with offenses regarding marriage.
Baudhāyana Śulvasūtra
Pythagorean theorem
The Baudhāyana Śulvasūtra states the rule referred to today in most of the world as the Pythagorean Theorem. Th
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https://en.wikipedia.org/wiki/Farkas%20Bolyai
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Farkas Bolyai (; 9 February 1775 – 20 November 1856; also known as Wolfgang Bolyai in Germany) was a Hungarian mathematician, mainly known for his work in geometry.
Biography
Bolyai was born in Bolya, a village near Hermannstadt, Grand Principality of Transylvania (now Buia, Sibiu County, Romania). His father was Gáspár Bolyai and his mother Krisztina Vajna. Farkas was taught at home by his father until the age of six when he was sent to the Calvinist school in Nagyszeben. His teachers recognized his talents in arithmetics and in learning languages. He learned Latin, Greek, Romanian, Hebrew and later also French, Italian and English. He easily multiplied, divided 13- or 14-digit numbers in his head, and was able to draw square and cubic roots from them. At the age of 12 he left school and was appointed as a tutor to the eight-year-old son of the count Kemény. This meant that Bolyai was now treated as a member of one of the leading families in the country, and he became not only a tutor but a real friend to the count's son. In 1790 Bolyai and his pupil both entered the Calvinist College in Kolozsvár (today Cluj-Napoca) where they spent five years.
The professor of philosophy at the College in Kolozsvár tried to turn Bolyai against mathematics and towards religious philosophy. Bolyai, however, decided to go abroad with Simon Kemény on an educational trip in 1796 and began to study mathematics systematically at German universities first at Jena and then at Göttingen. In these times Bolyai became a close friend of Carl Friedrich Gauss.
He returned home to Kolozsvár in 1799. It was there he met and married Zsuzsanna Benkő and where their son János Bolyai – later an even more famous mathematician than his father – was born in 1802. Soon thereafter he accepted a teaching position for mathematics and sciences at the Calvinist College in Marosvásárhely (today Târgu-Mureş), where he spent the rest of his life.
Mathematical work
Bolyai's main interests were the foundations of geometry and the parallel axiom.
His main work, Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi (An Attempt to Introduce Studious Youths to the Elements of Pure Mathematics; 1832), was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis. In this work, he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above. His study of the convergence of series includes a test equivalent to Raabe's test, which he discovered independently and at about the same time as Raabe. Other important ideas in the work include a general definition of a function and a definition of an equality between two plane figures if they can both be divided into a finite equal number of pairwise congruent pieces.
He first dissuaded his son from the study of non-Euclidean geometry, but by 183
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https://en.wikipedia.org/wiki/Chartered%20Mathematician
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Chartered Mathematician (CMath) is a professional qualification in Mathematics awarded to professional practising mathematicians by the Institute of Mathematics and its Applications (IMA) in the United Kingdom.
Chartered Mathematician is the IMA's highest professional qualification; achieving it is done through a rigorous peer-reviewed process. It provides formal recognition of a member’s qualifications in Mathematics, professional practise of Mathematics at an advanced level, technical standing, and commitment to remain at the forefront of Mathematics theory and practise throughout one's professional career.
The required standard for Chartered Mathematician registration is typically an accredited UK MMath degree, at least five years of peer-reviewed professional practise of advanced Mathematics, attainment of a senior-level of technical standing, and an ongoing commitment to Continuing Professional Development.
A Chartered Mathematician is entitled to use the post-nominal letters CMath, in accordance with the Royal Charter granted to the IMA by the Privy Council. The profession of Chartered Mathematician is a 'regulated profession' under the European professional qualification directives.
See also
Institute of Mathematics and its Applications
References
External links
Institute of Mathematics and its Applications website
Mathematics education in the United Kingdom
Mathematician
Mathematics
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https://en.wikipedia.org/wiki/ProBoards
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ProBoards is a free, remotely hosted message board service that facilitates online discussions by allowing people to create their own online communities.
Ownership and service statistics
ProBoards was founded by Patrick Clinger, who wrote the ProBoards software. Prior to launching ProBoards, Clinger had run HostedScripts, a company aimed at creating free web widgets.
The service hosts over 3,000,000 internet forums, which in turn have approximately 22,800,000 users worldwide. Currently, all ProBoards forums combined receive a total of over 600 million pageviews per month, making ProBoards one of the largest websites on the Internet.
However, according to TechCrunch writer Anthony Ha, those numbers have seemingly dropped as of 2014. In an interview, founder/owner Patrick Clinger stated "ProBoards has been used to create 3.5 million forums", but about 1.2 million of them are still active (i.e. resulting in the occasional page view). In October 2021, ProBoards was purchased by internet company VerticalScope Holdings.
Software history
Proboards is coded in Perl, a popular programming language with web developers. Previously, due to the remotely hosted nature of the service, users could not modify the software directly as with some forum systems, but some customisation was possible through the use of CSS or JavaScript codes. With the release of v.5, however, ProBoards gives Administrators and certain other members access to the HTML and CSS of the webpage, for easier coding purposes.
The first day of business for ProBoards was January 1, 2000. At first, ProBoards originally used software created by the owner, Patrick Clinger. In late 2001, though, ProBoards switched to the YaBB system. At the same time, other changes to the service made it the first remotely hosted service to offer a subdomain with each forum (e.g. username.proboards[servernumber].com)
On June 11, 2002, ProBoards Version 2 was launched. This was coded by Clinger and was a rewrite of the entire software rather than improvements to the existing YaBB based setup. The main goals of this rewrite were to improve the overall speed of the software and add new features to keep the product competitive.
In February 2003, version 3 of the ProBoards software was released, again making improvements on the overall speed of the software and including over 30 new features.
ProBoards upgraded to version 4 of its software on April 30, 2005. This time, the upgrade added over 100 new features and enhancements to the service. Despite this, bugs of varying levels of severity still existed.
The current version of the software is v5.
ProBoards' servers - physical machines running the ProBoards software - are hosted by SoftLayer. Previous to November 2010, ProBoards was hosted by ThePlanet.com, and previous to 2006, EV1 Servers. The servers are hosted in multiple SoftLayer datacenters in Texas.
In 2005, Patrick Clinger was invited by EV1 Servers to take part in a commercial for their business. The
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https://en.wikipedia.org/wiki/Raj%20Chandra%20Bose
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Raj Chandra Bose (or Basu) (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He also invented the notions of partial geometry, association scheme, and strongly regular graph and started a systematic study of difference sets to construct symmetric block designs.
He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that for no n do there exist two mutually orthogonal Latin squares of order 4n + 2.
Early life
Bose was born in Hoshangabad, India; he was the first of five children. His father was a physician and life was good until 1918 when his mother died in the influenza pandemic. His father died of a stroke the following year. Despite difficult circumstances, Bose continued to study securing first class in both the Masters examinations in Pure and Applied mathematics in 1925 and 1927 respectively at the Rajabazar Science College campus of University of Calcutta. His research was performed under the supervision of the geometry Professor Syamadas Mukhopadhyaya from Calcutta. Bose worked as a lecturer at Asutosh College, Calcutta. He published his work on the differential geometry of convex curves.
Academic life
Bose's course changed in December 1932 when P. C. Mahalanobis, director of the new (1931) Indian Statistical Institute, offered Bose a part-time job. Mahalanobis had seen Bose's geometrical work and wanted him to work on statistics. The day after Bose moved in, the secretary brought him all the volumes of Biometrika with a list of 50 papers to read and also Ronald Fisher's Statistical Methods for Research Workers. Mahalanobis told him, "You were saying that you do not know much statistics. You master the 50 papers ... and Fisher's book. This will suffice for your statistical education for the present." With Samarendra Nath Roy, who joined the ISI a little later, Bose was the chief mathematician at the Institute.
He first worked with multivariate analysis where he collaborated with Mahalanobis and Roy. In 1938–9 Fisher visited India and talked about the design of experiments. Roy had the idea of using the theory of finite fields and finite geometry to solve problems in design. The development of a mathematical theory of design would be Bose's main preoccupation until the mid-1950s.
In 1935 Bose had become full-time at the Institute. In 1940 joined the University of Calcutta where C. R. Rao and H. K. Nandi were in the first group of students he taught. In 1945 Bose became Head of the Department of Statistics.
University authorities in the United States told him he needed to have a doctorate. So he submitted his published papers on multivariate analysis and the design of experiments and was awarded a D. Litt. in 1947.
In 1947 Bose went to the United Sta
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https://en.wikipedia.org/wiki/A.%20A.%20Krishnaswami%20Ayyangar
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A. A. Krishnaswami Ayyangar (1892–1953) was an Indian mathematician. He received his M.A. in Mathematics at the age of 18 from Pachaiyappa's College, and subsequently taught mathematics there. In 1918, he joined the mathematics department of the University of Mysore and retired from there in 1947. He was born in a Tamil Brahmin family. He died in June 1953. He was the father of the Kannada poet and scholar A. K. Ramanujan.
Works
Ayyangar had a number of publications, including an article on the Chakravala method where he showed how the method differed from the method of continued fractions. He pointed out that this point was missed by André Weil, who thought that the Chakravala method was only an "experimental fact" to the Indians and attributed general proofs to Pierre de Fermat and Joseph-Louis Lagrange.
Professor Subhash Kak of Louisiana State University, Baton Rouge first noted that Ayyangar's presentations of the work of other Indian mathematicians was unique, and was instrumental in bringing it to the notice of the scientific community.
References
External links
Brief life and some papers
1892 births
1953 deaths
20th-century Indian mathematicians
Academic staff of the University of Mysore
Scientists from Karnataka
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https://en.wikipedia.org/wiki/Vijay%20Kumar%20Patodi
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Vijay Kumar Patodi (12 March 1945 – 21 December 1976) was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the index theorem for elliptic operators. He was a professor at Tata Institute of Fundamental Research, Mumbai (Bombay).
Education
Patodi was a graduate of Government High School, Guna, Madhya Pradesh. He received his bachelor's degree from Vikram University, Ujjain, his master's degree from the Benaras Hindu University, and his Ph.D. from the University of Bombay under the guidance of M. S. Narasimhan and S. Ramanan at the Tata Institute of Fundamental Research.
In the two papers based on his Ph.D. thesis, "Curvature and Eigenforms of the Laplace Operator" (Journal of Differential Geometry), and "An Analytical Proof of the Riemann-Roch-Hirzebruch Formula for Kaehler Manifolds" (also Journal of Differential Geometry), Patodi made his fundamental breakthroughs.
Research career
He was invited to spend 1971–1973 at the Institute for Advanced Study in Princeton, New Jersey, where he collaborated with Michael Atiyah, Isadore Singer, and Raoul Bott. The joint work led to a series of papers, "Spectral Asymmetry and Riemannian Geometry" (Math. Proc. Cambridge. Phil. Soc.) with Atiyah and Singer, in which the η-invariant was defined. This invariant was to play a major role in subsequent advances in the area in the 1980s.
Patodi was promoted to full professor at Tata Institute at age 30, however, he died at age 31, as a result of complications prior to surgery for a kidney transplant.
References
External links
Concise Biography
20th-century Indian mathematicians
Differential geometers
Indian topologists
Tata Institute of Fundamental Research alumni
Institute for Advanced Study visiting scholars
Vikram University alumni
Banaras Hindu University alumni
University of Mumbai alumni
1945 births
1976 deaths
Scientists from Madhya Pradesh
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https://en.wikipedia.org/wiki/Knot%20group
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In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,
Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in .
Properties
Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of that is isotopic to the identity and sends the first knot onto the second. Such a homeomorphism restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example).
The abelianization of a knot group is always isomorphic to the infinite cyclic group Z; this follows because the abelianization agrees with the first homology group, which can be easily computed.
The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm.
Examples
The unknot has knot group isomorphic to Z.
The trefoil knot has knot group isomorphic to the braid group B3. This group has the presentation
or
A (p,q)-torus knot has knot group with presentation
The figure eight knot has knot group with presentation
The square knot and the granny knot have isomorphic knot groups, yet these two knots are not equivalent.
See also
Link group
Further reading
Hazewinkel, Michiel, ed. (2001), "Knot and Link Groups", Encyclopedia of Mathematics, Springer,
Knot invariants
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https://en.wikipedia.org/wiki/ISTAT
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ISTAT may refer to:
International Society of Transport Aircraft Trading
National Institute of Statistics (Italy) or Istituto Nazionale di Statistica
i-STAT, a blood analyzer made by Abbott Laboratories
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https://en.wikipedia.org/wiki/David%20J.%20Simms
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David John Simms (13 January 1933 – 24 June 2018) was an Indian-born Irish mathematician who was a Fellow Emeritus and former Associate Professor of Mathematics at Trinity College, Dublin. Born in Sankeshwar, Mysore (the state now known as Karnataka), India, he specialized in differential geometry and geometric quantisation. He was a member of the Royal Irish Academy from 1978 and was a member of the Editorial Board of the journal Mathematical Proceedings of the Royal Irish Academy.
Academic career
Simms completed his undergraduate degree in Mathematics in Trinity College Dublin, graduating in 1955. He was elected a Scholar of the College in 1952, when he was just in the first year of his degree, a notable achievement. He went on to do a Ph.D. in the University of Cambridge under W. V. D. Hodge. Simms lectured in Glasgow University before returning to Trinity. He served as head of the Department of Pure and Applied Mathematics from 1991 to 1998.
Simms' research interests included differential geometry and geometric quantisation.
Books and select publications
Lie Groups and Quantum Mechanics, Springer Lecture Notes in Mathematics Number 52, 1968
Lectures on Geometric Quantization, (with N.M.J. Woodhouse) Springer Lecture Notes in Physics Number 53, 1976 professional papers.
Geometric quantization of energy levels in the Kepler problem, D.J. Simms - Symposia Mathematica, 1974
David Simms was a member of the Royal Irish Academy since 1978. He was a member of the Editorial Board of the journal Mathematical Proceedings of the Royal Irish Academy.
Personal life
Simms was married to Anngret Erichson, a former associate professor and head of geography at University College Dublin. They had three sons, one of whom, Brendan Simms, is a professor of international relations at Cambridge University. As a child Simms survived 13 days at sea following the sinking of the SS City of Cairo in November 1942. He was also the nephew of Irish communist Brian Goold-Verschoyle. He was also the nephew of Archbishop George Otto Simms.
Simms died on 25 June 2018 in Dublin.
References
External links
Professor Simms' Webpage
TCD Scholars Site
S.S. City of Cairo website
Flesh and Blood - RTE documentary on Goold-Verschoyle family
1933 births
2018 deaths
Academics of Trinity College Dublin
Academics of the University of Glasgow
Alumni of Trinity College Dublin
Alumni of the University of Cambridge
Differential geometers
Fellows of Trinity College Dublin
Members of the Royal Irish Academy
Scholars of Trinity College Dublin
20th-century Irish mathematicians
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https://en.wikipedia.org/wiki/Principle%20of%20distributivity
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The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other so that for any propositions A, B and C the equivalences
and
hold.
The principle of distributivity is valid in classical logic, but both valid and invalid in quantum logic.
The article "Is Logic Empirical?" discusses the case that quantum logic is the correct, empirical logic, on the grounds that the principle of distributivity is inconsistent with a reasonable interpretation of quantum phenomena.
References
Abstract algebra
Principles
Propositional calculus
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https://en.wikipedia.org/wiki/Is%20Logic%20Empirical%3F
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"Is Logic Empirical?" is the title of two articles (one by Hilary Putnam and another by Michael Dummett) that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical rendering of reality. The replacement derives from the work of Garrett Birkhoff and John von Neumann on quantum logic. In their work, they showed that the outcomes of quantum measurements can be represented as binary propositions and that these quantum mechanical propositions can be combined in a similar way as propositions in classical logic. However, the algebraic properties of this structure are somewhat different from those of classical propositional logic in that the principle of distributivity fails.
The idea that the principles of logic might be susceptible to revision on empirical grounds has many roots, including the work of W. V. Quine and the foundational studies of Hans Reichenbach.
W. V. Quine
What is the epistemological status of the laws of logic? What sort of arguments are appropriate for criticising purported principles of logic? In his seminal paper "Two Dogmas of Empiricism," the logician and philosopher W. V. Quine argued that all beliefs are in principle subject to revision in the face of empirical data, including the so-called analytic propositions. Thus the laws of logic, being paradigmatic cases of analytic propositions, are not immune to revision.
To justify this claim he cited the so-called paradoxes of quantum mechanics. Birkhoff and von Neumann proposed to resolve those paradoxes by abandoning the principle of distributivity, thus substituting their quantum logic for classical logic.
Quine did not at first seriously pursue this argument, providing no sustained argument for the claim in that paper. In Philosophy of Logic (the chapter titled "Deviant Logics"), Quine rejects the idea that classical logic should be revised in response to the paradoxes, being concerned with "a serious loss of simplicity", and "the handicap of having to think within a deviant logic". Quine, though, stood by his claim that logic is in principle not immune to revision.
Hans Reichenbach
Reichenbach considered one of the anomalies associated with quantum mechanics, the problem of complementary properties. A pair of properties of a system is said to be complementary if each one of them can be assigned a truth value in some experimental setup, but there is no setup which assigns a truth value to both properties. The classic example of complementarity is illustrated by the double-slit experiment in which a photon can be made to exhibit particle-like properties or wave-like properties, depending on the experimental setup used to detect its presence. Another example of complementary properties is that of having a precisely observed position or momentum.
Reichenbach ap
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https://en.wikipedia.org/wiki/Exponential%20formula
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In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures.
The exponential formula is a power-series version of a special case of Faà di Bruno's formula.
Algebraic statement
Here is a purely algebraic statement, as a first introduction to the combinatorial use of the formula.
For any formal power series of the form
we have
where
and the index runs through the list of all partitions of the set
. (When the product is empty and by definition equals .)
Formula in other expressions
One can write the formula in the following form:
and thus
where is the th complete Bell polynomial.
Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:where stands for the cycle index polynomial, for the symmetric group defined as:and denotes the number of cycles of of size . This is a consequence of the general relation between and Bell polynomials:
The combinatorial formula
In applications, the numbers count the number of some sort of "connected" structure on an -point set, and the numbers count the number of (possibly disconnected) structures. The numbers count the number of isomorphism classes of structures on points, with each structure being weighted by the reciprocal of its automorphism group, and the numbers count isomorphism classes of connected structures in the same way.
Examples
because there is one partition of the set that has a single block of size , there are three partitions of that split it into a block of size and a block of size , and there is one partition of that splits it into three blocks of size . This also follows from , since one can write the group as , using cyclic notation for permutations.
If is the number of graphs whose vertices are a given -point set, then is the number of connected graphs whose vertices are a given -point set.
There are numerous variations of the previous example where the graph has certain properties: for example, if counts graphs without cycles, then counts trees (connected graphs without cycles).
If counts directed graphs whose (rather than vertices) are a given point set, then counts connected directed graphs with this edge set.
In quantum field theory and statistical mechanics, the partition functions , or more generally correlation functions, are given by a formal sum over Feynman diagrams. The exponential formula shows that can be written as a sum over connected Feynman diagrams, in terms of connected correlation functions.
See also
References
Chapter 5 page 3
Exponentials
Enumerative combinatorics
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https://en.wikipedia.org/wiki/Magnetic%20reconnection
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Magnetic reconnection is a physical process occurring in electrically conducting plasmas, in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle acceleration. Magnetic reconnection involves plasma flows at a substantial fraction of the Alfvén wave speed, which is the fundamental speed for mechanical information flow in a magnetized plasma.
The concept of magnetic reconnection was developed in parallel by researchers working in solar physics and in the interaction between the solar wind and magnetized planets. This reflects the bidirectional nature of reconnection, which can either disconnect formerly connected magnetic fields or connect formerly disconnected magnetic fields, depending on the circumstances.
Ron Giovanelli is credited with the first publication invoking magnetic energy release as a potential mechanism for particle acceleration in solar flares. Giovanelli proposed in 1946 that solar flares stem from the energy obtained by charged particles influenced by induced electric fields within close proximity of sunspots. In the years 1947-1948, he published more papers further developing the reconnection model of solar flares. In these works, he proposed that the mechanism occurs at points of neutrality (weak or null magnetic field) within structured magnetic fields.
James Dungey is credited with first use of the term “magnetic reconnection” in his 1950 PhD thesis, to explain the coupling of mass, energy and momentum from the solar wind into Earth's magnetosphere. The concept was published for the first time in a seminal paper in 1961. Dungey coined the term "reconnection" because he envisaged field lines and plasma moving together in an inflow toward a magnetic neutral point (2D) or line (3D), breaking apart and then rejoining again but with different magnetic field lines and plasma, in an outflow away from the magnetic neutral point or line.
In the meantime, the first theoretical framework of magnetic reconnection was established by Peter Sweet and Eugene Parker at a conference in 1956. Sweet pointed out that by pushing two plasmas with oppositely directed magnetic fields together, resistive diffusion is able to occur on a length scale much shorter than a typical equilibrium length scale. Parker was in attendance at this conference and developed scaling relations for this model during his return travel.
Fundamental principles
Magnetic reconnection is a breakdown of "ideal-magnetohydrodynamics" and so of "Alfvén's theorem" (also called the "frozen-in flux theorem") which applies to large-scale regions of a highly-conducting magnetoplasma, for which the Magnetic Reynolds Number is very large: this makes the convective term in the induction equation dominate in such regions. The frozen-in flux theorem states that in such regions the field moves with the plasma velocity (the mean of the ion and electron velocities, weighted by their mass). The reconnection breakdown
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https://en.wikipedia.org/wiki/Reflexivity
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Reflexivity might mean:
Reflexivity (grammar)
Reflexivity (social theory)
Self-reflexivity (see Self-reference)
See also
Reflectivism
Reflexive (disambiguation)
Reflexive operator algebra
Reflexive pronoun
Reflexive relation
Reflexive space
Reflexive verb
Sesquilinear form
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https://en.wikipedia.org/wiki/Assist%20%28association%20football%29
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In association football, an assist is a contribution by a player which helps to score a goal. Statistics for assists made by players may be kept officially by the organisers of a competition, or unofficially by, for example, journalists or organisers of fantasy football competitions. Recording assists is not part of the official Laws of the Game and the criteria for an assist to be awarded may vary. Record of assists was virtually not kept at all until the end of the 20th century, although reports of matches commonly described a player as having "made" one or more goals. Since the 1990s, some leagues have kept official record of assists and based awards on them.
Criteria
Most commonly, an assist is credited to a player for passing or crossing the ball to the scorer. It may also be awarded to a player whose shot rebounds (off a defender, goalkeeper or goalpost) to a teammate who scores. Some systems may credit an assist to a player who wins a penalty kick or a free kick for another player to convert, or to an attacking player for contributing to an own goal. A goal may be unassisted, or have one assist; some systems allow for two assists.
FIFA World Cup
FIFA's Technical Study Group is responsible for awarding assist points at the FIFA World Cup. In the Technical Study Group's report on the 1986 World Cup, the authors calculated for the first time unofficial statistics for assists, developing the following criteria:
An assist was awarded to the player who had given the last pass to the goalscorer.
In addition, the last but two holder of the ball could get an assist provided that his action had decisive importance for the goal.
After goals from rebounds those players were awarded an assist who had shot on target.
After goals scored on penalty or by a directly converted free-kick the fouled player received a point.
In case that the goalscorer had laid on the goal for himself (dribble, solo run), no assists were awarded.
No assists were awarded, either, if the goalscorer took advantage of a missed pass by an opponent.
The 1990 World Cup technical report adopted similar criteria, but changed the free-kick/penalty criterion:
Where goals resulting from penalties are concerned, the player who is fouled in the area receives an assist point (unless, that is, the player who is fouled subsequently executes the penalty himself).
Planet World Cup has calculated some retrospective data on assists back to the 1966 World Cup, though the 1986 data differs from that of FIFA.
FIFA started officially keeping track of assists in World Cup tournaments at the 1994 edition. This was popularly ascribed to the popularity of detailed sports statistics among fans. 1994 was also the first World Cup in which assists were used as a tie-breaker in determining the Golden Shoe award for top scorer. In the event, both Hristo Stoichkov and Oleg Salenko tied with 19 points, from 6 goals and 1 assist.
France
The French league, Ligue 1, awards the Trophée de Meilleur Passe
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https://en.wikipedia.org/wiki/Nilpotent%20matrix
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In linear algebra, a nilpotent matrix is a square matrix N such that
for some positive integer . The smallest such is called the index of , sometimes the degree of .
More generally, a nilpotent transformation is a linear transformation of a vector space such that for some positive integer (and thus, for all ). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.
Examples
Example 1
The matrix
is nilpotent with index 2, since .
Example 2
More generally, any -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index . For example, the matrix
is nilpotent, with
The index of is therefore 4.
Example 3
Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,
although the matrix has no zero entries.
Example 4
Additionally, any matrices of the form
such as
or
square to zero.
Example 5
Perhaps some of the most striking examples of nilpotent matrices are square matrices of the form:
The first few of which are:
These matrices are nilpotent but there are no zero entries in any powers of them less than the index.
Example 6
Consider the linear space of polynomials of a bounded degree. The derivative operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix.
Characterization
For an square matrix with real (or complex) entries, the following are equivalent:
is nilpotent.
The characteristic polynomial for is .
The minimal polynomial for is for some positive integer .
The only complex eigenvalue for is 0.
The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities)
This theorem has several consequences, including:
The index of an nilpotent matrix is always less than or equal to . For example, every nilpotent matrix squares to zero.
The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible.
The only nilpotent diagonalizable matrix is the zero matrix.
See also: Jordan–Chevalley decomposition#Nilpotency criterion.
Classification
Consider the (upper) shift matrix:
This matrix has 1s along the superdiagonal and 0s everywhere else. As a linear transformation, the shift matrix "shifts" the components of a vector one position to the left, with a zero appearing in the last position:
This matrix is nilpotent with degree , and is the canonical nilpotent matrix.
Specifically, if is any nilpotent matrix, then is similar to a block diagonal matrix of the form
where each of the blocks is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices.
For example, any nonzero 2 × 2 nilp
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https://en.wikipedia.org/wiki/List%20of%20mathematical%20knots%20and%20links
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This article contains a list of mathematical knots and links. See also list of knots, list of geometric topology topics.
Knots
Prime knots
01 knot/Unknot - a simple un-knotted closed loop
31 knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together
41 knot/Figure-eight knot (mathematics) - a prime knot with a crossing number four
51 knot/Cinquefoil knot, (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a {5/2} star polygon (pentagram)
52 knot/Three-twist knot - the twist knot with three-half twists
61 knot/Stevedore knot (mathematics) - a prime knot with crossing number six, it can also be described as a twist knot with four twists
62 knot - a prime knot with crossing number six
63 knot - a prime knot with crossing number six
71 knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a {7/2} star polygon (heptagram)
74 knot, "endless knot"
818 knot, "carrick mat"
10161/10162, known as the Perko pair; this was a single knot listed twice in Dale Rolfsen's knot table; the duplication was discovered by Kenneth Perko
12n242/(−2,3,7) pretzel knot
(p, q)-torus knot - a special kind of knot that lies on the surface of an unknotted torus in R3
Composite
Square knot (mathematics) - a composite knot obtained by taking the connected sum of a trefoil knot with its reflection
Granny knot (mathematics) - a composite knot obtained by taking the connected sum of two identical trefoil knots
Links
0 link/Unlink - equivalent under ambient isotopy to finitely many disjoint circles in the plane
2 link/Hopf link - the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1)
4 link/Solomon's knot (a two component "link" rather than a one component "knot") - a traditional decorative motif used since ancient times (L4a1)
5 link/Whitehead link - two projections of the unknot: one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form (L5a1)
Brunnian link - a nontrivial link that becomes trivial if any component is removed
6 link/Borromean rings - three topological circles which are linked and form a Brunnian link (L6a4)
L10a140 link - presumably the simplest non-Borromean Brunnian link
Pretzel link - a Montesinos link with integer tangles
External links
Knots and links
Knot theory
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https://en.wikipedia.org/wiki/Pl%C3%BCcker%20coordinates
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In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe -dimensional linear subspaces, or flats, in an -dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.
Geometric intuition
A line in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points and The vector displacement from to is nonzero because the points are distinct, and represents the direction of the line. That is, every displacement between points on is a scalar multiple of . If a physical particle of unit mass were to move from to , it would have a moment about the origin. The geometric equivalent to this moment, is a vector whose direction is perpendicular to the plane containing and the origin, and whose length equals twice the area of the triangle formed by the displacement and the origin. Treating the points as displacements from the origin, the moment is , where "×" denotes the vector cross product. For a fixed line, , the area of the triangle is proportional to the length of the segment between and , considered as the base of the triangle; it is not changed by sliding the base along the line, parallel to itself. By definition the moment vector is perpendicular to every displacement along the line, so , where "⋅" denotes the vector dot product.
Although neither nor alone is sufficient to determine , together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between and . That is, the coordinates
may be considered homogeneous coordinates for , in the sense that all pairs , for , can be produced by points on and only , and any such pair determines a unique line so long as is not zero and . Furthermore, this approach extends to include points, lines, and a plane "at infinity", in the sense of projective geometry. In addition a point lies on the line if and only if .
Example. Let and . Then .
Alternatively, let the equations for points of two distinct planes containing be
Then their respective planes are perpendicular to vectors and , and the direction of must be perpendicular to both. Hence we may set , which is nonzero because are neither zero nor parallel (the planes being distinct and intersecting). If point satisfies both plane equations, then it also satisfies the linear combination
That is,
is a vector perpendicular to displac
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https://en.wikipedia.org/wiki/List%20of%20algebraic%20number%20theory%20topics
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This is a list of algebraic number theory topics.
Basic topics
These topics are basic to the field, either as prototypical examples, or as basic objects of study.
Algebraic number field
Gaussian integer, Gaussian rational
Quadratic field
Cyclotomic field
Cubic field
Biquadratic field
Quadratic reciprocity
Ideal class group
Dirichlet's unit theorem
Discriminant of an algebraic number field
Ramification (mathematics)
Root of unity
Gaussian period
Important problems
Fermat's Last Theorem
Class number problem for imaginary quadratic fields
Stark–Heegner theorem
Heegner number
Langlands program
General aspects
Different ideal
Dedekind domain
Splitting of prime ideals in Galois extensions
Decomposition group
Inertia group
Frobenius automorphism
Chebotarev's density theorem
Totally real field
Local field
p-adic number
p-adic analysis
Adele ring
Idele group
Idele class group
Adelic algebraic group
Global field
Hasse principle
Hasse–Minkowski theorem
Galois module
Galois cohomology
Brauer group
Class field theory
Class field theory
Abelian extension
Kronecker–Weber theorem
Hilbert class field
Takagi existence theorem
Hasse norm theorem
Artin reciprocity
Local class field theory
Iwasawa theory
Iwasawa theory
Herbrand–Ribet theorem
Vandiver's conjecture
Stickelberger's theorem
Euler system
p-adic L-function
Arithmetic geometry
Arithmetic geometry
Complex multiplication
Abelian variety of CM-type
Chowla–Selberg formula
Hasse–Weil zeta function
Mathematics-related lists
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https://en.wikipedia.org/wiki/Toroidal
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Toroidal describes something which resembles or relates to a torus or toroid:
Mathematics
Torus
Toroid, a surface of revolution which resembles a torus
Toroidal polyhedron
Toroidal coordinates, a three-dimensional orthogonal coordinate system
Toroidal and poloidal coordinates, directions relative to a torus of reference
Toroidal graph, a graph whose vertices can be placed on a torus such that no edges cross
Toroidal grid network, where an n-dimensional grid network is connected circularly in more than one dimension
Engineering
Toroidal inductors and transformers, a type of electrical device
Toroidal and poloidal, directions in magnetohydrodynamics
Toroidal engine, an internal combustion engine with pistons that rotate within a toroidal space
Toroidal CVT, a type of continuously variable transmission
Toroidal reflector, a parabolic reflector which has a different focal distance depending on the angle of the mirror
Toroidal propeller, an efficient propeller design
Other uses
Toroidal ring model in theoretical physics
Vortex ring, also known as a toroidal vortex; a toroidal flow in fluid mechanics
See also
Atoroidal
Torus (disambiguation)
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https://en.wikipedia.org/wiki/Adelic%20algebraic%20group
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In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear algebraic group. In the case of G being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.
In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelic algebraic group is taken to be the subspace topology in AN, the Cartesian product of N copies of the adele ring. In this case, is a topological group.
History of the terminology
Historically the idèles () were introduced by under the name "élément idéal", which is "ideal element" in French, which then abbreviated to "idèle" following a suggestion of Hasse. (In these papers he also gave the ideles a non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in terms of topological groups. defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of Idealelemente was the group of invertible elements of this ring. defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.
defined the ring of adeles in the function field case, under the name "repartitions"; the contemporary term adèle stands for 'additive idèles', and can also be a French woman's name. The term adèle was in use shortly afterwards and may have been introduced by André Weil. The general construction of adelic algebraic groups by followed the algebraic group theory founded by Armand Borel and Harish-Chandra.
Ideles
An important example, the idele group (ideal element group) I(K), is the case of . Here the set of ideles consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles. Instead, considering that lies in two-dimensional affine space as the 'hyperbola' defined parametrically by
the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A.
Inside AN, the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group
is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a com
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https://en.wikipedia.org/wiki/Tamagawa%20number
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In mathematics, the Tamagawa number of a semisimple algebraic group defined over a global field is the measure of , where is the adele ring of . Tamagawa numbers were introduced by , and named after him by .
Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on , defined over , the measure involved was well-defined: while could be replaced by with a non-zero element of , the product formula for valuations in is reflected by the independence from of the measure of the quotient, for the product measure constructed from on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.
Definition
Let be a global field, its ring of adeles, and a semisimple algebraic group defined over .
Choose Haar measures on the completions such that has volume 1 for all but finitely many places . These then induce a Haar measure on , which we further assume is normalized so that has volume 1 with respect to the induced quotient measure.
The Tamagawa measure on the adelic algebraic group is now defined as follows. Take a left-invariant -form on defined over , where is the dimension of . This, together with the above choices of Haar measure on the , induces Haar measures on for all places of . As is semisimple, the product of these measures yields a Haar measure on , called the Tamagawa measure. The Tamagawa measure does not depend on the choice of ω, nor on the choice of measures on the , because multiplying by an element of multiplies the Haar measure on by 1, using the product formula for valuations.
The Tamagawa number is defined to be the Tamagawa measure of .
Weil's conjecture on Tamagawa numbers
Weil's conjecture on Tamagawa numbers states that the Tamagawa number of a simply connected (i.e. not having a proper algebraic covering) simple algebraic group defined over a number field is 1. calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. found examples where the Tamagawa numbers are not integers, but the conjecture about the Tamagawa number of simply connected groups was proven in general by several works culminating in a paper by and for the analogue over function fields over finite fields by Lurie and Gaitsgory in 2011.
See also
Adelic algebraic group
References
.
Further reading
Aravind Asok, Brent Doran and Frances Kirwan, "Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics", February 22, 2013
J. Lurie, The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality posted June 8, 2012.
Algebraic groups
Algebraic number theory
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https://en.wikipedia.org/wiki/Zero%20matrix
|
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of matrices, and is denoted by the symbol or followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are
Properties
The set of matrices with entries in a ring K forms a ring . The zero matrix in is the matrix with all entries equal to , where is the additive identity in K.
The zero matrix is the additive identity in . That is, for all it satisfies the equation
There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents the linear transformation which sends all the vectors to the zero vector. It is idempotent, meaning that when it is multiplied by itself, the result is itself.
The zero matrix is the only matrix whose rank is 0.
Occurrences
The mortal matrix problem is the problem of determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.
In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.
See also
Identity matrix, the multiplicative identity for matrices
Matrix of ones, a matrix where all elements are one
Nilpotent matrix
Single-entry matrix, a matrix where all but one element is zero
References
Matrices
0 (number)
Sparse matrices
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https://en.wikipedia.org/wiki/Markov%20blanket
|
In statistics and machine learning, when one wants to infer a random variable with a set of variables, usually a subset is enough, and other variables are useless. Such a subset that contains all the useful information is called a Markov blanket. If a Markov blanket is minimal, meaning that it cannot drop any variable without losing information, it is called a Markov boundary. Identifying a Markov blanket or a Markov boundary helps to extract useful features. The terms of Markov blanket and Markov boundary were coined by Judea Pearl in 1988. A Markov blanket can be constituted by a set of Markov chains.
Markov blanket
A Markov blanket of a random variable in a random variable set is any subset of , conditioned on which other variables are independent with :
It means that contains at least all the information one needs to infer , where the variables in are redundant.
In general, a given Markov blanket is not unique. Any set in that contains a Markov blanket is also a Markov blanket itself. Specifically, is a Markov blanket of in .
Markov boundary
A Markov boundary of in is a subset of , that itself is a Markov blanket of , but any proper subset of is not a Markov blanket of . In other words, a Markov boundary is a minimal Markov blanket.
The Markov boundary of a node in a Bayesian network is the set of nodes composed of 's parents, 's children, and 's children's other parents. In a Markov random field, the Markov boundary for a node is the set of its neighboring nodes. In a dependency network, the Markov boundary for a node is the set of its parents.
Uniqueness of Markov boundary
The Markov boundary always exists. Under some mild conditions, the Markov boundary is unique. However, for most practical and theoretical scenarios multiple Markov boundaries may provide alternative solutions. When there are multiple Markov boundaries, quantities measuring causal effect could fail.
See also
Andrey Markov
Free energy minimisation
Moral graph
Separation of concerns
Causality
Causal inference
Notes
Bayesian networks
Markov networks
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https://en.wikipedia.org/wiki/Causal%20Markov%20condition
|
The Markov condition, sometimes called the Markov assumption, is an assumption made in Bayesian probability theory, that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents. Stated loosely, it is assumed that a node has no bearing on nodes which do not descend from it. In a DAG, this local Markov condition is equivalent to the global Markov condition, which states that d-separations in the graph also correspond to conditional independence relations. This also means that a node is conditionally independent of the entire network, given its Markov blanket.
The related Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. However, a network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition.
Motivation
Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of causation is that causes raise the probabilities of their effects, all else being equal.
A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer.
On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:).
Implications
Dependence and Causation
It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V. This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP)
Screening
It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.
Examples
In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall.
A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the
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https://en.wikipedia.org/wiki/Chirality%20%28mathematics%29
|
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.
A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek (enantios) 'opposite' + (morphe) 'form'.
Examples
Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.
Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out.
The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.
Chirality and symmetry group
A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. (In Euclidean geometry any isometry can be written as with an orthogonal matrix and a vector . The determinant of is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.
A general definition of chirality based on group theory exists. It does not refer to any orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.
Chirality in two dimensions
In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis of symmetry of a figure is a line , such that is invariant under the mapping , when is chosen to be the -axis of the coordinate system.) For that reason, a triangle is achiral if it is equilateral or isosceles, and is chiral if it is scalene.
Consider the following pattern:
This figure is chiral, as it is not identical to its mirror image:
But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection.
Chirality in three dimensions
In three dimensions, every figure that possesses a mirror plane of symmetry S1, an inversion center of symmetry S2, or a higher improper rotation (rotoreflection) Sn axis of symmetry is achiral. (A plane of symmetry of a figure is a plane , such that is invariant under the mapping , when is chosen to be the --plane of the coordinate system. A center of symmetry of a figure is a point , such that is invariant under the m
|
https://en.wikipedia.org/wiki/Disphenocingulum
|
In geometry, the disphenocingulum or pentakis elongated gyrobifastigium is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
Cartesian coordinates
Let a ≈ 0.76713 be the second smallest positive root of the polynomial
and and .
Then, Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane.
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Bilunabirotunda
|
In geometry, the bilunabirotunda is one of the Johnson solids ().
Geometry
It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Either one of the two clusters of two pentagons and two triangles can be aligned with a congruent patch of faces on the icosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the icosidodecahedron, then two vertices of the bilunabirotundae meet in the very center of the icosidodecahedron.
The other two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.
Each of the two pairs of adjacent pentagons (each pair of pentagons sharing an edge) can be aligned with the pentagonal faces of a metabidiminished icosahedron as well.
The bilunabirotunda has a weak relationship with the cuboctahedron, as it may be created by replacing four square faces of the cuboctahedron with pentagons.
Cartesian coordinates
The following define the vertices of a bilunabirotunda centered at the origin with edge length 1:
where is the golden ratio.
Related polyhedra and honeycombs
Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this six-bilunabirotunda model as 6J91(P4).
The bilunabirotunda can be used with the regular dodecahedron and cube as a space-filling honeycomb.
External links
Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)
Johnson solids
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https://en.wikipedia.org/wiki/Triangular%20hebesphenorotunda
|
In geometry, the triangular hebesphenorotunda is one of the Johnson solids ().
It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron.
The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the rhombicosidodecahedron: the three lunes, each lune consisting of a square and two antipodal triangles adjacent to the square.
The faces around each vertex can also be aligned with the corresponding faces of various diminished icosahedra.
Johnson uses the prefix hebespheno- to refer to a blunt wedge-like complex formed by three adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears structural resemblance to the pentagonal rotunda.
The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides.
Cartesian coordinates
Cartesian coordinates for the triangular hebesphenorotunda with edge length – 1 are given by the union of the orbits of the points
under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, = (sometimes written φ) is the golden ratio. The first point generates the triangle opposite the hexagon, the second point generates the bases of the triangles surrounding the previous triangle, the third point generates the tips of the pentagons opposite the first triangle, and the last point generates the hexagon.
One may then calculate the surface area of a triangular hebesphenorotunda of edge length a as
and its volume as
A second, inverted, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. This second polyhedron will be joined to the first at their common hexagonal face, and the pair will inscribe an icosidodecahedron. If the hexagonal face is scaled by the golden ratio, then the convex hull of the result will be the entire icosidodecahedron.
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/FOCAL%20%28programming%20language%29
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FOCAL (acronym for Formulating On-line Calculations in Algebraic Language, or FOrmula CALculator) is an interactive interpreted programming language based on JOSS and mostly used on Digital Equipment Corporation (DEC) Programmed Data Processor (PDP) series machines.
JOSS was designed to be a simple interactive language to allow programs to be easily written by non-programmers. FOCAL is very similar to JOSS in the commands it supports and the general syntax of the language. It differs in that many of JOSS' advanced features like ranges and user-defined functions were removed to simplify the parser. Some of the reserved words (keywords) were renamed so that they all start with a unique first letter. This allows users to type in programs using one-character statements, further reducing memory needs. This was an important consideration on the PDP-8, which was often limited to a few kilobytes (KB).
Like JOSS, and later BASICs, FOCAL on the PDP-8 was a complete environment that included a line editor, an interpreter, and input/output routines. The package as a whole was named FOCAL-8, which also ran on the PDP-5 and PDP-12. When ported to the PDP-11, the resulting FOCAL-11 relied on the underlying operating system, RT-11, to provide file support and editing. The language definition was updated twice, to FOCAL-69 and a very slightly modified FOCAL-71. A port to the Intel 8080 was also available.
FOCAL is notable as the language in which the original versions of the early video games Hamurabi and Lunar Lander were written. Both were later ported to BASIC, where they became much better known. FOCAL was not popular outside the PDP platform and largely disappeared during the move to the VAX-11. It had a strong revival in the Soviet Union where PDP-11 clones were used as educational and home computers (BK series).
History
JOSS
JOSS was released in May 1963 on the one-off JOHNNIAC computer at RAND Corporation. In RAND, use grew rapidly, and the machine, originally built in 1953, quickly ran out of capability. JOHNNIAC was decommissioned in 1966 and JOSS was reimplemented on a newly purchased PDP-6, Digital Equipment Corporation's (DEC) first "big" machine. Use continued to grow and by 1970, the system was being used by 500 to 600 users across the country and had spawned several innovations such as mobile computer terminals that could be wheeled from room to room and plugged in for quick access.
JOSS was highly influential. It emerged just as time-sharing was being introduced. There was significant interest in man-machine interaction and computers were seeing wider use. Whereas most time-sharing operating systems of the era concentrated on user account and file management, leaving the users to do their own programming, JOSS provided file editing and a programming language in one package. RAND showed the system to a parade of people in the industry.
FOCAL
The PDP-6 was DEC's first mainframe, and JOSS took full advantage of its power and memory capacity
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https://en.wikipedia.org/wiki/Augmented%20sphenocorona
|
In geometry, the augmented sphenocorona is one of the Johnson solids (), and is obtained by adding a square pyramid to one of the square faces of the sphenocorona. It is the only Johnson solid arising from "cut and paste" manipulations where the components are not all prisms, antiprisms or sections of Platonic or Archimedean solids.
Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -corona refers to a crownlike complex of 8 equilateral triangles. Finally, the descriptor augmented implies that another polyhedron, in this case a pyramid, is adjointed. Joining both complexes together with the pyramid results in the augmented sphenocorona.
Cartesian coordinates
To calculate Cartesian coordinates for the augmented sphenocorona, one may start by calculating the coordinates of the sphenocorona. Let k ≈ 0.85273 be the smallest positive root of the quartic polynomial
Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane. Calculating the centroid and the normal unit vector of one of the square faces gives the location of its last vertex as
One may then calculate the surface area of a snub square of edge length a as
and its volume as
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Pasch%27s%20theorem
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In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates.
Statement
The statement is as follows: [Here, for example, (, , ) means that point lies between points and .]
See also
Ordered geometry
Pasch's axiom
Notes
References
External links
Euclidean plane geometry
Foundations of geometry
Order theory
Theorems in plane geometry
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https://en.wikipedia.org/wiki/Geom
|
Geom may refer to:
Geom, a Korean sword
GEOM, a modular disk framework used in FreeBSD 5.0 and newer
An abbreviation of geometry
The God-Emperor of Mankind, a core character in the Warhammer 40,000 fictional universe
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https://en.wikipedia.org/wiki/Hendecagram
|
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices.
The name hendecagram combines a Greek numeral prefix, hendeca-, with the Greek suffix -gram. The hendeca- prefix derives from Greek ἕνδεκα (ἕν + δέκα, one + ten) meaning "eleven". The -gram suffix derives from γραμμῆς (grammēs) meaning a line.
Regular hendecagrams
There are four regular hendecagrams, which can be described by the notation
{11/2}, {11/3}, {11/4}, and {11/5}; in this notation, the number after the slash indicates the number of steps between pairs of points that are connected by edges. These same four forms can also be considered as stellations of a regular hendecagon.
Since 11 is prime, all hendecagrams are star polygons and not compound figures.
Construction
As with all odd regular polygons and star polygons whose orders are not products of distinct Fermat primes, the regular hendecagrams cannot be constructed with compass and straightedge.
However, describe folding patterns for making the hendecagrams {11/3}, {11/4}, and {11/5} out of strips of paper.
Applications
Prisms over the hendecagrams {11/3} and {11/4} may be used to approximate the shape of DNA molecules.
Fort Wood, now the base of the Statue of Liberty in New York City, is a star fort in the form of an irregular 11-point star.
The Topkapı Scroll contains images of an 11-pointed star Girih form used in Islamic art. The star in this scroll is not one of the regular forms of the hendecagram, but instead uses lines that connect the vertices of a hendecagon to nearly-opposite midpoints of the hendecagon's edges. 11-pointed star Girih patterns are also used on the exterior of the Momine Khatun Mausoleum; Eric Broug writes that its pattern "can be considered a high point in Islamic geometric design".
An 11-point star-shaped cross-section was used in the Space Shuttle Solid Rocket Booster, for the core of the forward section of the rocket (the hollow space within which the fuel burns). This design provided more surface area and greater thrust in the earlier part of a launch, and a slower burn rate and reduced thrust after the points of the star were burned away, at approximately the same time as the rocket passed the sound barrier.
See also
Hendecagrammic prism
References
External links
11
11 (number)
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https://en.wikipedia.org/wiki/Jessica%20Utts
|
Jessica Utts (born 1952) is a parapsychologist and statistics professor at the University of California, Irvine. She is known for her textbooks on statistics and her investigation into remote viewing.
Statistics education
In 2003, Utts published an article in American Statistician, a journal published by the American Statistical Association, calling for significant changes to collegiate level statistics education. In the article she argued that curricula do a fine job of covering the mathematical side of statistics, but do a poor job of teaching students the skills necessary to properly interpret statistical results in scientific studies. The argument continues that common errors found in news articles, such as the common misinterpretation that correlative studies show causation, would be reduced if there were significant changes made to standard statistics courses.
In 2016, Utts served as the 111th president of the American Statistical Association.
She is a Fellow of the American Statistical Association, and also a Fellow of the Institute of Mathematical Statistics.
Investigation of remote viewing
In 1995, the American Institutes for Research (AIR) appointed a panel consisting primarily of Utts and Ray Hyman to evaluate a project investigating remote viewing for espionage applications, the Stargate Project, which was funded by the Central Intelligence Agency and Defense Intelligence Agency, and carried out initially by Stanford Research Institute and subsequently by SAIC.
A report by Utts claimed the results were evidence of psychic functioning, however Hyman in his report argued Utts' conclusion that ESP had been proven to exist, especially precognition, was premature and the findings had not been independently replicated. According to Hyman "the overwhelming amount of data generated by the viewers is vague, general, and way off target. The few apparent hits are just what we would expect if nothing other than reasonable guessing and subjective validation are operating." Funding for the project was stopped after these reports were issued. Jessica Utts also co-authored papers with the parapsychologist Edwin May, who took over Stargate in 1985. The psychologist David Marks noted that because Utts had published papers with May "she was not independent of the research team. Her appointment to the review panel is puzzling; an evaluation is likely to be less than partial when an evaluator is not independent of the program under investigation."
The Stargate Project was terminated in 1995 with the conclusion that it was never useful in any intelligence operation. The information was vague and included a lot of irrelevant and erroneous data. There was also reason to suspect that the research managers had adjusted their project reports to fit the known background cues.
Utts is on the executive board of the International Remote Viewing Association (IRVA).
Publications
Books
Seeing Through Statistics, 3rd edition (2005) – The use of statistical
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https://en.wikipedia.org/wiki/Mathematics%20of%20three-phase%20electric%20power
|
In electrical engineering, three-phase electric power systems have at least three conductors carrying alternating voltages that are offset in time by one-third of the period. A three-phase system may be arranged in delta (∆) or star (Y) (also denoted as wye in some areas, as symbolically it is similar to the letter 'Y'). A wye system allows the use of two different voltages from all three phases, such as a 230/400 V system which provides 230 V between the neutral (centre hub) and any one of the phases, and 400 V across any two phases. A delta system arrangement provides only one voltage, but it has a greater redundancy as it may continue to operate normally with one of the three supply windings offline, albeit at 57.7% of total capacity. Harmonic current in the neutral may become very large if nonlinear loads are connected.
Definitions
In a star (wye) connected topology, with rotation sequence L1 - L2 - L3, the time-varying instantaneous voltages can be calculated for each phase A,C,B respectively by:
where:
is the peak voltage,
is the phase angle in radians
is the time in seconds
is the frequency in cycles per second and
voltages L1-N, L2-N and L3-N are referenced to the star connection point.
Diagrams
The below images demonstrate how a system of six wires delivering three phases from an alternator may be replaced by just three. A three-phase transformer is also shown.
Balanced loads
Generally, in electric power systems, the loads are distributed as evenly as is practical among the phases. It is usual practice to discuss a balanced system first and then describe the effects of unbalanced systems as deviations from the elementary case.
Constant power transfer
An important property of three-phase power is that the instantaneous power available to a resistive load, , is constant at all times. Indeed, let
To simplify the mathematics, we define a nondimensionalized power for intermediate calculations,
Hence (substituting back):
Since we have eliminated we can see that the total power does not vary with time. This is essential for keeping large generators and motors running smoothly.
Notice also that using the root mean square voltage , the expression for above takes the following more classic form:
.
The load need not be resistive for achieving a constant instantaneous power since, as long as it is balanced or the same for all phases, it may be written as
so that the peak current is
for all phases and the instantaneous currents are
Now the instantaneous powers in the phases are
Using angle subtraction formulae:
which add up for a total instantaneous power
Since the three terms enclosed in square brackets are a three-phase system, they add up to zero and the total power becomes
or
showing the above contention.
Again, using the root mean square voltage , can be written in the usual form
.
No neutral current
For the case of equal loads on each of three phases, no
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https://en.wikipedia.org/wiki/Greenway%20footpath%2C%20London
|
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https://en.wikipedia.org/wiki/Elongated%20triangular%20pyramid
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In geometry, the elongated triangular pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
The height is given by
If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.
Dual polyhedron
Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.
Related polyhedra and honeycombs
The elongated triangular pyramid can form a tessellation of space with square pyramids and/or octahedra.
References
External links
Johnson solids
Self-dual polyhedra
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Elongated%20square%20pyramid
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In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self-dual.
Formulae
The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length :
Dual polyhedron
The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal.
Related polyhedra and honeycombs
The elongated square pyramid can form a tessellation of space with tetrahedra, similar to a modified tetrahedral-octahedral honeycomb.
See also
Elongated square bipyramid
References
External links
Johnson solids
Self-dual polyhedra
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Elongated%20triangular%20bipyramid
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In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism is one of the Johnson solids (), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid () by inserting a triangular prism between its congruent halves.
The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.
Formulae
The following formulae for volume (), surface area () and height () can be used if all faces are regular, with edge length a:
Dual polyhedron
The dual of the elongated triangular bipyramid is called a triangular bifrustum and has 8 faces: 6 trapezoidal and 2 triangular.
References
External links
Johnson solids
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Elongated%20square%20bipyramid
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In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.
It has been named the pencil cube or 12-faced pencil cube due to its shape.
A zircon crystal is an example of an elongated square bipyramid.
Formulae
The following formulae for volume (), surface area () and height () can be used if all faces are regular, with edge length :
Dual polyhedron
The dual of the elongated square bipyramid is called a square bifrustum and has 10 faces: 8 trapezoidal and 2 square.
Related polyhedra and honeycombs
A special kind of elongated square bipyramid without all regular faces allows a self-tessellation of Euclidean space. The triangles of this elongated square bipyramid are not regular; they have edges in the ratio 2::.
It can be considered a transitional phase between the cubic and rhombic dodecahedral honeycombs. The cells are here colored white, red, and blue based on their orientation in space. The square pyramid caps have shortened isosceles triangle faces, with six of these pyramids meeting together to form a cube. The dual of this honeycomb is composed of two kinds of octahedra (regular octahedra and triangular antiprisms), formed by superimposing octahedra into the cuboctahedra of the rectified cubic honeycomb. Both honeycombs have a symmetry of [[4,3,4]].
Cross-sections of the honeycomb, through cell centers produces a chamfered square tiling, with flattened horizontal and vertical hexagons, and squares on the perpendicular polyhedra.
With regular faces, the elongated square bipyramid can form a tessellation of space with tetrahedra and octahedra. (The octahedra can be further decomposed into square pyramids.) This honeycomb can be considered an elongated version of the tetrahedral-octahedral honeycomb.
See also
Elongated square pyramid
References
External links
Johnson solids
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Elongated%20pentagonal%20bipyramid
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In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congruent halves.
Dual polyhedron
The dual of the elongated square bipyramid is a pentagonal bifrustum.
See also
Elongated pentagonal pyramid
External links
Johnson solids
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Elongated%20pentagonal%20cupola
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In geometry, the elongated pentagonal cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal cupola () by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola () with its "lid" (another pentagonal cupola) removed.
Formulas
The following formulas for the volume and surface area can be used if all faces are regular, with edge length a:
Dual polyhedron
The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Gyroelongated%20pentagonal%20cupola
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In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J46) with one pentagonal cupola removed.
Area and Volume
With edge length a, the surface area is
and the volume is
Dual polyhedron
The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Gyrobifastigium
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In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
It is also the vertex figure of the nonuniform duoantiprism (if and are greater than 2). Despite the fact that would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case which represents a uniform great duoantiprism.
Its dual, the elongated tetragonal disphenoid, can be found as cells of the duals of the duoantiprisms.
History and name
The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof. In the standard naming convention of the Johnson solids, bi- means two solids connected at their bases, and gyro- means the two halves are twisted with respect to each other.
The gyrobifastigium's place in the list of Johnson solids, immediately before the bicupolas, is explained by viewing it as a digonal gyrobicupola. Just as the other regular cupolas have an alternating sequence of squares and triangles surrounding a single polygon at the top (triangle, square or pentagon), each half of the gyrobifastigium consists of just alternating squares and triangles, connected at the top only by a ridge.
Honeycomb
The gyrated triangular prismatic honeycomb can be constructed by packing together large numbers of identical gyrobifastigiums.
The gyrobifastigium is one of five convex polyhedra with regular faces capable of space-filling (the others being the cube, truncated octahedron, triangular prism, and hexagonal prism) and it is the only Johnson solid capable of doing so.
Cartesian coordinates
Cartesian coordinates for the gyrobifastigium with regular faces and unit edge lengths may easily be derived from the formula of the height of unit edge length
as follows:
To calculate formulae for the surface area and volume of a gyrobifastigium with regular faces and with edge length a, one may simply adapt the corresponding formulae for the triangular prism:
Topologically equivalent polyhedra
Schmitt–Conway–Danzer biprism
The Schmitt–Conway–Danzer biprism (also called a SCD prototile) is a polyhedron topologically equivalent to the gyrobifastigium, but with parallelogram and irregular triangle faces instead of squares and equilateral triangles. Like the gyrobifastigium, it can fill space, but only aperiodically or with a screw symmetry, not with a full three-dimensional group of symmetries. Thus, it provides a partial solution to the three-dimensional einstein problem.
Dual
The dual polyhedron of the gyrobifastigium has 8 faces: 4 isosceles triangles, corresponding to the valence-3 vertices of the gyrobifastigium, and 4 parallelograms corresponding to the valence-4 equatorial vertices.
See also
Elongated gyrobifastigium
Elongated octahedron
References
External lin
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https://en.wikipedia.org/wiki/Pentagonal%20orthobicupola
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In geometry, the pentagonal orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by joining two pentagonal cupolae () along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola ().
The pentagonal orthobicupola is the third in an infinite set of orthobicupolae.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Pentagonal%20gyrobicupola
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In geometry, the pentagonal gyrobicupola is one of the Johnson solids (). Like the pentagonal orthobicupola (), it can be obtained by joining two pentagonal cupolae () along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.
The pentagonal gyrobicupola is the third in an infinite set of gyrobicupolae.
The pentagonal gyrobicupola is what you get when you take a rhombicosidodecahedron, chop out the middle parabidiminished rhombicosidodecahedron (), and paste the two opposing cupolae back together.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Elongated%20pentagonal%20orthobicupola
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In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola () by inserting a decagonal prism between its two congruent halves. Rotating one of the cupolae through 36 degrees before inserting the prism yields an elongated pentagonal gyrobicupola ().
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Elongated%20pentagonal%20gyrobicupola
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In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola () by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae () through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola ().
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Augmented%20triangular%20prism
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In geometry, the augmented triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid () to one of its equatorial faces. The resulting solid bears a superficial resemblance to the gyrobifastigium (), the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.
It is also the vertex figure of the nonuniform duoantiprism (if ). Despite the fact that would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices.
Its dual, a triangular bipyramid with one of its 4-valence vertices truncated, can be found as cells of the duoantitegums (duals of the duoantiprisms).
External links
Johnson solids
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https://en.wikipedia.org/wiki/Biaugmented%20triangular%20prism
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In geometry, the biaugmented triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a triangular prism by attaching square pyramids () to two of its equatorial faces.
It is related to the augmented triangular prism () and the triaugmented triangular prism ().
External links
Johnson solids
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https://en.wikipedia.org/wiki/Augmented%20pentagonal%20prism
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In geometry, the augmented pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid () to one of its equatorial faces.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Biaugmented%20pentagonal%20prism
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In geometry, the biaugmented pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a pentagonal prism by attaching square pyramids () to two of its nonadjacent equatorial faces. (The solid obtained by attaching pyramids to adjacent equatorial faces is not convex, and thus not a Johnson solid.)
External links
Johnson solids
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https://en.wikipedia.org/wiki/Augmented%20hexagonal%20prism
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In geometry, the augmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid () to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism (), a metabiaugmented hexagonal prism (), or a triaugmented hexagonal prism ().
External links
Johnson solids
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https://en.wikipedia.org/wiki/Parabiaugmented%20hexagonal%20prism
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In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, parallel (opposite) equatorial faces. Attaching the pyramids to nonadjacent, nonparallel equatorial faces yields a metabiaugmented hexagonal prism (). (The solid obtained by attaching pyramids to adjacent equatorial faces is not convex, and thus not a Johnson solid.)
External links
Johnson solids
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https://en.wikipedia.org/wiki/Metabiaugmented%20hexagonal%20prism
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In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, nonparallel equatorial faces. Attaching the pyramids to opposite equatorial faces yields a parabiaugmented hexagonal prism. (The solid obtained by attaching pyramids to adjacent equatorial faces is not convex, and thus not a Johnson solid.)
See also
Hexagonal prism
External links
Johnson solids
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https://en.wikipedia.org/wiki/Triaugmented%20hexagonal%20prism
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In geometry, the triaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids () to three of its nonadjacent equatorial faces.
See also
Hexagonal prism
References
External links
Johnson solids
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https://en.wikipedia.org/wiki/Augmented%20dodecahedron
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In geometry, the augmented dodecahedron is one of the Johnson solids (), consisting of a dodecahedron with a pentagonal pyramid () attached to one of the faces. When two or three such pyramids are attached, the result may be a parabiaugmented dodecahedron (), a metabiaugmented dodecahedron (), or a triaugmented dodecahedron ().
External links
Johnson solids
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https://en.wikipedia.org/wiki/Parabiaugmented%20dodecahedron
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In geometry, the parabiaugmented dodecahedron is one of the Johnson solids (). It can be seen as a dodecahedron with two pentagonal pyramids () attached to opposite faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (), a metabiaugmented dodecahedron (), a triaugmented dodecahedron (), or even a pentakis dodecahedron if the faces are made to be irregular.
The dual of this solid is the Gyroelongated pentagonal bifrustum.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Metabiaugmented%20dodecahedron
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In geometry, the metabiaugmented dodecahedron is one of the Johnson solids (). It can be viewed as a dodecahedron with two pentagonal pyramids () attached to two faces that are separated by one face. (The two faces are not opposite, but not adjacent either.) When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (), a parabiaugmented dodecahedron (), a triaugmented dodecahedron (), or even a pentakis dodecahedron if the faces are made to be irregular.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Triaugmented%20dodecahedron
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In geometry, the triaugmented dodecahedron is one of the Johnson solids (). It can be seen as a dodecahedron with three pentagonal pyramids () attached to nonadjacent faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (), a parabiaugmented dodecahedron (), a metabiaugmented dodecahedron (), or even a pentakis dodecahedron if the faces are made to be irregular.
External links
Johnson solids
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