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https://en.wikipedia.org/wiki/Dimitrios%20Karakasis
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Dimitrios Karakasis () was a Greek physician.
He was born in Siatista in 1734. He went to Halle, in Saxony, where he studied medicine, philosophy and mathematics. He took a Degree in medicine in 1760. He exercised his occupation as physician in Vienna, Larisa, Siatista, Kozani, Bucharest, and also taught in his birthplace, Siatista, in Macedonia.
See also
List of Macedonians (Greek)
External links
List of Great Macedonians (15th-19th century)
1734 births
Year of death unknown
People from Siatista
Greek Macedonians
18th-century Greek physicians
18th-century physicians from the Ottoman Empire
Macedonia under the Ottoman Empire
18th-century Greek educators
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https://en.wikipedia.org/wiki/Elliptic%20rational%20functions
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In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name).
Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:
where
cd(u,k) is the Jacobi elliptic cosine function.
K() is a complete elliptic integral of the first kind.
is the discrimination factor, equal to the minimum value of the magnitude of for .
For many cases, in particular for orders of the form n = 2a3b where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions.
Expression as a ratio of polynomials
For even orders, the elliptic rational functions may be expressed as a ratio of two polynomials, both of order n.
(for n even)
where are the zeroes and are the poles, and is a normalizing constant chosen such that . The above form would be true for even orders as well except that for odd orders, there will be a pole at x=∞ and a zero at x=0 so that the above form must be modified to read:
(for n odd)
Properties
The canonical properties
for
at
for
The slope at x=1 is as large as possible
The slope at x=1 is larger than the corresponding slope of the Chebyshev polynomial of the same order.
The only rational function satisfying the above properties is the elliptic rational function . The following properties are derived:
Normalization
The elliptic rational function is normalized to unity at x=1:
Nesting property
The nesting property is written:
This is a very important property:
If is known for all prime n, then nesting property gives for all n. In particular, since and can be expressed in closed form without explicit use of the Jacobi elliptic functions, then all for n of the form can be so expressed.
It follows that if the zeroes of for prime n are known, the zeros of all can be found. Using the inversion relationship (see below), the poles can also be found.
The nesting property implies the nesting property of the discrimination factor:
Limiting values
The elliptic rational functions are related to the Chebyshev polynomials of the first kind by:
Symmetry
for n even
for n odd
Equiripple
has equal ripple of in the interval . By the inversion relationship (see below), it follows that has equiripple in of .
Inversion relationship
The following inversion relationsh
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https://en.wikipedia.org/wiki/Facing%20Goya
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Facing Goya (2000) is an opera in four acts by Michael Nyman on a libretto by Victoria Hardie. It is an expansion of their one-act opera called Vital Statistics from 1987, dealing with such subjects as physiognomy, eugenics, and its practitioners, and also incorporates a musical motif from Nyman's art song, "The Kiss", inspired by a Paul Richards painting. Nyman also considers the work thematically tied to his other works, The Man Who Mistook His Wife for a Hat, The Ogre, and Gattaca, though he does not quote any of these musically, save a very brief passage of the latter. It was premièred at the Auditorio de Galicia, Santiago de Compostela, Spain on 3 August 2000. The revision with the cast heard on the album premiered at the Badisches Staatstheater Karlsruhe, Germany, on October 19, 2002. Vital Statistics has been withdrawn. The Santiago version included more material from Vital Statistics. The opera was most recently performed at the 2014 Spoleto Festival USA, located in Charleston, South Carolina.
The expanded opera deals with the elitism and prejudice of various movements in pseudosciences and art criticism, wrapped around a thread of a desire to make a clone of Francisco Goya through use of his long-lost skull, which he hid from the likes of Paul Broca, and which the Art Banker finds under a floorboard in a "degenerate art" gallery in Act II. This skull is the object of numerous fights in the second and third acts, often with one character snatching it from another. The opera is non-realistic in its presentation, with only one through-character, the Art Banker. Indeed, when Goya does appear, it is not the result of cloning, but a purely fantastical device. Four other performers play different roles in each section who are thematically connected. In addition, two actors are called for in non-speaking roles. The Art Banker also speaks narration into a dictaphone, but this was omitted from the studio recording, though the lines are reprinted in the booklet.
Roles
Art Banker, a widow (contralto), loves Goya, but is corrupted by money. She foolishly wants to patent Goya's talent gene. Despite this, she is the most charismatic and sympathetic figure of the satire. She is a time tripper. An art banker is a person who deals in exchange of famous artworks among museums. This character is currently a specialist in the work of Goya.
Soprano 1 (coloratura), obsessed with science, she lives in her head, and is the one who ultimately cracks the human genome. (Craniometrist 1, Eugenicist/Art Critic 1, Microbiologist). At one point she nearly chokes herself with a tape measure, but continues to sing.
Soprano 2 (lyric), unhappy individualist who sees the dangers of racism in gene control. She is opposed to cloning and State ownership of genetic readouts. She does not believe that recreating a person recreates that person's talent. (Craniometery Assistant 2, Art Critic 2, Genetic Research Doctor)
Tenor, a shallow opportunist who beli
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https://en.wikipedia.org/wiki/Philippe%20Avril
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Philippe Avril (1654 – 1698 (presumed)) was a Jesuit explorer of the Far East. He was born at Angoulême, France on 16 September 1654.
Avril was a professor of philosophy and mathematics at Paris when he was dispatched to the Jesuit missions of China. Following the instructions of Ferdinand Verbiest, another Jesuit, then at Peking, he attempted an overland journey, and traveled for six years through Kurdistan, Armenia, Astrakhan, Persia, and other countries of eastern Asia.
Arriving at Moscow, Avril was refused permission to pass through Tatary, and was sent by the Government to Poland, whence he made his way to Istanbul and from there went back to France. Though exhausted by disease, he set out again on a vessel, which was lost at sea. Avril presumably died in a 1698 shipwreck.
Avril's journal and writings provide a significant amount of useful material for modern historians and demographers.
References
Sources
1654 births
French explorers
17th-century French Jesuits
1698 deaths
French Roman Catholic missionaries
Jesuit missionaries in China
French expatriates in China
Jesuit missionaries
Deaths due to shipwreck at sea
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https://en.wikipedia.org/wiki/Grimm%27s%20conjecture
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In number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.
Formal statement
If n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 ≤ i ≤ k.
Weaker version
A weaker, though still unproven, version of this conjecture states: If there is no prime in the interval , then has at least k distinct prime divisors.
See also
Prime gap
References
Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 3rd ed., Springer Science+Business Media, pp. 133–134, 2004.
External links
Prime Puzzles #430
Conjectures about prime numbers
Unsolved problems in number theory
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https://en.wikipedia.org/wiki/List%20of%20Real%20Madrid%20CF%20records%20and%20statistics
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Real Madrid Club de Fútbol is a Spanish professional association football club based in Madrid. The club was formed in 1902 as Madrid Football Club, and played its first competitive match on 13 May 1902, when it entered the semi-final of the Campeonato de Copa de S.M. Alfonso XIII. Real Madrid currently plays in the Spanish top-tier La Liga, having become one of the founding members of that league in 1929, and is one of three clubs, the others being Barcelona and Athletic Bilbao, to have never been relegated from the league. They have also been involved in European football ever since they became the first Spanish club to enter the European Cup in 1955, except for the 1977–78 and 1996–97 seasons.
This list encompasses the major honours won by Real Madrid and records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first team competitions. It also records notable achievements by Real Madrid players on the international stage, and the highest transfer fees paid and received by the club.
The club currently holds the record for the most European Cup / UEFA Champions League triumphs, with 14, and the most La Liga titles, with 35. Additionally, Real has won the Copa del Rey 20 times, the Supercopa de España 12 times, the Copa de la Liga once, the Copa Eva Duarte once, the UEFA Cup twice, the European/UEFA Super Cup five times, the Intercontinental Cup three time, the FIFA Club World Cup five times, the Latin Cup twice and Copa Iberoamericana once. Powered by its fourteen European Cups, Real Madrid have a distinction of being the most successful club in terms of international titles, having amassed 32 pieces of silverware, more than any other team in the world. On the domestic front, its 69 titles rank second to Barcelona. The club's record appearance maker is Raúl, who made 741 appearances from 1994 to 2010; the club's record goalscorer is Portuguese forward Cristiano Ronaldo, who scored 450 goals in all competitions from 2009 to 2018.
Players
Appearances
Competitive, professional matches only. Players in italics are still active outside the club.
As of 4 June 2023.
Others
Player with most major trophies at Real Madrid: 25 – Marcelo
Youngest first-team player: – Martin Ødegaard v Getafe, 2014–15 La Liga, 23 May 2015
Youngest first-team player (including friendly matches): – José Gandarias v Deportivo Auténtico, Friendly match, 17 December 1916
Oldest post-Second World War player: – Ferenc Puskás v Real Betis, 1965–66 Copa del Generalísimo, 8 May 1966
Most appearances in La Liga: 550 – Raúl
Most appearances in Copa del Rey: 84 – Santillana
Most appearances in Copa de la Liga: 13 – Isidoro San José
Most appearances in Supercopa de España: 15 – Sergio Ramos
Most appearances in International competitions: 162 – Iker Casillas
Most appearances in UEFA club competitions: 157 – Iker Casillas
Most appearances
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https://en.wikipedia.org/wiki/COV
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COV, Cov, CoV or Co-V may refer to:
Cash-Over-Valuation
City of Villains, a multiplayer online video game
Coefficient of variation, a statistical measure
Covariance, a measure in probability theory and statistics
Calculus of variations, a field of mathematical analysis
Abbreviation of Coventry, a city in the United Kingdom
COV, the ICAO airline designator for Helicentre Coventry, United Kingdom
COV, the station code for Coventry railway station
Coventry R.F.C., often abbreviated to just "Cov"
Coventry City F.C., which is also sometimes known by the shorter form
The Amtrak station code for Connellsville station, Pennsylvania, United States
The LRT station abbreviation for Cove LRT station, Punggol, Singapore
The NYSE abbreviation for Covidien Ltd, a medical technology and pharmaceutical company
CoV or Co-V, an abbreviation for Coronavirus
Cao Miao language (ISO 639 language code: cov)
See also
nCoV (disambiguation)
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https://en.wikipedia.org/wiki/Isohedral%20figure
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In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces and , there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces.
The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex-, edge-, and face-transitive (i.e. isogonal, isotoxal, and isohedral).
A form that is isohedral, has regular vertices, and is also edge-transitive (i.e. isotoxal) is said to be a quasiregular dual. Some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted.
A polyhedron which is isohedral and isogonal is said to be noble.
Not all isozonohedra are isohedral. For example, a rhombic icosahedron is an isozonohedron but not an isohedron.
Examples
Classes of isohedra by symmetry
k-isohedral figure
A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domains. Similarly, a k-isohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k, or only for some m < k). ("1-isohedral" is the same as "isohedral".)
A monohedral polyhedron or monohedral tiling (m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An m-hedral polyhedron or tiling has m different face shapes ("dihedral", "trihedral"... are the same as "2-hedral", "3-hedral"... respectively).
Here are some examples of k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions:
Related terms
A cell-transitive or isochoric figure is an n-polytope (n ≥ 4) or n-honeycomb (n ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs, duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells.
A facet-transitive or isotopic figure is an n-dimensional polytope or honeycomb with its facets ((n−1)-faces) congruent and transitive. The dual of an isotope is an isogonal polytope. By definition, this isotopic property is common to the duals of the uniform polytopes.
An isotopic 2-dimensional figure is isotoxal, i.e. edge-transitive.
An isot
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https://en.wikipedia.org/wiki/Isotoxal%20figure
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In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
Isotoxal polygons
An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal -gons are centrally symmetric, so are also zonogons.
In general, an isotoxal -gon has dihedral symmetry. For example, a rhombus is an isotoxal "×-gon" (quadrilateral) with symmetry. All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular -gon has dihedral symmetry.
An isotoxal -gon with outer internal angle can be labeled as The inner internal angle may be greater or less than degrees, making convex or concave polygons.
Star polygons can also be isotoxal, labeled as with and with the greatest common divisor where is the turning number or density. Concave inner vertices can be defined for If then is "reduced" to a compound of rotated copies of
Caution: The vertices of are not always placed like those of whereas the vertices of the regular are placed like those of the regular
A set of "uniform tilings", actually isogonal tilings using isotoxal polygons as less symmetric faces than regular ones can be defined.
Isotoxal polyhedra and tilings
Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).
Quasiregular polyhedra, like the cuboctahedron and the icosidodecahedron, are isogonal and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not isogonal.
Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) is not isotoxal, as it has two edge types: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge.
An isotoxal polyhedron has the same dihedral angle for all edges.
The dual of a convex polyhedron is also a convex polyhedron.
The dual of a non-convex polyhedron is also a non-convex polyhedron. (By contraposition.)
The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)
There are nine convex isotoxal polyhedra: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.
There are fourteen non-convex isotoxal polyhedra: the four (regular) Kepler–Poinsot polyhedra, the two (quasiregular) common cores of dual Kepler–Poinsot p
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https://en.wikipedia.org/wiki/Whitehead%27s%20point-free%20geometry
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In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory.
Point-free geometry was first formulated in Whitehead (1919, 1920), not as a theory of geometry or of spacetime, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical.
Formalizations
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first-order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain of discourse for both theories consists of "regions." All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom requires more than three quantified variables; hence a translation of first-order theories into relation algebra is possible. Each set of axioms has but four existential quantifiers.
Inclusion-based point-free geometry (mereology)
The fundamental primitive binary relation is inclusion, denoted by the infix operator "≤", which corresponds to the binary Parthood relation that is a standard feature in mereological theories. The intuitive meaning of x ≤ y is "x is part of y." Assuming that equality, denoted by the infix operator "=", is part of the background logic, the binary relation Proper Part, denoted by the infix operator "<", is defined as:
The axioms are:
Inclusion partially orders the domain.
G1. (reflexive)
G2. (transitive) WP4.
G3. (antisymmetric)
Given any two regions, there exists a region that includes both of them. WP6.
G4.
Proper Part densely orders the domain. WP5.
G5.
Both atomic regions and a universal region do not exist. Hence the domain has neither an upper nor a lower bound. WP2.
G6.
Proper Parts Principle. If all the proper parts of x are proper parts of y, then x is included in y. WP3.
G7.
A model of G1–G7 is an inclusion space.
Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space S, an abstractive class is a class G of regions such that S\G is totally ordered by inclusion. Moreover, there does not exist a region included in all of the regions included in G.
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the Euclidean plane, then the corresponding abstractive classes are points and lines.
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's (1987: 83) system W. In turn, W formalizes a theory in Whitehead (1919) whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons (1987) did not repair this defect, i
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https://en.wikipedia.org/wiki/Cubic%20reciprocity
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Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
History
Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, 62 years after his death.
Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae (1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) he said that he was publishing these proofs because their techniques (Gauss's lemma and Gaussian sums, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.
From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814. Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.
Jacobi published several theorems about cubic residuacity in 1827, but no proofs. In his Königsberg lectures of 1836–37 Jacobi presented proofs. The first published proofs were by Eisenstein (1844).
Integers
A cubic residue (mod p) is any number congruent to the third power of an integer (mod p). If x3 ≡ a (mod p) does not have an integer solution, a is a cubic nonresidue (mod p).
As is often the case in number theory, it is easier to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to be positive odd primes.
We first note that if q ≡ 2 (mod 3) is a prime then every number is a cubic residue modulo q. Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat's little theorem,
Multiplying the two congruences we have
Now substituting 3n + 2 for q we have:
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3). In this case the non-zero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic non-residue. The first set is the cubic residues; the second one is e times the numbers in the first set, and the third is e2 times the numbers in the first set. Another way to describe this division is to let e be a primitive root (mod p); then the first (resp. second, third) set is the numbers whose indices with respect to this root are congruent to 0 (resp. 1, 2) (mod 3). In the vocabulary o
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https://en.wikipedia.org/wiki/National%20Cipher%20Challenge
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The National Cipher Challenge is an annual cryptographic competition organised by the University of Southampton School of Mathematics. Competitors attempt to break cryptograms published on the competition website. In the 2017, more than 7,500 students took part in the competition. Participants must be in full-time school level education in order to qualify for prizes.
Format
The competition is organised into eight to ten challenges, which are further subdivided into parts A and B. The part A challenge consists of a comparatively simpler cryptogram, and usually provides some useful information to assist in the solving of part B. Part B is usually more complex. In later challenges the cryptograms become harder to break. In the past, part A cryptograms have been encrypted with the Caesar cipher, the Affine cipher, the Keyword cipher, the Transposition cipher, the Vigenère cipher and the 2x2 Hill cipher.
The part B challenges are intended to be harder. These begin with relatively simple substitution ciphers, including the Bacon cipher and Polybius square, before moving on to transposition ciphers, Playfair ciphers and polyalphabetic ciphers such as the Vigenère cipher, the Autokey cipher and the Alberti cipher. In the later stages of the competition, the ADFGVX cipher, the Solitaire cipher, the Double Playfair cipher, the Hill cipher, the Book cipher and versions of the Enigma and Fialka cipher machines have all been used. The 2009 challenge ended with a Jefferson Disk cipher, the 2012 challenge ended with the ADFGVX Cipher, the 2014 with the Playfair Cipher, and the most recent challenge ended with a sectioned Cadenus transposition.
Prizes
£25 cash prizes are awarded to eight random entrants who submit a correct solution for each part A of the challenge. Leaderboards for the part B challenges are also compiled, based on how accurate solutions are and how quickly the entrant broke the cipher. Prizes are awarded to the top three entrants at the end of the challenge. In the 2009/10 challenge, the sponsors provided several prizes: IBM provided iPod Touches to each member of the team winning the Team Prize, Trinity College provided a cash prize of £700, and GCHQ provided a cash prize of £1000. In previous years prizes such as an IBM Thinkpad laptop have been awarded.
After the challenge the winners of the top prizes and other randomly selected entrants are invited to a day held at Bletchley Park consisting of lectures (with subjects such as the Semantic Web, World War II cryptography and computer programming) and the prize-giving ceremony.
Current sponsors of the competition include GCHQ, IBM, British Computer Society, Trinity College, Cambridge, Cambridge University Press, Winton Capital Management and EPSRC.
References
External links
The official challenge website
The website of the 2016/17 challenge (slightly broken)
The website of the 2015/16 challenge
The websites for the challenges earlier than this are no longer available.
Computer s
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https://en.wikipedia.org/wiki/Congrua
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Congrua may refer to:
the plural of congruum, in mathematics, the difference of an arithmetic progression of squares
congrua portio, the lowest sum proper for the yearly income of a cleric
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https://en.wikipedia.org/wiki/Outline%20of%20logic
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Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments.
Foundations of logic
Philosophy of logic
Analytic-synthetic distinction
Antinomy
A priori and a posteriori
Definition
Description
Entailment
Identity (philosophy)
Inference
Logical form
Logical implication
Logical truth
Logical consequence
Name
Necessity
Material conditional
Meaning (linguistic)
Meaning (non-linguistic)
Paradox (list)
Possible world
Presupposition
Probability
Quantification
Reason
Reasoning
Reference
Semantics
Strict conditional
Syntax (logic)
Truth
Truth value
Validity
Branches of logic
Affine logic
Alethic logic
Aristotelian logic
Boolean logic
Buddhist logic
Bunched logic
Categorical logic
Classical logic
Computability logic
Deontic logic
Dependence logic
Description logic
Deviant logic
Doxastic logic
Epistemic logic
First-order logic
Formal logic
Free logic
Fuzzy logic
Higher-order logic
Infinitary logic
Informal logic
Intensional logic
Intermediate logic
Interpretability logic
Intuitionistic logic
Linear logic
Many-valued logic
Mathematical logic
Metalogic
Minimal logic
Modal logic
Non-Aristotelian logic
Non-classical logic
Noncommutative logic
Non-monotonic logic
Ordered logic
Paraconsistent logic
Philosophical logic
Predicate logic
Propositional logic
Provability logic
Quantum logic
Relevance logic
Sequential logic
Strict logic
Substructural logic
Syllogistic logic
Symbolic logic
Temporal logic
Term logic
Topical logic
Traditional logic
Zeroth-order logic
Philosophical logic
Informal logic and critical thinking
Informal logic
Critical thinking
Argumentation theory
Argument
Argument map
Accuracy and precision
Ad hoc hypothesis
Ambiguity
Analysis
Attacking Faulty Reasoning
Belief
Belief bias
Bias
Cognitive bias
Confirmation bias
Credibility
Critical pedagogy
Critical reading
Decidophobia
Decision making
Dispositional and occurrent belief
Emotional reasoning
Evidence
Expert
Explanation
Explanatory power
Fact
Fallacy
Higher-order thinking
Inquiry
Interpretive discussion
Occam's razor
Opinion
Practical syllogism
Precision questioning
Propaganda
Propaganda techniques
Problem Solving
Prudence
Pseudophilosophy
Reasoning
Relevance
Rhetoric
Rigour
Socratic questioning
Source credibility
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https://en.wikipedia.org/wiki/Cosma%20Shalizi
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Cosma Rohilla Shalizi (born February 28, 1974) is an associate professor in the Department of Statistics at Carnegie Mellon University in Pittsburgh.
Life
Cosma Rohilla Shalizi is of Indian Tamil, Afghan (Rohilla) and Italian heritage and was born in Boston, where he lived for the first two years of his life. He grew up in Bethesda, Maryland.
In 1990, he was accepted as a Chancellor's Scholar at the University of California, Berkeley, and completed a bachelor's degree in Physics. Subsequently, he attended the University of Wisconsin–Madison where he received a doctorate in physics in May 2001. From 1998 to 2002, he worked at the Santa Fe Institute, in the Evolving Cellular Automata Project and the Computation, Dynamics and Inference group. From 2002 to 2005, he worked at the Center for the Study of Complex Systems at the University of Michigan in Ann Arbor. In August 2006, he became an assistant professor in the Department of Statistics at Carnegie Mellon University in Pittsburgh.
Shalizi is co-author of the CSSR algorithm, which exploits entropy properties to efficiently extract Markov models from time-series data without assuming a parametric form for the model.
Shalizi was interviewed at the Institute for New Economic Thinking in November 2011 on "Why Economics Needs Data Mining." He "urge[d] economists to stop doing what they are doing: Fitting large complex models to a small set of highly correlated time series data. Once you add enough variables, parameters, bells and whistles, your model can fit past data very well, and yet fail miserably in the future. Shalizi tells us how to separate the wheat from the chaff, how to compensate for overfitting and prevent models from memorizing noise. He introduces techniques from data mining and machine learning to economics — this is new economic thinking."
Shalizi gave an invited "Distinguished Lecture" at the University of California at Santa Barbara Data Science Initiative in May 2019. There he presented analysis of the statistical weaknesses of utilizing observational data to infer neighborhood effects in social networks. Id.
Shalizi's scholarly work has been cited more than 17,000 times, according to Google Scholar.
Shalizi writes a popular science blog "Three-Toed Sloth".
References
External links
1974 births
Living people
University of Wisconsin–Madison College of Letters and Science alumni
Carnegie Mellon University faculty
Cellular automatists
Probability theorists
University of Michigan faculty
Santa Fe Institute people
American academics of Indian descent
Rohilla
Network scientists
Academics from Boston
University of California, Berkeley alumni
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https://en.wikipedia.org/wiki/Lo%C5%A1onec
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Lošonec () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
https://web.archive.org/web/20070427022352/http://www.statistics.sk/mosmis/eng/run.html
http://www.losonec.com/ - official web page
private web page
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Majcichov
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Majcichov () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
Statistics.sk
En.e-obce.sk
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Mal%C5%BEenice
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Malženice () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
https://www.webcitation.org/5QjNYnAux?url=http://www.statistics.sk/mosmis/eng/run.html
http://en.e-obce.sk/obec/malzenice/malzenice.html
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Pavlice%2C%20Trnava%20District
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Pavlice () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Rado%C5%A1ovce%2C%20Trnava%20District
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Radošovce () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
http://en.e-obce.sk/obec/radosovce/radosovce.html
http://www.statistics.sk/mosmis/eng/run.html
https://web.archive.org/web/20101113055135/http://www.mesta-obce.sk/trnavsky-kraj/okres-trnava/radosovce/
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Ru%C5%BEindol
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Ružindol () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
https://www.ruzindol.sk/
http://en.e-obce.sk/obec/ruzindol/ruzindol.html
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/Slovensk%C3%A1%20Nov%C3%A1%20Ves
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Slovenská Nová Ves () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
http://en.e-obce.sk/obec/slovenskanovaves/slovenska-nova-ves.html
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/%C5%A0elpice
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Šelpice () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
http://en.e-obce.sk/obec/selpice/selpice.html
https://www.selpice.eu/
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/%C5%A0%C3%BArovce
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Šúrovce () is a village and municipality of Trnava District in the Trnava region of Slovakia.
References
External links
http://en.e-obce.sk/obec/sturovce/surovce.html
http://www.statistics.sk/mosmis/eng/run.html
http://www.surovce.sk/
Villages and municipalities in Trnava District
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https://en.wikipedia.org/wiki/201%20%28number%29
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201 (two hundred [and] one) is the natural number following 200 and preceding 202.
In mathematics
As the two proper factors of 201 are both Gaussian primes, 201 is a Blum integer.
In computing
201 is an HTTP status code indicating a new resource was successfully created in response to the request, with the textual part of the response line indicating the URL of the newly created document.
In astronomy
201 is a Saros cycle; the next solar eclipse in this cycle is predicted to take place in AD 3223..
The New General Catalogue object NGC201 is a magnitude 15 spiral galaxy in the constellation Cetus.
201 Penelope is a large Main belt asteroid discovered in 1879.
In other fields
A 201 file is the term used in the U.S. Army for the set of documents maintained by the US government for members of the Armed Forces recording their service history. It is also referred to as the Official Military Personnel File.
201 in binary (11001001) is the title of an episode of Star Trek: The Next Generation.
Area code 201 is the area code assigned to northern New Jersey in the United States.
201 is the course number of basic or entry-level courses at some Canadian universities (such as the University of Calgary and Athabasca University), especially if the number 101 is allocated to remedial courses.
201 is also short for 201 Poplar, the jail in Memphis, Tennessee, and alluded to in many rap songs from Memphis artists.
in Philippine employment, a 201 file is a file detailing an employee's history and records with a particular employer
The 201 Class diesel locomotive used by Iarnród Éireann and NI Railways.
The 201 series is a Japanese commuter train type
The EMI 201 television camera use in the 1960s
"201" is the title of an episode of South Park.
Event 201, a pandemic exercise
Integers
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https://en.wikipedia.org/wiki/202%20%28number%29
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202 (two hundred [and] two) is the natural number following 201 and preceding 203.
In mathematics
202 is a Smith number, meaning that its digit sum and the sum of digits of its prime factors are equal. It is also a strobogrammatic number, meaning that when shown on a seven-segment display, turning the display upside-down shows the same number.
There are exactly 202 partitions of 32 (a power of two) into smaller powers of two. There are also 202 distinct (non-congruent) polygons that can be formed by connecting all eight vertices of a regular octagon into a cycle, and 202 distinct (non-isomorphic) directed graphs on four unlabeled vertices, not having any isolated vertices.
See also
Area code 202, the area code assigned to Washington D.C.
HTTP status code 202 meaning the request was accepted but has not yet been fulfilled
List of highways numbered 202
The Peugeot 202 automobile
Potassium sorbate, a preservative whose E number is 202
References
Integers
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https://en.wikipedia.org/wiki/203%20%28number%29
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203 (two hundred [and] three) is the natural number following 202 and preceding 204.
In mathematics
203 is the seventh Bell number, giving the number of partitions of a set of size 6. 203 different triangles can be made from three rods with integer lengths of at most 12, and 203 integer squares (not necessarily of unit size) can be found in a staircase-shaped polyomino formed by stacks of unit squares of heights ranging from 1 to 12.
In other fields
203 is the HTTP status code for non-authoritative information, indicating that the request was successful but the enclosed payload has been modified from that of the origin server's 200 (OK) response by a transforming proxy.
See also
Area code 203, in Connecticut
The year 203
Hill 203, near Lüshunkou, China
References
Integers
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https://en.wikipedia.org/wiki/204%20%28number%29
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204 (two hundred [and] four) is the natural number following 203 and preceding 205.
In mathematics
204 is a refactorable number. 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. Its square, 2042 = 41616, is the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated triangular pyramid number. 204 is a member of the Mian-Chowla sequence.
There are exactly 204 irreducible quintic polynomials over a four-element field, exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board, exactly 204 squares of an infinite chess move that are eight knight's moves from the center, exactly 204 strings of length 11 over a three-letter alphabet with no consecutively-repeated substring, and exactly 204 ways of immersing an oriented circle into the oriented plane so that it has four double points.
Both 204 and its square are sums of a pair of twin primes: 204 = 101 + 103 and 2042 = 41616 = 20807 + 20809. The only smaller numbers with the same property are 12 and 84.
204 is a sum of all the perfect squares from 1 to 64 (i.e. 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 = 204).
In other fields
In telecommunications, area code 204 is a North American telephone area code for the Canadian province of Manitoba. 204 is one of the original 86 area codes assigned in 1947 in the contiguous United States and the then-nine-province extent of Canada. More recently a second area code (431) was added to allow for the expanding phone number distribution within the province.
204 is the HTTP status code indicating the request was successfully fulfilled and that there is no additional content to send in the response payload body.
In a poker deck with a single wild joker, there are 204 hands that are at least as good as a straight flush.
Model 204 is a database management system.
References
Integers
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https://en.wikipedia.org/wiki/205%20%28number%29
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205 (two hundred [and] five) is the natural number following 204 and preceding 206.
In mathematics
205 is a lucky number, and a Wolstenholme number.
On an infinite chessboard, a knight can reach exactly 205 squares within four moves. There are 205 different ways of forming a connected graph by adding six edges to a set of five labeled vertices.
In other fields
The atomic number of an element temporarily called Binilpentium
See also
List of highways numbered 205
205 Martha, a large Main belt asteroid
205 Yonge Street, a building in Toronto
205 series, a commuter train type in Japan
Peugeot 205, a French car
WWE 205 Live, an American professional wrestling program
References
Integers
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https://en.wikipedia.org/wiki/206%20%28number%29
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206 (two hundred [and] six) is the natural number following 205 and preceding 207.
In mathematics
206 is both a nontotient and a noncototient. 206 is an untouchable number. It is the lowest positive integer (when written in English as "two hundred and six") to employ all of the vowels once only, not including Y. The other numbers sharing this property are 230, 250, 260, 602, 640, 5000, 8000, 9000, 80,000 and 90,000. 206 and 207 form the second pair of consecutive numbers (after 14 and 15) whose sums of divisors are equal. There are exactly 206 different linear forests on five labeled nodes, and exactly 206 regular semigroups of order four up to isomorphism and anti-isomorphism.
In science
There are 206 bones in the typical adult human body.
See also
The Year 206 AD
Cessna 206, a single engine light aircraft
Bell 206, a light helicopter
The Peugeot 206, a French supermini automobile
US Area code 206, and The 206 slang terminology for the urban part of the greater Seattle area
206 (Ulster) Battery Royal Artillery (Volunteers) "The Ulster Gunners", part of British Army's 105th Regiment Royal Artillery (Volunteers)
206 Hersilia, a fairly large Main belt asteroid
206 Bones, a novel by Kathy Reichs
References
Integers
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https://en.wikipedia.org/wiki/207%20%28number%29
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207 (two hundred [and] seven) is the natural number following 206 and preceding 208. It is an odd composite number with a prime factorization of .
In Mathematics
207 is a Wedderburn-Etherington number. There are exactly 207 different matchstick graphs with eight edges. 207 is also a deficient number, as 207's proper divisors (divisors not including the number itself) only add up to 105:
See also
Peugeot 207
List of highways numbered 207
References
Integers
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https://en.wikipedia.org/wiki/209%20%28number%29
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209 (two hundred [and] nine) is the natural number following 208 and preceding 210.
In mathematics
There are 209 spanning trees in a 2 × 5 grid graph, 209 partial permutations on four elements, and 209 distinct undirected simple graphs on 7 or fewer unlabeled vertices.
209 is the smallest number with six representations as a sum of three positive squares. These representations are:
209 .
By Legendre's three-square theorem, all numbers congruent to 1, 2, 3, 5, or 6 mod 8 have representations as sums of three squares, but this theorem does not explain the high number of such representations for 209.
, one less than the product of the first four prime numbers. Therefore, 209 is a Euclid number of the second kind, also called a Kummer number. One standard proof of Euclid's theorem that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer number. However, the Kummer numbers are not all prime, and as a semiprime (the product of two smaller prime numbers ), 209 is the first example of a composite Kummer number.
References
Integers
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https://en.wikipedia.org/wiki/214%20%28number%29
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214 (two hundred [and] fourteen) is the natural number following 213 and preceding 215.
In mathematics
214 is a composite number (with prime factorization 2 * 107) and a triacontakaiheptagonal number (37-gonal number).
214!! − 1 is a 205-digit prime number.
The 11th perfect number 2106×(2107−1) has 214 divisors.
Number of regions into which a figure made up of a row of 5 adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.
In other fields
214 is a song by Rivermaya.
214 Aschera is a Main belt asteroid.
E214 is the E number of Ethylparaben.
The Bell 214 is a helicopter.
The Tupolev 214 is an airliner.
Type 214 submarine
There are several highways numbered 214.
Form DD 214 documents discharge from the U.S. Armed Forces.
The number of Wainwright-listed summits of the English Lake District
214 is also:
The first area code of metropolitan Dallas, Texas
The number of Chinese radicals for the writing of Chinese characters according to the 1716 Kangxi Dictionary.
SMTP status code for a reply message to a help command
The Dewey Decimal Classification for Theodicy (the problem of evil).
References
Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (p. 143). London: Penguin Group.
Integers
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https://en.wikipedia.org/wiki/215%20%28number%29
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215 (two hundred [and] fifteen) is the natural number following 214 and preceding 216.
In mathematics
215 is a composite number and a semiprime ().
215 is the second smallest integer (after 5) such that is twice a square: .
215 is a vertically symmetric number on a calculator display .
There are 215 sequences of four integers, counting re-arrangements as distinct, such that the sum of their reciprocals is 1. These are
24 arrangements of (2,3,7,42), (2,3,8,24), (2,3,9,18), (2,3,10,15), (2,4,5,20) and (2,4,6,12).
12 arrangements of (3,3,4,12), (3,4,4,6), (2,3,12,12), (2,4,8,8) and (2,5,5,10).
6 arrangements of (3,3,6,6).
4 arrangements of (2,6,6,6).
1 arrangement of (4,4,4,4).
In other fields
215 Oenone is a main belt asteroid.
E215 is the E number of Sodium ethyl para-hydroxybenzoate.
There are several highways numbered 215.
215 is also:
The Dewey Decimal Classification for Science and religion.
The year AD 215 or 215 BC
215 is often used as slang for marijuana, from California Proposition 215, legalizing it for medical use.
The first area code of metropolitan Philadelphia, Pennsylvania
References
Integers
ca:Nombre 210#Nombres del 211 al 219
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https://en.wikipedia.org/wiki/217%20%28number%29
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217 (two hundred [and] seventeen) is the natural number following 216 and preceding 218.
In mathematics
217 is a centered hexagonal number, a 12-gonal number, a centered 36-gonal number, a Fermat pseudoprime to base 5, and a Blum integer. It is both the sum of two positive cubes and the difference of two positive consecutive cubes in exactly one way: . When written in binary, it is a non-repetitive Kaprekar number. It is also the sum of all the divisors of .
See also
217, the year
References
Integers
ca:Nombre 210#Nombres del 211 al 219
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https://en.wikipedia.org/wiki/218%20%28number%29
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218 (two hundred [and] eighteen) is the natural number following 217 and preceding 219.
In mathematics
Mertens function(218) = 3, a record high.
218 is nontotient and also noncototient.
218 is the number of inequivalent ways to color the 12 edges of a cube using at most 2 colors, where two colorings are equivalent if they differ only by a rotation of the cube.
There are 218 nondegenerate Boolean functions of 3 variables.
The number of surface points on a 73 cube.
In other fields
218 is the current number of votes in the US House of Representatives a party or coalition needs to win in order to achieve a majority.
The years 218 and 218 BC
Area code 218, for northern Minnesota.
References
Integers
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https://en.wikipedia.org/wiki/226%20%28number%29
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226 (two hundred [and] twenty-six) is the natural number following 225 and preceding 227.
In mathematics
226 is a happy number, and a semiprime (2×113),
and a member of Aronson's sequence.
At most 226 different permutation patterns can occur within a single 9-element permutation.
References
Integers
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https://en.wikipedia.org/wiki/232%20%28number%29
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232 (two hundred [and] thirty-two) is the natural number following 231 and preceding 233.
In mathematics
232 is both a central polygonal number and a cake number.
It is both a decagonal number and a centered 11-gonal number. It is also
a refactorable number,
a Motzkin sum,
an idoneal number, a Riordan number and a noncototient.
232 is a telephone number: in a system of seven telephone users, there are 232 different ways of pairing up some of the users.
There are also exactly 232 different eight-vertex connected indifference graphs, and 232 bracelets with eight beads of one color and seven of another. Because this number has the form , it follows that there are exactly 232 different functions from a set of four elements to a proper subset of the same set.
References
Integers
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https://en.wikipedia.org/wiki/238%20%28number%29
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238 (two hundred [and] thirty-eight) is the natural number following 237 and preceding 239.
In mathematics
238 is an untouchable number.
There are 238 2-vertex-connected graphs on five labeled vertices, and 238 order-5 polydiamonds (polyiamonds that can partitioned into 5 diamonds). Out of the 720 permutations of six elements, exactly 238 of them have a unique longest increasing subsequence.
There are 238 compact and paracompact hyperbolic groups of ranks 3 through 10.
References
Integers
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https://en.wikipedia.org/wiki/252%20%28number%29
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252 (two hundred [and] fifty-two) is the natural number following 251 and preceding 253.
In mathematics
252 is:
the central binomial coefficient , the largest one divisible by all coefficients in the previous line
, where is the Ramanujan tau function.
, where is the function that sums the cubes of the divisors of its argument:
a practical number,
a refactorable number,
a hexagonal pyramidal number.
a member of the Mian-Chowla sequence.
There are 252 points on the surface of a cuboctahedron of radius five in the face-centered cubic lattice, 252 ways of writing the number 4 as a sum of six squares of integers, 252 ways of choosing four squares from a 4×4 chessboard up to reflections and rotations, and 252 ways of placing three pieces on a Connect Four board.
References
Integers
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https://en.wikipedia.org/wiki/253%20%28number%29
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253 (two hundred [and] fifty-three) is the natural number following 252 and preceding 254.
In mathematics
253 is:
a semiprime since it is the product of 2 primes.
a triangular number.
a star number.
a centered heptagonal number.
a centered nonagonal number.
a Blum integer.
a member of the 13-aliquot tree.
References
Integers
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https://en.wikipedia.org/wiki/258%20%28number%29
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258 (two hundred [and] fifty-eight) is the natural number following 257 and preceding 259.
In mathematics
258 is:
a sphenic number
a nontotient
the sum of four consecutive prime numbers because 258 = 59 + 61 + 67 + 71
63 + 62 + 6
an Ulam number
References
Integers
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https://en.wikipedia.org/wiki/259%20%28number%29
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259 (two hundred [and] fifty-nine) is the natural number following 258 and preceding 260.
In mathematics
259 is:
a semiprime
63 + 62 + 6 + 1, so 259 is a repdigit in base 6 (11116)
a lucky number
References
Integers
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https://en.wikipedia.org/wiki/Cantellation%20%28geometry%29
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In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tilings and honeycombs. Cantellating a polyhedron is also rectifying its rectification.
Cantellation (for polyhedra and tilings) is also called expansion by Alicia Boole Stott: it corresponds to moving the faces of the regular form away from the center, and filling in a new face in the gap for each opened edge and for each opened vertex.
Notation
A cantellated polytope is represented by an extended Schläfli symbol t0,2{p,q,...} or r or rr{p,q,...}.
For polyhedra, a cantellation offers a direct sequence from a regular polyhedron to its dual.
Example: cantellation sequence between cube and octahedron:
Example: a cuboctahedron is a cantellated tetrahedron.
For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope to its birectified form.
Examples: cantellating polyhedra, tilings
See also
Uniform polyhedron
Uniform 4-polytope
Chamfer (geometry)
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp.145-154 Chapter 8: Truncation, p 210 Expansion)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
Polyhedra
4-polytopes
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https://en.wikipedia.org/wiki/276%20%28number%29
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276 (two hundred [and] seventy-six) is the natural number following 275 and preceding 277.
In mathematics
276 is the sum of 3 consecutive fifth powers (276 = 15 + 25 + 35). As a figurate number it is a triangular number, a hexagonal number, and a centered pentagonal number, the third number after 1 and 6 to have this combination of properties.
276 is the size of the largest set of equiangular lines in 23 dimensions. The maximal set of such lines, derived from the Leech lattice, provides the highest dimension in which the "Gerzon bound" of is known to be attained; its symmetry group is the third Conway group, Co3.
276 is the smallest number for which it is not known if the corresponding aliquot sequence either terminates or ends in a repeating cycle.
In the Bible
In Acts 27 verses 37-44 the Bible refers to 276 people on board a ship all of which made it to safety after the ship ran aground.
In other fields
In the Christian calendar, there are 276 days from the Annunciation on March 25 to Christmas on December 25, a number considered significant by some authors.
References
Integers
ca:Nombre 270#Nombres del 271 al 279
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https://en.wikipedia.org/wiki/288%20%28number%29
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288 (two hundred [and] eighty-eight) is the natural number following 287 and preceding 289.
Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".
In mathematics
Factorization properties
Because its prime factorization contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number. This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization. Among the highly abundant numbers, numbers with record-setting sums of divisors, it is one of only 13 such numbers with an odd divisor sum.
Both 288 and are powerful numbers, numbers in which all exponents of the prime factorization are larger than one. This property is closely connected to being highly abundant with an odd divisor sum: all sufficiently large highly abundant numbers have an odd prime factor with exponent one, causing their divisor sum to be even. 288 and 289 form only the second consecutive pair of powerful numbers after
Factorial properties
288 is a superfactorial, a product of consecutive factorials, since Coincidentally, as well as being a product of descending powers, 288 is a sum of ascending powers:
288 appears prominently in Stirling's approximation for the factorial, as the denominator of the second term of the Stirling series
Figurate properties
288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number and a dodecagonal number. Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number:
Enumerative properties
There are 288 different ways of completely filling in a sudoku puzzle grid. For square grids whose side length is the square of a prime number, such as 4 or 9, a completed sudoku puzzle is the same thing as a "pluperfect Latin square", an array in which every dissection into rectangles of equal width and height to each other has one copy of each digit in each rectangle. Therefore, there are also 288 pluperfect Latin squares of order 4. There are 288 different invertible matrices modulo six, and 288 different ways of placing two chess queens on a board with toroidal boundary conditions so that they do not attack each other. There are 288 independent sets in a 5-dimensional hypercube, up to symmetries of the hypercube.
In other areas
In early 20th-century molecular biology, some mysticism surrounded the use of 288 to count protein structures, largely based on the fact that it is a smooth number.
A common mathematical pun involves the fact that and that 144 is named as a gross: "Q: Why should the number 288 never be mentioned? A: it is two gross."
References
Integers
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https://en.wikipedia.org/wiki/Boolean%20algebras%20canonically%20defined
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Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions.
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.
Just as there are basic examples of groups, such as the group of integers and the symmetric group of permutations of objects, there are also basic examples of Boolean algebras such as the following.
The algebra of binary digits or bits 0 and 1 under the logical operations including disjunction, conjunction, and negation. Applications include the propositional calculus and the theory of digital circuits.
The algebra of sets under the set operations including union, intersection, and complement. Applications are far-reaching because set theory is the standard foundations of mathematics.
Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic and digital logic.
Unlike groups of finite order, which exhibit complexity and diversity and whose first-order theory is decidable only in special cases, all finite Boolean algebras share the same theorems and have a decidable first-order theory. Instead, the intricacies of Boolean algebra are divided between the structure of infinite algebras and the algorithmic complexity of their syntactic structure.
Definition
Boolean algebra treats the equational theory of the maximal two-element finitary algebra, called the Boolean prototype, and the models of that theory, called Boolean algebras. These terms are defined as follows.
An algebra is a family of operations on a set, called the underlying set of the algebra. We take the underlying set of the Boolean prototype to be {0,1}.
An algebra is finitary when each of its operations takes only finitely many arguments. For the prototype each argument of an operation is either or , as is the result of the operation. The maximal such algebra consists of all finitary operations on {0,1}.
The number of arguments taken by each operation is called the arity of the operation. An operation on {0,1} of arity , or -ary operation, can be applied to any of possible values for its arguments. For each choice of arguments, the operation may return or , whence there are -ary operations.
The prototype therefore has two operations taking no arguments, called zeroary or nullary operations, namely zero and one. It has four unary operations, two of which are constant operations, another
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https://en.wikipedia.org/wiki/Albert%20William%20Recht
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Albert William Recht (1898–1962) was an American mathematician and astronomer.
Initially he applied to work as a Spanish instructor at University of Denver. Instead he was hired by the Mathematics Department. He became chair of the mathematics department in 1943–44 and 1947–49.
While at the university he pursued his interest in astronomy, working at the Chamberlin Observatory under the instruction of Herbert Howe. In 1926 Recht became director of the observatory, and full director in 1928.
Over the following eleven years he studied during the summer months to earn his Ph.D. from the University of Chicago in 1939, working at Yerkes Observatory. He did his thesis work on the 6P/d'Arrest comet. Following his graduation, he was most noted for his work on the popularization of astronomy. During the 1950s he began a popular program of public viewing nights at the observatory. However his efforts to preserve the observatory at the time were unsuccessful.
The crater Recht on the far side of the Moon is named after him.
References
External links
Chamberlin Observatory Tour
1898 births
1962 deaths
American astronomers
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https://en.wikipedia.org/wiki/Fort%20Smith%20Region%2C%20Northwest%20Territories
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Fort Smith Region was a former Statistics Canada census division, one of two in the Northwest Territories, Canada. It was abolished in the 2011 census, along with the other census division of Inuvik Region, and the land area of the Northwest Territories was divided into new census divisions named Region 1, Region 2, Region 3, Region 4, Region 5, Region 6.
Its former territory covered all of the modern-day Regions 3 through 6, as well as a part of Region 2. For example, its border with the old Inuvik Region ran through the middle of Great Bear Lake, which is now entirely within the modern-day Region 2.
It contained more than 77 percent of the population and more than 54 percent of the land area of the Northwest Territories. Its main economic centre was the territorial capital of Yellowknife; it also contained the town of Fort Smith. The 2006 census reported a population of 32,272 spread over a land area of .
Communities
City
Yellowknife
Towns
Fort Smith
Hay River
Village
Fort Simpson
Hamlets
Fort Liard
Fort Providence
Behchokǫ̀
Whatì
Settlements
Dettah
Enterprise
Fort Resolution
Jean Marie River
Kakisa
Łutselk'e
Nahanni Butte
Gamèti
Fort Reliance
Trout Lake
Wekweeti
Wrigley
Indian reserves
Hay River Reserve (Hay River Dene)
Salt River First Nation
References
Regions of the Northwest Territories
Census divisions of the Canadian territories
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https://en.wikipedia.org/wiki/TIFR%20Centre%20for%20Applicable%20Mathematics
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The TIFR Centre for Applicable Mathematics is part of the School of Mathematics of the Tata Institute of Fundamental Research.
The centre originated from the school's efforts since the mid-1970s to develop areas in applicable mathematics. In fact, B. V. Sreekantan had proposed setting up this centre during the "Fifth Plan period". Along with the National Centre for Radio Astrophysics and the National Centre for Biological Sciences, the centre was established in the two decades following the 1970s. Many renowned mathematicians from India and overseas have contributed to the centre's development.
Research and consulting
At the centre, research at advanced level is currently pursued in both theoretical and numerical aspects of differential equations, especially the following subtopics:
Applications of Nonlinear Functional Analysis in the Study of Differential Equations
Control Aspects of Partial Differential Equations
Hyperbolic Equations and Conservation Laws
Homogenization and Solid Fluid Interactions
Numerical Analysis of PDE (Special Reference to Atmospheric Dynamics)
Microlocal Analysis
The centre offers consulting at the individual and institutional levels in applying theory to engineering-related problems and in exposition of underlying mathematics.
Education and programs
The centre has had an active role in training students in areas of applicable mathematics through the IISc-TIFR Joint Programme. Following the deemed university status of TIFR, the TIFR Centre has offered its own programs since 2004, which are:
Ph.D.
Integrated Ph.D. (M.Sc. and Ph.D.)
Among its other doctoral programs, the center has an integrated PhD program in mathematics. Eligible students for this program receive a monthly fellowship of Rs 21000. Admissions for this integrated program usually begins each year in August.
The centre has a program to invite visiting professors, both for disseminating new topics through lecture courses and for research collaboration.
The centre also offers post-doctoral fellowships and possibilities of short-term visits.
Notable people
In September 2015, together with Ritabrata Munshi (from the Tata Institute of Fundamental Research), K. Sandeep (from the TIFR Centre for Applied Mathematics) obtained the Shanti Swarup Bhatnagar Prize for Science and Technology in the field of mathematical sciences.
In January 2019, the Indian Academy of Sciences announced that 23 scientists were elected as fellows of the academy, including Sandeep Kunnath (who studies partial differential equation, variational methods, and nonlinear functional analysis) and G D Veerappa Gowda, both from the TIFR Centre for Applicable Mathematics.
References
External links
TIFR Centre for applicable mathematics
TIFR-CAM vision
Research institutes in Bangalore
Tata Institute of Fundamental Research
Educational institutions in India with year of establishment missing
Mathematical institutes
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https://en.wikipedia.org/wiki/Dense%20order
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In mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable.
Example
The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real numbers, the dyadic rationals and the decimal fractions. In fact, every Archimedean ordered ring extension of the integers is a densely ordered set.
On the other hand, the linear ordering on the integers is not dense.
Uniqueness for total dense orders without endpoints
Georg Cantor proved that every two non-empty dense totally ordered countable sets without lower or upper bounds are order-isomorphic. This makes the theory of dense linear orders without bounds an example of an ω-categorical theory where ω is the smallest limit ordinal. For example, there exists an order-isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proofs of these results use the back-and-forth method.
Minkowski's question mark function can be used to determine the order isomorphisms between the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals.
Generalizations
Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related. Formally:
Alternatively, in terms of composition of R with itself, the dense condition may be expressed as R ⊆ R ; R.
Sufficient conditions for a binary relation R on a set X to be dense are:
R is reflexive;
R is coreflexive;
R is quasireflexive;
R is left or right Euclidean; or
R is symmetric and semi-connex and X has at least 3 elements.
None of them are necessary. For instance, there is a relation R that is not reflexive but dense.
A non-empty and dense relation cannot be antitransitive.
A strict partial order < is a dense order if and only if < is a dense relation. A dense relation that is also transitive is said to be idempotent.
See also
Dense set — a subset of a topological space whose closure is the whole space
Dense-in-itself — a subset of a topological space such that does not contain an isolated point
Kripke semantics — a dense accessibility relation corresponds to the axiom
References
Further reading
David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, , p. 6ff
Binary relations
Order theory
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https://en.wikipedia.org/wiki/Sturmian%20sequence
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In mathematics, a Sturmian sequence may refer to:
A Sturmian word: a sequence with minimal complexity function
A sequence used to determine the number of distinct real roots of a polynomial by Sturm's theorem
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https://en.wikipedia.org/wiki/Observational%20study
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In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concerns or logistical constraints. One common observational study is about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator. This is in contrast with experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group. Observational studies, for lacking an assignment mechanism, naturally present difficulties for inferential analysis.
Motivation
The independent variable may be beyond the control of the investigator for a variety of reasons:
A randomized experiment would violate ethical standards. Suppose one wanted to investigate the abortion – breast cancer hypothesis, which postulates a causal link between induced abortion and the incidence of breast cancer. In a hypothetical controlled experiment, one would start with a large subject pool of pregnant women and divide them randomly into a treatment group (receiving induced abortions) and a control group (not receiving abortions), and then conduct regular cancer screenings for women from both groups. Needless to say, such an experiment would run counter to common ethical principles. (It would also suffer from various confounds and sources of bias, e.g. it would be impossible to conduct it as a blind experiment.) The published studies investigating the abortion–breast cancer hypothesis generally start with a group of women who already have received abortions. Membership in this "treated" group is not controlled by the investigator: the group is formed after the "treatment" has been assigned.
The investigator may simply lack the requisite influence. Suppose a scientist wants to study the public health effects of a community-wide ban on smoking in public indoor areas. In a controlled experiment, the investigator would randomly pick a set of communities to be in the treatment group. However, it is typically up to each community and/or its legislature to enact a smoking ban. The investigator can be expected to lack the political power to cause precisely those communities in the randomly selected treatment group to pass a smoking ban. In an observational study, the investigator would typically start with a treatment group consisting of those communities where a smoking ban is already in effect.
A randomized experiment may be impractical. Suppose a researcher wants to study the suspected link between a certain medication and a very rare group of symptoms arising as a side effect. Setting aside any ethical considerations, a randomized experiment would be impractical because of the rarity of the effect. There may not be a subject pool large enough for the symptoms to be obs
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https://en.wikipedia.org/wiki/Jacques%20Cast%C3%A9r%C3%A8de
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Jacques Castérède (10 April 1926 – 6 April 2014) was a French composer and pianist.
Life
Born in Paris, Castérède studied at Lycée Buffon. He earned his baccalauréat in elementary mathematics, then entered the Paris Conservatory in 1944 and began studying piano under Armand Ferté, composition under Tony Aubin, and analysis under Olivier Messiaen. While at the Conservatory, between 1948 and 1953 he received five first prizes (in piano, chamber music, analysis, composition, and harmony). He also won the Grand Prix de Rome in 1953 with his cantata La Boîte de Pandore (Pandora's Box). The following year, he went to Rome, where he stayed at the Villa Medici until 1958.
In 1960, he was appointed professor of solfège in the Paris Conservatory, then counseller of piano studies (Conseilleur aux Études) in 1966, and analysis in 1971. In addition, he taught composition at the École Normale from 1983 to 1988, and analysis from 1988 to 1998. On an invitation from the Chinese government, he became a professor of composition at the Central Academy in Beijing. He received numerous awards as a composer, among them the Paris Civil Award in 1991, the Charles Cros Award, and the Record Academy Award in 1995. His many works, which include symphonies, concertos, ballets, and ensemble and chamber music, have been performed throughout France, Germany, and Italy as well as in the United States and Canada.
His music is essentially melodic, often using modal scales over rich and varied structures.
Main works
Stage
Chamber opera La Cour des miracles (1954) {sop, msp, alto, btn, orch}
Oratorio Le Livre de Job (1958) {recit, tnr, btn, bss, choir, orch}
Ballet Basketball (1959) {hrp, p, perc, orch}
Ballet But (1959)
Orchestral
1ère Symphonie pour cordes (1952) {strings}
La Folle nuit de n'importe ou (1955) {orch, clvcn, hrp, perc}
La Grande peur (1955) {martenot, 2p, child-choir, sop, fl, orch}
Cinq Danses symphoniques (1956) {orch, p, hrp}
Musique pour un conte d'Edgar Allan Poe (1957) {hrp, clsta, xylo, vib, p, recit, orch}
Suite en trois mouvements à la mémoire d'Honegger (1957)
La Déscente de croix de Rubens (1958)
Le Fil d'Ariane (1959), orchestra
Symphonie No. 2 (1960)
Prélude et fugue (1960) {strings}
Pamplemousse (1962)
Sophie-Dorothée (1962)
Promenade printaniere (1963)
Divertissement d'été (1965) {p, hrp, brass, choir, perc?}
Concertante works
Concerto No. 1 pour piano et orchestre à cordes (1954) {p, strings}
Concertino (1958) {tp, tb, p, perc, strings}
Concerto No. 2 pour piano et orchestre (1970) {p, orch}
Concerto pour guitare et orchestre (1973) {g, orch}
Trois paysages d'automne (1982) {vc, strings}
2ème Concerto pour guitare et orchestre (1988) {g, orch}
Chamber
Intermezzo (1953) {ob, p}
Pastorale (1953) {as, p}
Scherzo (1953) {as, p}
Wind Quintet (1953) {fl, ob, cl, hrn, bssn}
Sonatine pour trompette et piano (1953) {tp, p}
Suite mythologique (1954) {3martenot, p}
Sonate pour violon et piano (1955) {vln, p}
Sonate en fo
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https://en.wikipedia.org/wiki/Star%C3%A1%20Huta
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Stará Huta () is a village and municipality in Detva District, in the Banská Bystrica Region of central Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Detva District
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https://en.wikipedia.org/wiki/Legendre%20rational%20functions
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In mathematics the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.
A rational Legendre function of degree n is defined as:
where is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:
with eigenvalues
Properties
Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.
Recursion
and
Limiting behavior
It can be shown that
and
Orthogonality
where is the Kronecker delta function.
Particular values
References
Rational functions
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https://en.wikipedia.org/wiki/LAPACK%2B%2B
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LAPACK++, the Linear Algebra PACKage in C++, is a computer software library of algorithms for numerical linear algebra that solves systems of linear equations and eigenvalue problems.
It supports various matrix classes for vectors, non-symmetric matrices, SPD matrices, symmetric matrices, banded, triangular, and tridiagonal matrices. However, it does not include all of the capabilities of original LAPACK library.
History
The original LAPACK++ (up to v1.1a) was written by R. Pozo et al. at the University of Tennessee and Oak Ridge National Laboratory.
In 2000, R. Pozo et al. left the project, with the projects' web page stating LAPACK++ would be superseded by the Template Numerical Toolkit (TNT).
The current LAPACK++ (versions 1.9 onwards) started off as a fork from the original LAPACK++. There are extensive fixes and changes, such as more wrapper functions for LAPACK and BLAS routines.
See also
List of numerical analysis software
List of numerical libraries
External links
old LAPACK++ Homepage (version 1.1a)
new LAPACK++ Homepage (versions 1.9 onwards)
C++ numerical libraries
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https://en.wikipedia.org/wiki/Hyperbolic%20distribution
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The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distributions.
The origin of the distribution is the observation by Ralph Bagnold, published in his book The Physics of Blown Sand and Desert Dunes (1941), that the logarithm of the histogram of the empirical size distribution of sand deposits tends to form a hyperbola. This observation was formalised mathematically by Ole Barndorff-Nielsen in a paper in 1977, where he also introduced the generalised hyperbolic distribution, using the fact the a hyperbolic distribution is a random mixture of normal distributions.
References
Continuous distributions
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https://en.wikipedia.org/wiki/George%20Paine%20%28civil%20servant%29
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George Paine CB DFC (14 April 1918 – 2 March 1992) (known as "Toby") was a statistician in the British Civil Service. He rose to become Director of Statistics and Intelligence at the Inland Revenue, Registrar General of England and Wales, and Director of Office of Population Censuses and Surveys from November 1972.
He was born in Kent and was schooled at home, Ovingdean Hall School and Bradfield College before attending Peterhouse, Cambridge.
He navigated in De Havilland Mosquitos during World War II and earned a DFC. He took early retirement in 1978. to farm in Wiltshire.
Honours and awards
19 September 1944 – Flying Office Robert Lyle James Barbour (125456), RAFVR, 264 Sqn. Flying Officer George Paine (129167), RAFVR, 264 Sqn. Have been awarded the Distinguished Flying Cross (DFC) – These officers have completed very many sorties as pilot and observer respectively. They have displayed great skill and co-operation and have destroyed 3 enemy aircraft at night. Their keenness and devotion to duty have been most commendable.
15 June 1974 – George Paine, DFC, Director of the Office of Population Censuses and Surveys and Registar General of England and Wales is appointed a Companion of the Order of the Bath
References
Philip Redfern, "Obituary: George ('Toby') Paine CB, DFC, 1918-92", Journal of the Royal Statistical Society. Series A (Statistics in Society) 156 (1993), pp. 121–122
1918 births
1992 deaths
English statisticians
People educated at Bradfield College
Recipients of the Distinguished Flying Cross (United Kingdom)
Royal Air Force officers
English aviators
Companions of the Order of the Bath
Registrars-General for England and Wales
Civil servants in the Office of Population Censuses and Surveys
Members of HM Government Statistical Service
Civil servants in the Board of Inland Revenue
People from the Borough of Maidstone
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https://en.wikipedia.org/wiki/Spin-weighted%20spherical%20harmonics
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In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree , just like ordinary spherical harmonics, but have an additional spin weight that reflects the additional symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics , and are typically denoted by , where and are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight are simply the standard spherical harmonics:
Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group . They were subsequently and independently rediscovered by and applied to describe gravitational radiation, and again by as so-called "monopole harmonics" in the study of Dirac monopoles.
Spin-weighted functions
Regard the sphere as embedded into the three-dimensional Euclidean space . At a point on the sphere, a positively oriented orthonormal basis of tangent vectors at is a pair of vectors such that
where the first pair of equations states that and are tangent at , the second pair states that and are unit vectors, the penultimate equation that and are orthogonal, and the final equation that is a right-handed basis of .
A spin-weight function is a function accepting as input a point of and a positively oriented orthonormal basis of tangent vectors at , such that
for every rotation angle .
Following , denote the collection of all spin-weight functions by . Concretely, these are understood as functions on } satisfying the following homogeneity law under complex scaling
This makes sense provided is a half-integer.
Abstractly, is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle of the Serre twist on the complex projective line . A section of the latter bundle is a function on } satisfying
Given such a , we may produce a spin-weight function by multiplying by a suitable power of the hermitian form
Specifically, is a spin-weight function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.
The operator
The spin weight bundles are equipped with a differential operator (eth). This operator is essentially the Dolbeault operator, after suitable identifi
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https://en.wikipedia.org/wiki/Multicomplex%20number
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In mathematics, the multicomplex number systems are defined inductively as follows: Let C0 be the real number system. For every let in be a square root of −1, that is, an imaginary unit. Then . In the multicomplex number systems one also requires that (commutativity). Then is the complex number system, is the bicomplex number system, is the tricomplex number system of Corrado Segre, and is the multicomplex number system of order n.
Each forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system
The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute ( when for Clifford).
Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: despite and , and despite and . Any product of two distinct multicomplex units behaves as the of the split-complex numbers, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane.
With respect to subalgebra , k = 0, 1, ..., , the multicomplex system is of dimension over
References
G. Baley Price (1991) An Introduction to Multicomplex Spaces and Functions, Marcel Dekker.
Corrado Segre (1892) "The real representation of complex elements and hyperalgebraic entities" (Italian), Mathematische Annalen 40:413–67 (see especially pages 455–67).
Hypercomplex numbers
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https://en.wikipedia.org/wiki/Emory%20McClintock
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Emory McClintock (1840–1916), born John Emory McClintock was an American actuary, born in Carlisle, Pennsylvania. He graduated from Columbia University, where he was tutor in mathematics in 1859–1860. From 1863 to 1866 he served as United States consular agent at Bradford, England. He served as president of the American Mathematical Society in 1890–1894 and of the Actuarial Society of America in 1895–1897.
Early life and career
He was born to John and Caroline Augusta Wakeman McClintock. His father was a minister of the Methodist Episcopal Church, who was also a professor of mathematics, and ancient languages at Dickinson College in Carlisle. His father was also involved in an 1847 a riot over slavery, as he tried to prevent slavechasers from taking African-American citizens of Pennsylvania into slavery.
He was actuary of the Asbury Life Insurance Company, New York (1867–1871), of the Northwestern Mutual Life Insurance Company, Milwaukee, Wisconsin, (1871–1889), and of the Mutual Life Insurance Company, New York (1889–1911).
At the Mutual, he was vice president from 1905 to 1911, a trustee after 1905, and a consulting actuary after 1911.
References
Thomas S. Fiske, Emory McClintock, Bulletin of the American Mathematical Society, 23, (1917), pp. 353–357. (includes a list of his publications)
External links
Archives and Special Collections, Dickinson College, J. Emory McClintock, Family Papers 1853-1918
Columbia University faculty
Columbia University alumni
1840 births
People from Carlisle, Pennsylvania
1916 deaths
People from Milwaukee
American actuaries
Presidents of the American Mathematical Society
19th-century American businesspeople
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https://en.wikipedia.org/wiki/Expansion%20%28geometry%29
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In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.
The expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation (which represents expansion with the letter ). For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, and new square faces in place of the original edges.
Expansion of regular polytopes
According to Coxeter, this multidimensional term was defined by Alicia Boole Stott for creating new polytopes, specifically starting from regular polytopes to construct new uniform polytopes.
The expansion operation is symmetric with respect to a regular polytope and its dual. The resulting figure contains the facets of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements.
It has somewhat different meanings by dimension. In a Wythoff construction, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e2 is the same as t0,2 in any dimension.
By dimension:
A regular {p} polygon expands into a regular 2n-gon.
The operation is identical to truncation for polygons, e{p} = e1{p} = t0,1{p} = t{p} and has Coxeter-Dynkin diagram .
A regular {p,q} polyhedron (3-polytope) expands into a polyhedron with vertex figure p.4.q.4.
This operation for polyhedra is also called cantellation, e{p,q} = e2{p,q} = t0,2{p,q} = rr{p,q}, and has Coxeter diagram .
For example, a rhombicuboctahedron can be called an expanded cube, expanded octahedron, as well as a cantellated cube or cantellated octahedron.
A regular {p,q,r} 4-polytope (4-polytope) expands into a new 4-polytope with the original {p,q} cells, new cells {r,q} in place of the old vertices, p-gonal prisms in place of the old faces, and r-gonal prisms in place of the old edges.
This operation for 4-polytopes is also called runcination, e{p,q,r} = e3{p,q,r} = t0,3{p,q,r}, and has Coxeter diagram .
Similarly a regular {p,q,r,s} 5-polytope expands into a new 5-polytope with facets {p,q,r}, {s,r,q}, {p,q}×{ } prisms, {s,r}×{ } prisms, and {p}×{s} duoprisms.
This operation is called sterication, e{p,q,r,s} = e4{p,q,r,s} = t0,4{p,q,r,s} = 2r2r{p,q,r,s} and has Coxeter diagram .
The general operator for expansion of a regular n-polytope is t0,n-1{p,q,r,...}. New regular facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ... ridge, etc.
See also
Conway polyhedron notation
Notes
References
Coxeter, H. S. M., Regular Polytopes. 3rd edition, Dover, (1973) .
Nor
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https://en.wikipedia.org/wiki/William%20S.%20Hatcher
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William S. Hatcher (1935–2005) was a mathematician, philosopher, educator and a member of the Baháʼí Faith. He held a doctorate in mathematics from the University of Neuchatel, Switzerland, and bachelor's and master's degrees from Vanderbilt University in Nashville, Tennessee. A specialist in the philosophical alloying of science and religion, for over thirty years he held university positions in North America, Europe, and Russia.
Biography
He was born in Charlotte, North Carolina, United States on 20 September 1935, and died on 27 November 2005.
Work and achievements
Hatcher is one of eight Platonist philosophers listed for the second half of the twentieth century in the Encyclopedie Philosophique Universelle.
Hatcher was the author of over fifty monographs, books, and articles in the mathematical sciences, logic and philosophy. Among the publications of which he is author or coauthor are:
The Foundations of Mathematics (1968)
Absolute Algebra (with Stephen Whitney, 1978)
The Science of Religion (1980)
The Logical Foundations of Mathematics (1982)
The Baha'i Faith: The Emerging Global Religion (1984)
Logic and Logos: Essays on Science, Religion and Philosophy (1990)
The Law of Love Enshrined (1996)
The Ethics of Authenticity (1997)
Love, Power, and Justice (1998)
Minimalism: A Bridge between Classical Philosophy and the Baha'i Revelation (2002)
Relationship to the Baháʼí Faith
He served on National Spiritual Assembly of the Baha'is of Canada (1983–91) as well as on the inaugural National Spiritual Assemblies of Switzerland (1962–65) and the Russian Federation (1996). He lived in Russia from 1993 to 1998. He was also a founding member of the Association for Baháʼí Studies.
See also
Proof of the Truthful
Notes
References
External links
1935 births
2005 deaths
American Bahá'ís
20th-century American mathematicians
21st-century American mathematicians
American philosophy academics
Philosophers of religion
Vanderbilt University alumni
University of Neuchâtel alumni
Converts to the Bahá'í Faith
20th-century Bahá'ís
21st-century Bahá'ís
Academic staff of Université Laval
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https://en.wikipedia.org/wiki/Albert%20Lautman
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Albert Lautman (8 February 1908 – 1 August 1944) was a French philosopher of mathematics, born in Paris. An escaped prisoner of war, he was shot by the Nazi authorities in Toulouse on 1 August 1944.
Family
His father was a Jewish emigrant from Vienna who became a medical doctor after he was seriously wounded in the First World War.
Selected bibliography
Essai sur les Notions de Structure et d'Existence en Mathématiques
Essai sur l'Unité des Sciences Mathématiques
Symétrie et Dissymétrie en Mathématiques et en Physique
Les Mathématiques, les idées et le réel physique
Translations
Mathematics, Ideas and the Physical Real (2011) - this volume advertises itself as "the first English collection of the work of Albert Lautman"
Notes
External links
Fractal Ontology (English) with translations of Lautman's work by Taylor Adkins and Joseph Weissman.
1908 births
1944 deaths
Writers from Paris
Jews in the French resistance
École Normale Supérieure alumni
Philosophers of mathematics
20th-century French philosophers
World War II prisoners of war held by Germany
French prisoners of war in World War II
Deaths by firearm in France
People executed by Germany by firearm
Resistance members killed by Nazi Germany
French people executed by Nazi Germany
French male non-fiction writers
20th-century French male writers
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https://en.wikipedia.org/wiki/Representation%20%28mathematics%29
|
In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform, in some consistent way, to those existing among the corresponding represented objects xi. More specifically, given a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphism that preserves Π. The label representation is sometimes also applied to the homomorphism itself (such as group homomorphism in group theory).
Representation theory
Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representation theory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.
Other examples
Although the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout mathematics.
Graph theory
An active area of graph theory is the exploration of isomorphisms between graphs and other structures.
A key class of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets
(or, more precisely, non-disjointness) is a symmetric relation.
This gives rise to the study of intersection graphs for innumerable families of sets.
One foundational result here, due to Paul Erdős and his colleagues, is that every n-vertex graph may be represented in terms of intersection among subsets of a set of size no more than n2/4.
Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the field of spectral graph theory.
Order theory
Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set (also known as poset) is isomorphic to a collection of sets ordered by the inclusion (or containment) relation ⊆.
Some posets that arise as the inclusion orders for natural classes of objects include the Boolean lattices and the orders of dimension n.
Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them are the n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circle orders—the posets representable in terms of containment among disks in the plane. A particularly nice result in this field is the characterization of the planar graphs, as those graphs whose vertex-edge incidence relations are circle orders.
There are also geometric representations that are not based on inclusion. Indeed, one of the best studied classes among these are the interval orders, which represent the partial order in terms of wh
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https://en.wikipedia.org/wiki/353%20%28number%29
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353 (three hundred fifty-three) is the natural number following 352 and preceding 354. It is a prime number.
In mathematics
353 is a palindromic prime, an irregular prime, a super-prime, a Chen prime, a Proth prime, and an Eisentein prime.
In connection with Euler's sum of powers conjecture, 353 is the smallest number whose 4th power is equal to the sum of four other 4th powers, as discovered by R. Norrie in 1911:
In a seven-team round robin tournament, there are 353 combinatorially distinct outcomes in which no subset of teams wins all its games against the teams outside the subset; mathematically, there are 353 strongly connected tournaments on seven nodes.
353 is one of the solutions to the stamp folding problem: there are exactly 353 ways to fold a strip of eight blank stamps into a single flat pile of stamps.
353 in Mertens Function returns 0.
353 is an index of a prime Lucas number.
References
Integers
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https://en.wikipedia.org/wiki/David%20Pingree
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David Edwin Pingree (January 2, 1933, New Haven, Connecticut – November 11, 2005, Providence, Rhode Island) was an American historian of mathematics in the ancient world. He was a University Professor and Professor of History of Mathematics and Classics at Brown University.
Life
Pingree graduated from Phillips Academy in Andover, Massachusetts in 1950. He studied at Harvard University, where he earned his doctorate in 1960 with a dissertation on the supposed transmission of Hellenistic astrology to India. His dissertation was supervised by Daniel Henry Holmes Ingalls, Sr. and Otto Eduard Neugebauer. After completing his PhD, Pingree remained at Harvard three more years as a member of its Society of Fellows before moving to the University of Chicago to accept the position of Research Associate at the Oriental Institute.
He joined the History of Mathematics Department at Brown University in 1971, eventually holding the chair until his death.
As successor to Otto Neugebauer (1899–1990) in Brown's History of Mathematics Department (which Neugebauer established in 1947), Pingree numbered among his colleagues men of extraordinary learning, including Abraham Sachs and Gerald Toomer.
Career
Jon McGinnis of the University of Missouri, St. Louis, describes Pingree's life-work thus:
... Pingree devoted himself to the study of the exact sciences, such as mathematics, mathematical astronomy and astral omens. He was also acutely interested in the transmission of those sciences across cultural and linguistic boundaries. His interest in the transmission of the exact sciences came from two fronts or, perhaps more correctly, his interest represents two sides of the same coin. On the one hand, he was concerned with how one culture might appropriate, and so alter, the science of another (earlier) culture in order to make that earlier scientific knowledge more accessible to the recipient culture. On the other hand, Pingree was also interested in how scientific texts surviving from a later culture might be used to reconstruct or cast light on our fragmentary records of earlier sciences. In this quest, Pingree would, with equal facility use ancient Greek works to clarify Babylonian texts on divination, turn to Arabic treatises to illuminate early Greek astronomical and astrological texts, seek Sanskrit texts to explain Arabic astronomy, or track the appearance of Indian astronomy in medieval Europe.
In June 2007, the Brown University Library acquired Pingree's personal collection of scholarly materials. The collection focuses on the study of mathematics and exact sciences in the ancient world, especially India, and the relationship of Eastern mathematics to the development of mathematics and related disciplines in the West. The collection contains some 22,000 volumes, 700 fascicles, and a number of manuscripts. The holdings consist of both antiquarian and recent materials published in Sanskrit, Arabic, Hindi, Persian and Western languages.
Awards
Recipient of
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https://en.wikipedia.org/wiki/GMS%20%28software%29
|
GMS (Groundwater Modeling System) is water modeling application for building and simulating groundwater models from Aquaveo. It features 2D and 3D geostatistics, stratigraphic modeling and a unique conceptual model approach. Currently supported models include MODFLOW, MODPATH, MT3DMS, RT3D, FEMWATER, SEEP2D, and UTEXAS.
Version 6 introduced the use of XMDF (eXtensible Model Data Format), which is a compatible extension of HDF5. The purpose of this is to allow internal storage and management of data in a single HDF file, rather than using many flat files.
History
GMS was initially developed in the late 1980s and early 1990s on Unix workstations by the Engineering Computer Graphics Laboratory at Brigham Young University. The development of GMS was funded primarily by The United States Army Corps of Engineers and was known—until version 4.0, released in late 1999—as the Department of Defense Groundwater Modeling System, or DoD GMS. It was ported to Microsoft Windows in the mid 1990s. Version 3.1 was the last version that supported HP-UX, IRIX, OSF/1, and Solaris platforms. Development of GMS—along with WMS and SMS—was transferred to Aquaveo when it formed in April 2007.
A study published in the Journal of Agricultural and Applied Economics in August 2000 stated that "GMS provides an interface to the groundwater flow model, MODFLOW, and the contaminant transport model, MT3D. MODFLOW is a three-dimensional, cell-centered, finite-difference, saturated-flow model capable of both steady-state and transient analyses...These two models, when put together, provide a comprehensive tool for examining groundwater flow and nitrate transport and accumulation". The study was designed to help develop a "permit scheme to effectively manage nitrate pollution of groundwater supplies for communities in rural areas without hindering agricultural production in watersheds".
Version history
Reception
A 2001 report prepared for the Iowa Comprehensive Petroleum Underground Storage Tank Fund Board stated that GMS was "a very user-friendly software package with strong technical support." Raymond H. Johnson, a hydrogeologist with the US Geological Survey, called GMS 6.0 "a useful all around groundwater modeling package that offers the advantages of modular purchases, multiple model support, linkages to ArcGIS, conceptual model development, and integrated inversion routines." A 2006 report from the Center for Nuclear Waste Regulatory Analyses in San Antonio, Texas called GMS "the most sophisticated groundwater modeling software available".
References
External links
GMS Wiki
Scientific simulation software
Science software for Windows
Hydrogeology software
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https://en.wikipedia.org/wiki/%C5%BD%C3%ADp
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Žíp () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Linear%20programming%20relaxation
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In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable.
For example, in a 0–1 integer program, all constraints are of the form
.
The relaxation of the original integer program instead uses a collection of linear constraints
The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program.
Example
Consider the set cover problem, the linear programming relaxation of which was first considered by . In this problem, one is given as input a family of sets F = {S0, S1, ...}; the task is to find a subfamily, with as few sets as possible, having the same union as F.
To formulate this as a 0–1 integer program, form an indicator variable xi for each set Si, that takes the value 1 when Si belongs to the chosen subfamily and 0 when it does not. Then a valid cover can be described by an assignment of values to the indicator variables satisfying the constraints
(that is, only the specified indicator variable values are allowed) and, for each element ej of the union of F,
(that is, each element is covered). The minimum set cover corresponds to the assignment of indicator variables satisfying these constraints and minimizing the linear objective function
The linear programming relaxation of the set cover problem describes a fractional cover in which the input sets are assigned weights such that the total weight of the sets containing each element is at least one and the total weight of all sets is minimized.
As a specific example of the set cover problem, consider the instance F = {{a, b}, {b, c}, {a, c}}. There are three optimal set covers, each of which includes two of the three given sets. Thus, the optimal value of the objective function of the corresponding 0–1 integer program is 2, the number of sets in the optimal covers. However, there is a fractional solution in which each set is assigned the weight 1/2, and for which the total value of the objective function is 3/2. Thus, in this example, the linear programming relaxation has a value differing from that of the unrelaxed 0–1 integer program.
Solution quality of relaxed and original programs
The linear programming relaxation of an integer program may be solved using any standard linear programming technique. If it happens that, in the optimal solution, all variables have integer values, then it will also be an optimal solution to the original integer program. However, this is generally not true, except for some special cases (e.g. problems with totally unimodular matrix specifications.)
In all cases, though, the solution quality of the linear program is at least as good as that of the integer program, because
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https://en.wikipedia.org/wiki/Ren%C3%A9%20Renno
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René Renno (born 19 February 1979) is a German former professional footballer who played as a goalkeeper.
Career statistics
References
External links
1979 births
Living people
Footballers from Berlin
German men's footballers
Men's association football goalkeepers
Bundesliga players
2. Bundesliga players
Hertha BSC II players
Tennis Borussia Berlin players
SG Wattenscheid 09 players
Rot-Weiss Essen players
VfL Bochum players
VfL Bochum II players
FC Energie Cottbus players
FC Energie Cottbus II players
3. Liga players
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https://en.wikipedia.org/wiki/Roland%20Dobrushin
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Roland Lvovich Dobrushin () (July 20, 1929 – November 12, 1995) was a mathematician who made important contributions to probability theory, mathematical physics, and information theory.
Life and work
Dobrushin received his Ph.D. at Moscow State University under the supervision of Andrey Kolmogorov.
In statistical mechanics, he introduced (simultaneously with Lanford and Ruelle) the DLR equations for the Gibbs measure. Together with Kotecký and Shlosman, he studied the formation of droplets in Ising-type models, providing mathematical justification of the Wulff construction.
He was a foreign member of the American Academy of Arts and Sciences, Academia Europæa and US National Academy of Sciences.
The Dobrushin prize was established in his honour.
Notes
References
External links
Memorial website.
Biography (in Russian)
Obituary from The Independent
1929 births
1995 deaths
Soviet mathematicians
20th-century Russian mathematicians
Probability theorists
Moscow State University alumni
Members of Academia Europaea
Foreign associates of the National Academy of Sciences
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https://en.wikipedia.org/wiki/Thomas%20Rathgeber
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Thomas Rathgeber (born 30 April 1985 in Kempten, West Germany) is a German footballer who plays for FC Kempten.
Career statistics
Club
References
External links
Thomas Rathgeber at FuPa
1985 births
Living people
German men's footballers
VfL Bochum players
VfL Bochum II players
SpVgg Unterhaching players
Kickers Offenbach players
1. FC Saarbrücken players
SpVgg Unterhaching II players
FC Schalke 04 II players
SSV Ulm 1846 players
Bundesliga players
2. Bundesliga players
3. Liga players
Regionalliga players
Men's association football forwards
Sportspeople from Kempten im Allgäu
Footballers from Swabia (Bavaria)
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https://en.wikipedia.org/wiki/Keith%20Clark%20%28computer%20scientist%29
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Keith Leonard Clark (born 29 March 1943) is an Emeritus Professor in the Department of Computing at Imperial College London, England.
Education
Clark studied Mathematics at Durham University (Hatfield College), graduating in 1964 with a first-class degree. Clark then continued his studies at Cambridge University, taking a second undergraduate degree in Philosophy in 1966. He earned a Ph.D. in 1980 from the University of London with thesis titled Predicate logic as a computational formalism.
Career
Clark undertook Voluntary Service Overseas from 1967 to 1968 as a teacher of Mathematics at a school in Sierra Leone. He lectured in Computer Science at the Mathematics Department of Queen Mary College from 1969 to 1975. In 1975 he moved to Imperial College London, where he became a Senior Lecturer in the Department of Computer Science and joined Robert Kowalski in setting up the logic programming group. From 1987 to 2009 he was Professor of Computational Logic at Imperial College.
Clark's key contributions have been in the field of logic programming. His current research interests include multi-agent systems, cognitive robotics and multi-threading.
Business Interests
In 1980, with colleague Frank McCabe, he founded an Imperial College spin-off company, Logic Programming Associates, to develop and market Prolog systems for microcomputers (micro-Prolog) and to provide consultancy on expert systems and other logic programming applications. The company's star product was MacProlog. It had a user interface exploiting all the graphic user interface primitives of the Mac's OS, and primitives allowing bespoke Prolog-based applications to be built with application specific interfaces. Clark has also acted as a consultant to IBM, Hewlett-Packard and Fujitsu among other companies.
Selected publications
K. L. Clark, D. Cowell, Programs, Machines and Computation, McGraw-Hill, London, 1976.
K. L. Clark, S-A. Tarnlund, A first order theory of data and programs, Proc. IFIP Congress, Toronto, 939–944 pp, 1977.
K. L. Clark, Negation as failure, Logic and Data Bases (eds. Gallaire & Minker) Plenum Press, New York, 293–322 pp, 1978. (Also in Readings in Nonmonotonic Reasoning, (ed. M. Ginsberg), Morgan Kaufmann, 311–325, 1987.)
K. L. Clark, S. Gregory, A relational language for parallel programming, Proc. ACM Conference on Functional Languages and Computer Architecture, ACM, New York, 171–178 pp, 1981. (Also in Concurrent Prolog, (ed. E Shapiro), MIT Press, 9–26 pp, 1987.)
K. L. Clark, S-A. Tarnlund (eds), Logic Programming, Academic Press, London, 1982.
K. L. Clark, F. G. McCabe, micro-PROLOG: Programming in Logic, Prentice-Hall International, 1984.
K. L. Clark, I. Foster, A Declarative Environment for Concurrent Logic Programming, Proceedings of Colloquium on Functional and Logic Programming and Specification, LNCS 250, Springer-Verlag, 212 - 242 pp, 1987
K. L. Clark, Logic Programming Schemes and their Implementations, Computational Logic (ed Lasse
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https://en.wikipedia.org/wiki/Martin%20Amedick
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Martin Amedick (born 6 September 1982) is a German retired professional footballer who plays as a centre back.
Career statistics
.
Honours
Club
Borussia Dortmund
DFB-Pokal runner-up: 2007–08
References
External links
1982 births
German men's footballers
Living people
Sportspeople from Paderborn
Footballers from Detmold (region)
Eintracht Braunschweig players
Arminia Bielefeld players
SC Paderborn 07 players
Borussia Dortmund players
Borussia Dortmund II players
1. FC Kaiserslautern players
Eintracht Frankfurt players
Bundesliga players
2. Bundesliga players
Men's association football defenders
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https://en.wikipedia.org/wiki/Weil%20group
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In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/ (where the superscript c denotes the commutator subgroup).
For more details about Weil groups see or or .
Class formation
The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program.
If E/F is a normal layer, then the (relative) Weil group WE/F of E/F is the extension
1 → AF → WE/F → Gal(E/F) → 1
corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class uE/F in H2(Gal(E/F), AF). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers
G/F, for F an open subgroup of G.
The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
Archimedean local field
For archimedean local fields the Weil group is easy to describe: for C it is the group C× of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C× ∪ j C× of the non-zero quaternions.
Finite field
For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse).
Local field
For a local field of characteristic p > 0, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).
For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.
More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology. This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group. (The resulting topology is "locally profinite".)
Function field
For global fields of characteristic p>0 (function fields), the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).
Number field
For number fields there is no known "natural" construction of the Weil group without using cocycles to constru
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https://en.wikipedia.org/wiki/Gauss%20sum
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In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
where the sum is over elements of some finite commutative ring , is a group homomorphism of the additive group into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit , where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function.
Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet -functions, where for a Dirichlet character the equation relating and ) (where is the complex conjugate of ) involves a factor
History
The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for the field of residues modulo a prime number , and the Legendre symbol. In this case Gauss proved that or for congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration).
An alternate form for this Gauss sum is
.
Quadratic Gauss sums are closely connected with the theory of theta functions.
The general theory of Gauss sums was developed in the early 19th century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. Gauss sums over a residue ring of integers are linear combinations of closely related sums called Gaussian periods.
The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where is a field of elements and is nontrivial, the absolute value is . The determination of the exact value of general Gauss sums, following the result of Gauss on the quadratic case, is a long-standing issue. For some cases see Kummer sum.
Properties of Gauss sums of Dirichlet characters
The Gauss sum of a Dirichlet character modulo is
If is also primitive, then
in particular, it is nonzero. More generally, if is the conductor of and is the primitive Dirichlet character modulo that induces , then the Gauss sum of is related to that of by
where is the Möbius function. Consequently, is non-zero precisely when is squarefree and relatively prime to .
Other relations between and Gauss sums of other characters include
where is the complex conjugate Dirichlet character, and if is a Dirichlet character modulo such that and are relatively prime, then
The relation among , , and when and are of the same modulus (and is primitive) is measured by the Jacobi sum . Specifically,
Further properties
Gauss sums can be used to prove quadratic reciprocity, cubic reciprocity, and quartic reciprocity.
Gauss sums can be used to calculate the number of solutions of polynomial equations over finite fields, and thus can be used to calculate certain zeta functions.
See also
Quadratic Gauss sum
Elliptic Gauss sum
Jacobi sum
Kummer sum
Kloosterman sum
Gaussian period
Hasse–Davenport relation
Chowl
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https://en.wikipedia.org/wiki/Heap%20%28mathematics%29
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In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted that satisfies a modified associativity property:
A biunitary element h of a semiheap satisfies [h,h,k] = k = [k,h,h] for every k in H.
A heap is a semiheap in which every element is biunitary.
The term heap is derived from груда, Russian for "heap", "pile", or "stack". Anton Sushkevich used the term in his Theory of Generalized Groups (1937) which influenced Viktor Wagner, promulgator of semiheaps, heaps, and generalized heaps. Груда contrasts with группа (group) which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.)
Examples
Two element heap
Turn into the cyclic group , by defining the identity element, and . Then it produces the following heap:
Defining as the identity element and would have given the same heap.
Heap of integers
If are integers, we can set to produce a heap. We can then choose any integer to be the identity of a new group on the set of integers, with the operation
and inverse
.
Heap of a groupoid with two objects
One may generalize the notion of the heap of a group to the case of a groupoid which has two objects A and B when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms x, y, z define a heap operation according to:
This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.
Heterogeneous relations
Let A and B be different sets and the collection of heterogeneous relations between them. For define the ternary operator
where qT is the converse relation of q. The result of this composition is also in so a mathematical structure has been formed by the ternary operation. Viktor Wagner was motivated to form this heap by his study of transition maps in an atlas which are partial functions. Thus a heap is more than a tweak of a group: it is a general concept including a group as a trivial case.
Theorems
Theorem: A semiheap with a biunitary element e may be considered an involuted semigroup with operation given by ab = [a, e, b] and involution by a–1 = [e, a, e].
Theorem: Every semiheap may be embedded in an involuted semigroup.
As in the study of semigroups, the structure of semiheaps is described in terms of ideals with an "i-simple semiheap" being one with no proper ideals. Mustafaeva translated the Green's relations of semigroup theory to semiheaps and defined a ρ class to be those elements generating the same principle two-sided ideal. He then proved that no i-simple semiheap can have more than two ρ classes.
He also described regularity classes of a semiheap S:
where n and m have the same parity and the ternary operation of the semiheap ap
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https://en.wikipedia.org/wiki/Nilpotent%20ideal
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In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that Ik = 0. By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0.
The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem. The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
Relation to nil ideals
The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than one reason. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.
In a right Artinian ring, any nil ideal is nilpotent. This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis), the result follows. In fact, this can be generalized to right Noetherian rings; this result is known as Levitzky's theorem.
See also
Köthe conjecture
Nilpotent element
Nilradical
Jacobson radical
Notes
References
Ideals (ring theory)
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https://en.wikipedia.org/wiki/Cantellated%205-cell
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In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the regular 5-cell.
Cantellated 5-cell
The cantellated 5-cell or small rhombated pentachoron is a uniform 4-polytope. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 cuboctahedra, 5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; the vertex figure is a nonuniform triangular prism.
Alternate names
Cantellated pentachoron
Cantellated 4-simplex
(small) prismatodispentachoron
Rectified dispentachoron
Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)
Configuration
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Images
Coordinates
The Cartesian coordinates of the vertices of the origin-centered cantellated 5-cell having edge length 2 are:
The vertices of the cantellated 5-cell can be most simply positioned in 5-space as permutations of:
(0,0,1,1,2)
This construction is from the positive orthant facet of the cantellated 5-orthoplex.
Related polytopes
The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70 octahedra (10 rectified tetrahedra, 20 triangular antiprisms, 40 triangular antipodiums), 30 tetrahedra (as tetragonal disphenoids), and 60 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.
Vertex figure
Cantitruncated 5-cell
The cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5 truncated octahedra, 10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.
Configuration
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Alternative names
Cantitruncated pentachoron
Cantitruncated 4-simplex
Great prismatodispentachoron
Truncated dispentachoron
Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)
Images
Cartesian coordinates
The Cartesian coordinates of an origin-centered cantitruncated 5-cell having edge length 2 are:
These vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:
(0,0,1,2,3)
This construction is from the positive orthant facet of the cantitruncated 5-orthoplex.
Related polytopes
A double symmetry construction can be made by placing truncated tetrahedra on the
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https://en.wikipedia.org/wiki/Cantellated%2024-cells
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In four-dimensional geometry, a cantellated 24-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 24-cell.
There are 2 unique degrees of cantellations of the 24-cell including permutations with truncations.
Cantellated 24-cell
The cantellated 24-cell or small rhombated icositetrachoron is a uniform 4-polytope.
The boundary of the cantellated 24-cell is composed of 24 truncated octahedral cells, 24 cuboctahedral cells and 96 triangular prisms. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.
Construction
When the cantellation process is applied to 24-cell,
each of the 24 octahedra becomes a small rhombicuboctahedron.
In addition however, since each octahedra's edge was previously shared with two
other octahedra, the separating edges form the three parallel edges of a
triangular prism - 96 triangular prisms, since the 24-cell contains 96 edges.
Further, since each vertex was previously shared with 12 faces,
the vertex would split into 12 (24*12=288) new vertices.
Each group of 12 new vertices forms a cuboctahedron.
Coordinates
The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:
(0, , , 2+2)
(1, 1+, 1+, 1+2)
The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract.
The dual configuration has all permutations and signs of:
(0,2,2+,2+)
(1,1,1+,3+)
Structure
The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra via their axial square faces, and to the triangular prisms via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.
Cantic snub 24-cell
A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.
The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:
Images
Related polytopes
The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.
Cantitruncated 24-cell
The cantitruncated 24-cell or great rhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24-cell, 24 truncated cubes corresponding with the cells of
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https://en.wikipedia.org/wiki/Runcinated%2024-cells
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In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell.
There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations.
Runcinated 24-cell
In geometry, the runcinated 24-cell or small prismatotetracontoctachoron is a uniform 4-polytope bounded by 48 octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual.
E. L. Elte identified it in 1912 as a semiregular polytope.
Alternate names
Runcinated 24-cell (Norman W. Johnson)
Runcinated icositetrachoron
Runcinated polyoctahedron
Small prismatotetracontoctachoron (spic) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the runcinated 24-cell having edge length 2 is given by all permutations of sign and coordinates of:
(0, 0, , 2+)
(1, 1, 1+, 1+)
The permutations of the second set of coordinates coincide with the vertices of an inscribed cantellated tesseract.
Projections
Related regular skew polyhedron
The regular skew polyhedron, {4,8|3}, exists in 4-space with 8 square around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed. The dual regular skew polyhedron, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell.
Runcitruncated 24-cell
The runcitruncated 24-cell or prismatorhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated octahedra, corresponding with the cells of a 24-cell, 24 rhombicuboctahedra, corresponding with the cells of the dual 24-cell, 96 triangular prisms, and 96 hexagonal prisms.
Coordinates
The Cartesian coordinates of an origin-centered runcitruncated 24-cell having edge length 2 are given by all permutations of coordinates and sign of:
(0, , 2, 2+3)
(1, 1+, 1+2, 1+3)
The permutations of the second set of coordinates give the vertices of an inscribed omnitruncated tesseract.
The dual configuration has coordinates generated from all permutations and signs of:
(1,1,1+,5+)
(1,3,3+,3+)
(2,2,2+,4+)
Projections
Runcicantic snub 24-cell
A half-symmetry construction of the runcitruncated 24-cell (or runcicantellated 24-cell), as , also called a runcicantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. Like the snub 24-cell, it has symmetry [3+,4,3], order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons.
The difference can be seen in the vertex figures:
Runcic snub 24-cell
A related 4-polytope is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s3{3,4,3}, . It is not uniform, but it is vertex-transitive and has all regular polygon faces. It is
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https://en.wikipedia.org/wiki/Truncated%20120-cells
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In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell.
There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 600-cell.
Truncated 120-cell
The truncated 120-cell or truncated hecatonicosachoron is a uniform 4-polytope, constructed by a uniform truncation of the regular 120-cell 4-polytope.
It is made of 120 truncated dodecahedral and 600 tetrahedral cells. It has 3120 faces: 2400 being triangles and 720 being decagons. There are 4800 edges of two types: 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron. Each vertex has 3 truncated dodecahedra and one tetrahedron around it. Its vertex figure is an equilateral triangular pyramid.
Alternate names
Truncated 120-cell (Norman W. Johnson)
Tuncated hecatonicosachoron / Truncated dodecacontachoron / Truncated polydodecahedron
Truncated-icosahedral hexacosihecatonicosachoron (Acronym thi) (George Olshevsky, and Jonathan Bowers)
Images
Bitruncated 120-cell
The bitruncated 120-cell or hexacosihecatonicosachoron is a uniform 4-polytope. It has 720 cells: 120 truncated icosahedra, and 600 truncated tetrahedra. Its vertex figure is a digonal disphenoid, with two truncated icosahedra and two truncated tetrahedra around it.
Alternate names
Bitruncated 120-cell / Bitruncated 600-cell (Norman W. Johnson)
Bitruncated hecatonicosachoron / Bitruncated hexacosichoron / Bitruncated polydodecahedron / Bitruncated polytetrahedron
Truncated-icosahedral hexacosihecatonicosachoron (Acronym Xhi) (George Olshevsky, and Jonathan Bowers)
Images
Truncated 600-cell
The truncated 600-cell or truncated hexacosichoron is a uniform 4-polytope. It is derived from the 600-cell by truncation. It has 720 cells: 120 icosahedra and 600 truncated tetrahedra. Its vertex figure is a pentagonal pyramid, with one icosahedron on the base, and 5 truncated tetrahedra around the sides.
Alternate names
Truncated 600-cell (Norman W. Johnson)
Truncated hexacosichoron (Acronym tex) (George Olshevsky, and Jonathan Bowers)
Truncated tetraplex (Conway)
Structure
The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra. The truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra.
Images
Related polytopes
Notes
References
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
J.H. Conway a
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https://en.wikipedia.org/wiki/Cantellated%20120-cell
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In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell.
There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.
Cantellated 120-cell
The cantellated 120-cell is a uniform 4-polytope. It is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra, 1200 triangular prisms, 600 octahedra. Its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.
Alternative names
Cantellated 120-cell Norman Johnson
Cantellated hecatonicosachoron / Cantellated dodecacontachoron / Cantellated polydodecahedron
Small rhombated hecatonicosachoron (Acronym srahi) (George Olshevsky and Jonathan Bowers)
Ambo-02 polydodecahedron (John Conway)
Images
Cantitruncated 120-cell
The cantitruncated 120-cell is a uniform polychoron.
This 4-polytope is related to the regular 120-cell. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.
The image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagonal faces are hidden, allowing us to see the elemented projected inside.
Alternative names
Cantitruncated 120-cell Norman Johnson
Cantitruncated hecatonicosachoron / Cantitruncated polydodecahedron
Great rhombated hecatonicosachoron (Acronym grahi) (George Olshevsky and Jonthan Bowers)
Ambo-012 polydodecahedron (John Conway)
Images
Cantellated 600-cell
The cantellated 600-cell is a uniform 4-polytope. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms. Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.
Alternative names
Cantellated 600-cell Norman Johnson
Cantellated hexacosichoron / Cantellated tetraplex
Small rhombihexacosichoron (Acronym srix) (George Olshevsky and Jonathan Bowers)
Ambo-02 tetraplex (John Conway)
Construction
This 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time:
There are 1440 pentagonal faces between the icosidodecahedra and pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.
There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.
Images
Cantitruncated 600-cell
The cantitruncate
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https://en.wikipedia.org/wiki/Runcinated%20120-cells
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In four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.
There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.
The runcinated 120-cell can be seen as an expansion applied to a regular 4-polytope, the 120-cell or 600-cell.
Runcinated 120-cell
The runcinated 120-cell or small disprismatohexacosihecatonicosachoron is a uniform 4-polytope. It has 2640 cells: 120 dodecahedra, 720 pentagonal prisms, 1200 triangular prisms, and 600 tetrahedra. Its vertex figure is a nonuniform triangular antiprism (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.
Alternate names
Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson)
Runcinated hecatonicosachoron / Runcinated dodecacontachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron
Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George Olshevsky, Jonathan Bowers)
Images
Runcitruncated 120-cell
The runcitruncated 120-cell or prismatorhombated hexacosichoron is a uniform 4-polytope. It contains 2640 cells: 120 truncated dodecahedra, 720 decagonal prisms, 1200 triangular prisms, and 600 cuboctahedra. Its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.
Alternate names
Runcicantellated 600-cell (Norman W. Johnson)
Prismatorhombated hexacosichoron (Acronym: prix) (George Olshevsky, Jonathan Bowers)
Images
Runcitruncated 600-cell
The runcitruncated 600-cell or prismatorhombated hecatonicosachoron is a uniform 4-polytope. It is composed of 2640 cells: 120 rhombicosidodecahedron, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).
Alternate names
Runcicantellated 120-cell (Norman W. Johnson)
Prismatorhombated hecatonicosachoron (Acronym: prahi) (George Olshevsky, Jonathan Bowers)
Images
Omnitruncated 120-cell
The omnitruncated 120-cell or great disprismatohexacosihecatonicosachoron is a convex uniform 4-polytope, composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform 4-polytope.
The vertices and edges form the Cayley graph of the Coxeter group H4.
Alternate names
Omnitruncated 120-cell / Omnitruncated 600-cell (Norman W. Johnson)
Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron
Great diprismatohexacosihecatonicosac
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https://en.wikipedia.org/wiki/Maryam%20Mirzakhani
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Maryam Mirzakhani (, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics, becoming the first woman to win the prize, as well as the first Iranian. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".
On 14 July 2017, Mirzakhani died of breast cancer at the age of 40.
Early life and education
Mirzakhani was born on 12 May 1977 in Tehran, Iran. As a child, she attended Tehran Farzanegan School, part of the National Organization for Development of Exceptional Talents (NODET). In her junior and senior years of high school, she won the gold medal for mathematics in the Iranian National Olympiad, thus allowing her to bypass the national college entrance exam. In 1994, Mirzakhani became the first Iranian woman to win a gold medal at the International Mathematical Olympiad in Hong Kong, scoring 41 out of 42 points. The following year, in Toronto, she became the first Iranian to achieve the full score and to win two gold medals in the International Mathematical Olympiad. Later in her life, she collaborated with friend, colleague, and Olympiad silver medalist, Roya Beheshti Zavareh (), on their book 'Elementary Number Theory, Challenging Problems', (in Persian) which was published in 1999. Mirzakhani and Zavareh together were the first women to compete in the Iranian National Mathematical Olympiad and won gold and silver medals in 1995, respectively.
On 17 March 1998, after attending a conference consisting of gifted individuals and former Olympiad competitors, Mirzakhani and Zavareh, along with other attendees, boarded a bus in Ahvaz en route to Tehran. The bus was involved in an accident wherein it fell off a cliff, killing seven of the passengers—all Sharif University students. This incident is widely considered a national tragedy in Iran. Mirzakhani and Zavareh were two of the few survivors.
In 1999, she obtained a Bachelor of Science in mathematics from the Sharif University of Technology. During her time there, she received recognition from the American Mathematical Society for her work in developing a simple proof of the theorem of Schur. She then went to the United States for graduate work, earning a PhD in 2004 from Harvard University, where she worked under the supervision of the Fields Medalist, Curtis T. McMullen. At Harvard, she is said to have been "distinguished by determination and relentless questioning". She used to take her class notes in her native language Persian.
Career
Mirzakhani was a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University. In 2009, she became a professor at Stanford University.
Research work
Mirzakhani made several contributions to th
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https://en.wikipedia.org/wiki/Logarithmic%20convolution
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In mathematics, the scale convolution of two functions and , also known as their logarithmic convolution is defined as the function
when this quantity exists.
Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from to :
Define and and let , then
Logarithms
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https://en.wikipedia.org/wiki/Donald%20Shell
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Donald L. Shell (March 1, 1924 – November 2, 2015) was an American computer scientist who designed the Shellsort sorting algorithm. He acquired his Ph.D. in mathematics from the University of Cincinnati in 1959, and published the Shellsort algorithm in the Communications of the ACM in July that same year.
Career
Donald Shell acquired a B.S. in Civil Engineering from the Michigan College of Mining and Technology which is now Michigan Technological University. This was a four-year degree which he acquired in three years with the highest GPA given in the college's history. A record which persisted for more than 30 years. After acquiring his degree he went into the Army Corps of Engineers, and from there to the Philippines to help repair damages during World War II. When he returned after the war, he married Alice McCullough and returned to Michigan Technological University, where he taught mathematics. In 1949 they moved to Cincinnati, Ohio, for Don to work for General Electric's engines division, where he developed a convergence algorithm and wrote a program to perform performance cycle calculations for GE's first aircraft jet engines. He also attended the University of Cincinnati, where in 1951 he acquired a M.S. in mathematics and, in 1959, acquired his Ph.D. in Mathematics. In July of that year he published the Shellsort algorithm and "The Share 709 System: A Cooperative Effort". In 1958, he and A. Spitzbart had published "A Chebycheff Fitting Criterion".
Although he is most widely known for his Shellsort algorithm, his Ph.D. is also considered by some to be the first major investigation of the convergence of infinite exponentials, with some very deep results of the convergence into the complex plane. This area has grown considerably and research related to it is now investigated in what is more commonly called tetration. In October 1962 he wrote "On the Convergence of Infinite Exponentials" in the Proceedings of the American Mathematical Society.
After acquiring his Ph.D., Shell moved to Schenectady, New York, to become Manager of Engineering for General Electric's new Information Services Department, the first commercial enterprise to link computers together using the client–server architecture. This architecture is the fundamental design for the Internet. He worked with John George Kemeny and Thomas Eugene Kurtz to commercialize the Dartmouth Time-Sharing System in 1963.
In 1971 Shell wrote "Optimizing the Polyphase Sort" in the Communications of the ACM, and in 1972 he joined with a colleague, Ralph Mosher (who designed the walking truck), to start a business, Robotics Inc., where he was the General Manager and chief software engineer. Four years later, in 1976, they sold the company and Shell returned to General Electric Information Services Corporation.
In 1984 he retired and moved to North Carolina.
Marriages and family
Donald Shell married Alice McCullough after returning from World War II. They had two sons. Alice became ill
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https://en.wikipedia.org/wiki/Ovoid%20%28polar%20space%29
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In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank intersects O in exactly one point.
Cases
Symplectic polar space
An ovoid of (a symplectic polar space of rank n) would contain points.
However it only has an ovoid if and only and q is even. In that case, when the polar space is embedded into the classical way, it is also an ovoid in the projective geometry sense.
Hermitian polar space
Ovoids of and would contain points.
Hyperbolic quadrics
An ovoid of a hyperbolic quadricwould contain points.
Parabolic quadrics
An ovoid of a parabolic quadric would contain points. For , it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid.
If q is even, is isomorphic (as polar space) with , and thus due to the above, it has no ovoid for .
Elliptic quadrics
An ovoid of an elliptic quadric would contain points.
See also
Ovoid (projective geometry)
References
Incidence geometry
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https://en.wikipedia.org/wiki/China%2C%20Nuevo%20Le%C3%B3n
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China is a municipality in the Mexican state of Nuevo León. China is approximately northeast of Monterrey.
According to a 2010 census done by the National Institute of Statistics and Geography (INEGI), China had 10,867 inhabitants. The town is home to the Presa El Cuchillo reservoir and has different theme parks.
Localities
San Bernardo
References
Municipalities of Nuevo León
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https://en.wikipedia.org/wiki/Gisbert%20W%C3%BCstholz
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Gisbert Wüstholz (born June 4, 1948, in Tuttlingen, Germany) is a German mathematician internationally known for his fundamental contributions to number theory (in the field of transcendental number theory, Diophantine approximation) and arithmetic geometry.
Early life and education
Gisbert Wüstholz was born in 1948 in Tuttlingen and studied from 1967 to 1973 at the University of Freiburg where he finished his PhD under the supervision of Theodor Schneider in 1978.
Career
On the invitation of Friedrich Hirzebruch Wüstholz stayed for a year as a Postdoc at the University of Bonn and then he got a Postdoc position at the University of Wuppertal where he worked with Walter Borho from 1979 till 1984 and then moved to Bonn to become associate professor at the newly founded Max Planck Institute for Mathematics. From 1985 to 1987 he was full Professor for Mathematics at Wuppertal and in 1987 elected for a chair in Mathematics at ETH Zurich. He founded the Zurich Graduate School in Mathematics in 2003 and served as the director since then until 2008. Since 2013 he is a professor emeritus at ETH Zurich.
He is Member of the German National Academy of Sciences Leopoldina (since 2000), of the Berlin-Brandenburg Academy of Sciences and Humanities (since 2003), of the Academia Europaea (since 2008) where he was chairman of the Mathematics Section from 2011 to 2013, and of the European Academy of Sciences and Arts (since 2016). From 1999 he was an Honorary Advisory Professor at the Tongji University, Shanghai. From 2011 he was Senator for Mathematics at the Leopoldina. He is an Honorary Professor at Graz University of Technology, Austria (since 2017).
Gisbert Wüstholz stayed for extended periods at a number of universities and research institutes such as the University of Michigan at Ann Arbor (1984,1988) and the Institut des Hautes Études Scientifiques in Bures-sur-Yvette (1987). He was member of the Institute for Advanced Study in Princeton (1986, 1990, 1994/95, 2011), in 1992 Visiting Fellow Commoner at Trinity College in Cambridge for research projects with Alan Baker and visited in the following year the Mathematical Sciences Research Institute in Berkeley (1993). He was frequently guest at the Max Planck Institute for Mathematics at Bonn and the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) at Vienna. Since 2015 he is staying as a guest at the University of Zurich. In the academic year 2017/18 he was Senior Research Fellow at the Freiburg Institute for Advanced Studies (FRIAS).
Since 1980 Gisbert Wüstholz has close connections to a number of universities in Asia: he stayed for a couple of months each at Kyushu University at Fukuoka (1992), the Morningside Center of Mathematics of the Chinese Academy of Sciences at Beijing, at the Hong Kong University of Science and Technology (HKUST) (1996, 1997, 2006, 2010) and at the University of Hong Kong (HKU) (1999, 2011, 2012). Several visits took him to the Vietnam Institute for
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https://en.wikipedia.org/wiki/Zamora%20Municipality%2C%20Miranda
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Zamora is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 187,075. The town of Guatire is the municipal seat of the Zamora Municipality.
Name
The municipality is one of several named "Zamora Municipality" for the 19th century Venezuelan soldier Ezequiel Zamora.
History
The region was historically an agricultural area that cultivated various export products. The population of Guatire has increased rapidly in recent years as an east suburb of Caracas.
Geography
The municipality's northern border, separating it from Vargas, is the east part of the Costa (or Caribe) Mountain Range, while the lower Caraballo Ridge comprises its southern border. Grande River (also called Guarenas River or Caucagua River) runs from the west to the southeast. Guatire, the municipality's shire town, is located near the western border.
Demographics
The Zamora Municipality, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, has a population of 187,075 (up from 152,422 in 2001). This amounts to 7% of the state's population. The municipality's population density in 2011 was .
Government
The mayor of the Zamora Municipality is Raziel Rodriguez, elected in 2021 for 2021-2025 tenure with 46,97% of the vote. The municipality is divided into two parishes; Guatire and Bolívar.
References
External links
official website
Municipalities of Miranda (state)
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https://en.wikipedia.org/wiki/Betti
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Betti may refer to:
People
Betti (given name)
Betti (surname)
Other uses
Betti number in topology, named for Enrico Betti
Betti's theorem in engineering theory, named for Enrico Betti
Betti reaction, a chemical addition reaction
See also
Beti (disambiguation)
Betty (disambiguation)
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https://en.wikipedia.org/wiki/HNN
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HNN may refer to:
HLN (TV channel), an American television news channel
HNN extension, in combinatorial group theory
Hanunó'o language, spoken in the Philippines
Hawaii News Now, a television program
History News Network, a project of George Washington University
HNN, Henderson VORTAC, located near Henderson, West Virginia.
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https://en.wikipedia.org/wiki/Floor%20effect
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In statistics, a floor effect (also known as a basement effect) arises when a data-gathering instrument has a lower limit to the data values it can reliably specify. This lower limit is known as the "floor". The "floor effect" is one type of scale attenuation effect; the other scale attenuation effect is the "ceiling effect". Floor effects are occasionally encountered in psychological testing, when a test designed to estimate some psychological trait has a minimum standard score that may not distinguish some test-takers who differ in their responses on the test item content. Giving preschool children an IQ test designed for adults would likely show many of the test-takers with scores near the lowest standard score for adult test-takers (IQ 40 on most tests that were currently normed as of 2010). To indicate differences in current intellectual functioning among young children, IQ tests specifically for young children are developed, on which many test-takers can score well above the floor score. An IQ test designed to help assess intellectually disabled persons might intentionally be designed with easier item content and a lower floor score to better distinguish among individuals taking the test as part of an assessment process.
See also
Ceiling effect (statistics)
References
Further reading
Everitt, B.S. (2002) The Cambridge dictionary of Statistics, Second Edition. CUP.
Psychometrics
Psychological testing
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https://en.wikipedia.org/wiki/Hexadecagon
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In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon.
Regular hexadecagon
A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.
Construction
As 16 = 24 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians.
Measurements
Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees.
The area of a regular hexadecagon with edge length t is
Because the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius R by truncating Viète's formula:
Since the area of the circumcircle is the regular hexadecagon fills approximately 97.45% of its circumcircle.
Symmetry
The regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups: Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups: Z16, Z8, Z4, Z2, and Z1, the last implying no symmetry.
On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
The most common high symmetry hexadecagons are d16, an isogonal hexadecagon constructed by eight mirrors can alternate long and short edges, and p16, an isotoxal hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexadecagon.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g16 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexadecagon, m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petrie polygon projection of an 8-cube, with 28 of 1792 faces. The list enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.
Skew hexadecagon
A skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A skew zig-zag hexadecagon has vertices alternating between two parallel planes.
A regular sk
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https://en.wikipedia.org/wiki/%C5%BDiar%2C%20Rev%C3%BAca%20District
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Žiar () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Magnezitovce
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Magnezitovce () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20071116010355/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Licince
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Licince () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Počet obyvateľov: 711
Rozloha: 1830 ha
Prvá písomná zmienka: v roku 1263
Starosta: Ladislav Miklóš (ehm..chh)
Villages and municipalities in Revúca District
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