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https://en.wikipedia.org/wiki/David%20Mumford
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David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.
Early life
Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started an experimental school in Tanzania and worked for the then newly created United Nations.
He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. Mumford then went to Harvard University, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956. He completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He married Erika, an author and poet, in 1959 and they had four children, Stephen, Peter, Jeremy, and Suchitra. He currently has seven grandchildren.
Work in algebraic geometry
Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, and on algebraic surfaces.
His books Abelian Varieties (with C. P. Ramanujam) and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction. They are now available as The Red Book of Varieties and Schemes ().
Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and a study of Goro Shimura's papers from the 1960s.
Mumford's research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, and enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group. This work on the equations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory.
He also was one of the founders of the toroidal embedding theory; and sought to apply the theory to Gröbner basis techniques, through students who worked in algebraic computation.
Work on pathologies in algebraic geometry
In a sequence of four papers published in the American Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior in algebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples. These pathologies fall into two types: (a) bad behavior in characteristic p and (b) bad behavior in moduli spaces.
C
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https://en.wikipedia.org/wiki/36%20%28number%29
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36 (thirty-six) is the natural number following 35 and preceding 37.
In mathematics
36 is both the square of six, and the eighth triangular number or sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them; in fact, it is the smallest number with exactly nine divisors, which leads 36 to be the 7th highly composite number. It is the sum of the fourth pair of twin-primes (17 + 19), and the 18th Harshad number in decimal, as it is divisible by the sum of its digits (9).
It is the smallest number with exactly eight solutions (37, 57, 63, 74, 76, 108, 114, 126) to the Euler totient function . Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is also the eighth semiperfect number.
This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers.
36 is the number of degrees in the interior angle of each tip of a regular pentagram.
The thirty-six officers problem is a mathematical puzzle with no solution.
The number of possible outcomes (not summed) in the roll of two distinct dice.
36 is the largest numeric base that some computer systems support because it exhausts the numerals, 0–9, and the letters, A-Z. See Base 36.
The truncated cube and the truncated octahedron are Archimedean solids with 36 edges.
The number of domino tilings of a 4×4 checkerboard is 36.
Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number.
The sum of the integers from 1 to 36 is 666 (see number of the beast).
Measurements
The number of inches in a yard (3 feet).
In the UK, a standard beer barrel is 36 UK gallons, about 163.7 litres.
3 dozen, or a quarter of a gross.
In science
The atomic number of krypton
Many early computers featured a 36-bit word length
36 is the number of characters required to store the display name of a UUID or GUID (e.g., 00000000-0000-0000-C000-000000000046).
In religion
Jewish tradition holds that the number 36 has had special significance since the beginning of time: According to the Midrash, the light created by God on the first day of creation shone for exactly 36 hours; it was replaced by the light of the Sun that was created on the Fourth Day. The Torah commands 36 times to love, respect and protect the stranger. Furthermore, in every generation there are 36 righteous people (the "Lamed Vav Tzadikim") in whose merit the world continues to exist. In the modern celebration of Hanukkah, 36 candles are kindled in the menorah over the eight days of that holiday (not includ
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https://en.wikipedia.org/wiki/37%20%28number%29
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37 (thirty-seven) is the natural number following 36 and preceding 38.
In mathematics
37 is the 12th prime number, and the 3rd isolated prime without a twin prime.
37 is the first irregular prime.
The sum of the squares of the first 37 primes is divisible by 37.
Every positive integer is the sum of at most 37 fifth powers (see Waring's problem).
It is the third cuban prime following 7 and 19.
37 is the fifth Padovan prime, after the first four prime numbers 2, 3, 5, and 7.
It is also the fifth lucky prime, after 3, 7, 13, and 31.
37 is the third star number and the fourth centered hexagonal number.
There are exactly 37 complex reflection groups.
The smallest magic square, using only primes and 1, contains 37 as the value of its central cell:
Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11).
In decimal 37 is a permutable prime with 73, which is the 21st prime number. By extension, the mirroring of their digits and prime indexes makes 73 the only Sheldon prime. In moonshine theory, whereas all p ⩾ 73 are non-supersingular primes, the smallest such prime is 37.
37 requires twenty-one steps to return to 1 in the Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary {16, 8, 4, 2, 1} Collatz pathway are 5 and 32, whose sum is 37. The trajectories for 3 and 21 both require seven steps to reach 1.
The first two integers that return for the Mertens function (2 and 39) have a difference of 37. Their product (2 × 39) is the twelfth triangular number 78. Their sum is 41, which is the constant term in Euler's lucky numbers that yield prime numbers of the form k2 − k + 41; the largest of which (1601) is a difference of 78 from the second-largest prime (1523) generated by this quadratic polynomial.
In decimal
For a three-digit number that is divisible by 37, a rule of divisibility is that another divisible by 37 can be generated by transferring first digit onto the end of a number. For example: 37|148 ➜ 37|481 ➜ 37|814.
Any multiple of 37 can be mirrored and spaced with a zero each for another multiple of 37. For example, 37 and 703, 74 and 407, and 518 and 80105 are all multiples of 37.
Any multiple of 37 with a three-digit repunit inserted generates another multiple of 37. For example, 30007, 31117, 74, 70004 and 78884 are all multiples of 37.
In science
The atomic number of rubidium
The normal human body temperature in degrees Celsius
Astronomy
NGC 2169 is known as the 37 Cluster, due to its resemblance of the numerals.
In other fields
Thirty-seven is:
The number of the French department Indre-et-Loire
The number of slots in European roulette (numbered 0 to 36, the 00 is not used in European roulette as it is in American roulette)
The RSA public exponent used by PuTTY
Richard Nixon, 37th president of the United States.
See also
List of highways numbered 37
Number Thirty-Seven, Pennsylvania, uni
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https://en.wikipedia.org/wiki/39%20%28number%29
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39 (thirty-nine) is the natural number following 38 and preceding 40.
In mathematics
39 is the 12th distinct semiprime and the 4th in the (3.q) family. It is the last member of the third distinct semiprime pair (38,39).
39 has an aliquot sum of 17, which is a prime. 39 is the 4th member of the 17-aliquot tree within an aliquot sequence of one composite numbers (39,17,1,0) to the Prime in the 17-aliquot tree.
It is a perfect totient number.*39 is the sum of five consecutive primes (3 + 5 + 7 + 11 + 13) and also is the product of the first and the last of those consecutive primes. Among small semiprimes only three other integers (10, 155, and 371) share this attribute. 39 also is the sum of the first three powers of 3 (3 + 3 + 3). Given 39, the Mertens function returns 0.
39 is the smallest natural number which has three partitions into three parts which all give the same product when multiplied: {25, 8, 6}, {24, 10, 5}, {20, 15, 4}.
39 is a Perrin number, coming after 17, 22, 29 (it is the sum of the first two mentioned).
Since the greatest prime factor of 392 + 1 = 1522 is 761, which is more than 39 twice, 39 is a Størmer number.
The F26A graph is a symmetric graph with 39 edges.
In science
The atomic number of yttrium
Astronomy
Messier object Open Cluster M39, a magnitude 5.5 open cluster in the constellation Cygnus
The New General Catalogue object NGC 39, a spiral galaxy in the constellation Andromeda
In religion
The number of the 39 categories of activity prohibited on Shabbat according to Halakha
The number of mentions of work or labor in the Torah
The actual number of lashes given by the Sanhedrin to a person meted the punishment of 40 lashes
The number of books in the Old Testament according to Protestant canon
The number of statements on Anglican Church doctrine, Thirty-Nine Articles
In other fields
Arts and entertainment
In the title of the John Buchan novel and subsequent films (one by Alfred Hitchcock), The Thirty-Nine Steps
The age American comedian Jack Benny claimed to be for more than 40 years
"39" is a song by the Cure on their album Bloodflowers
"39" is a song by Tenacious D on their album Rize of the Fenix
"'39" is a song by Queen on their album A Night at the Opera
The book series The 39 Clues revolves around 39 clues hidden around the world
Glorious 39 is a 2009 drama film set at the beginning of World War II
In the CBS reality show Survivor, contestants compete for 39 days
The number of episodes done during its one season in 1955–1956 of The Honeymooners television series is commonly referred to as the "Classic 39"
Japanese wordplay and slang:
Internet chat slang for "Thank You" when written with numbers (3=San, 9=Kyuu)
This number is also considered to be the lucky number of Hatsune Miku due to the digits 3 and 9 being pronounceable as "mi" and "ku", respectively.
History
The number of signers to the United States Constitution, out of 55 members of the Philadelphia Convention delegates
The traditional number of ti
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https://en.wikipedia.org/wiki/47%20%28number%29
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47 (forty-seven) is the natural number following 46 and preceding 48. It is a prime number.
In mathematics
Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, the fourth isolated prime, and the sixth Lucas prime. Forty-seven is a highly cototient number. It is an Eisenstein prime with no imaginary part and real part of the form .
It is a Lucas number. It is also a Keith number because its digits appear as successive terms earlier in the series of Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, ...
It is the number of trees on 9 unlabeled nodes.
Forty-seven is a strictly non-palindromic number.
Its representation in binary being 101111, 47 is a prime Thabit number, and as such is related to the pair of amicable numbers {17296, 18416}.
In science
47 is the atomic number of silver.
Astronomy
The 47-year cycle of Mars: after 47 years – 22 synodic periods of 780 days each – Mars returns to the same position among the stars and is in the same relationship to the Earth and Sun. The ancient Mesopotamians discovered this cycle.
Messier object M47, a magnitude 4.5 open cluster in the constellation Puppis
47 Tucanae, the second brightest globular cluster in the sky, located in the constellation Tucana.
The New General Catalogue object NGC 47, a barred spiral galaxy in the constellation Cetus. This object is also designated as NGC 58.
In popular culture
Pomona College
Calendar years
47 BC
AD 47
See also
List of highways numbered 47
Other
Telephone dialing country code for Norway
The AK-47, also known as a Kalashnikov rifle, is one of the most widely used military weapons in the world.
The CH-47 Chinook, a helicopter.
47 is the number of the French department Lot-et-Garonne.
The P-47 Thunderbolt was a fighter plane in World War II.
There are Forty-seven Ronin in the famous Japanese story.
There are 47 Prefectures of Japan.
The player protagonist of the Hitman video game franchise is called Agent 47.
References
Integers
In-jokes
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https://en.wikipedia.org/wiki/Gymnasium%20Jur%20Hronec
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Gymnázium Jura Hronca (GJH) is a gymnasium (grammar school) located in Bratislava, Slovakia.
The school has a focus on the study of natural sciences, mathematics, and computer sciences. However its affiliation with the International Baccalaureate, an active bi-lingual (English – Slovak) programme and the option to study several foreign languages such as French and German, the school has a strong reputation for the study of foreign languages.
In the school year 2005/2006, GJH launched lower programs of IB Primary Years Programme and IB Middle Years Programme.
The school has been recently known as the "Spojená škola Gymnázium Jura Hronca a ZŠ Košická" (United school of the Gymnázium Jura Hronca and the Košická Primary School) after a merge with the primary school Základná škola a osemročné gymnázium Košická sharing the same building.
History
The school was founded on January 9, 1959 as an 11-year secondary school. In the school year 1969/70 the school is granted the status of a Gymnasium, named after the Slovak mathematician Jur Hronec.
International Baccalaureate
Spojená škola Novohradská has been an IB World School since June 1994. It offers the IB Primary Years Programme (since 2009), IB Middle Years Programme (since 2009) and IB Diploma Programme (since 1994). It's the only school in Slovakia to offer all 3 programmes (as of 2023).
Alumni
2024 - Adam Gergely
Student Activities
Bratislava Model United Nations
Students from Gymnazium Jura Hronca organize the BratMUN conference held in Bratislava.
The Jur Hronec Cup
The Jur Hronec Cup (súťaž o džbán Jura Hronca) is a competition between all classes of the school.
During the year, classes earn points in different activities (Sports day, different competitions) and in the end of the year, the winner receives money and school-free days for a class-trip and the right to hold The Jur Hronec Cup for the next year.
Eschenbach
The Gymnazium Jura Hronca organises every year a school exchange with the Gymnasium Eschenbach in Bavaria, Germany. This school event takes place every year, the 2011 trip being the 20th, with a special trip to Belgium included in the program. This exchange is organized by Dr. Jan Mayer from the Slovak side, and from Dr. Hans Schmid from the German side.
References
External links
Homepage
GJH Evaluation Report 2010/2011
GJH 2010/2011 Leavers Higher Education Statistics
Official BratMUN Homepage
An article about the 2010 BratMUN in the SME daily.
Education in Bratislava
International Baccalaureate schools
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https://en.wikipedia.org/wiki/Subbase
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In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
Definition
Let be a topological space with topology A subbase of is usually defined as a subcollection of satisfying one of the two following equivalent conditions:
The subcollection generates the topology This means that is the smallest topology containing : any topology on containing must also contain
The collection of open sets consisting of all finite intersections of elements of together with the set forms a basis for This means that every proper open set in can be written as a union of finite intersections of elements of Explicitly, given a point in an open set there are finitely many sets of such that the intersection of these sets contains and is contained in
(If we use the nullary intersection convention, then there is no need to include in the second definition.)
For subcollection of the power set there is a unique topology having as a subbase. In particular, the intersection of all topologies on containing satisfies this condition. In general, however, there is no unique subbasis for a given topology.
Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
Alternative definition
Less commonly, a slightly different definition of subbase is given which requires that the subbase cover In this case, is the union of all sets contained in This means that there can be no confusion regarding the use of nullary intersections in the definition.
However, this definition is not always equivalent to the two definitions above. There exist topological spaces with subcollections of the topology such that is the smallest topology containing , yet does not cover . (An example is given at the end of the next section.) In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the T1 separation axiom must be a cover of that space. But as seen below, to prove the Alexander Subbase Theorem, one must assume that covers
Examples
The topology generated by any subset (including by the empty set ) is equal to the trivial topology
If is a topology on and is a basis for then the topology generated by is Thus any basis for a topology is also a subbasis for
If is any subset of then the topology generated by will be a subset of
The usual topology on the real numbers has a subbase consisting of all semi-infinite open in
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https://en.wikipedia.org/wiki/Linear%20algebraic%20group
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In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .
Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.
One of the first uses for the theory was to define the Chevalley groups.
Examples
For a positive integer , the general linear group over a field , consisting of all invertible matrices, is a linear algebraic group over . It contains the subgroups
consisting of matrices of the form, resp.,
and .
The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the Borel subgroup of . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of is conjugated into . Any unipotent subgroup can be conjugated into .
Another algebraic subgroup of is the special linear group of matrices with determinant 1.
The group is called the multiplicative group, usually denoted by . The group of -points is the multiplicative group of nonzero elements of the field . The additive group , whose -points are isomorphic to the additive group of , can also be expressed as a matrix group, for example as the subgroup in :
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every representation of the multiplicative group is a direct sum of irreducible representations. (Its irreducible representations all have dimension 1, of the form for an integer .) By contrast, the only irreducible representation of the additive group is the trivial representation. So every representation of (such as the 2-dimensional representation above) is an iterated extension of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.
Definitions
For an algebraically closed field k, much of the structure of an algebraic variety X over k is encoded in its set X(k) of k-rational
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https://en.wikipedia.org/wiki/Local%20zeta%20function
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In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
where is a non-singular -dimensional projective algebraic variety over the field with elements and is the number of points of defined over the finite field extension of .
Making the variable transformation gives
as the formal power series in the variable .
Equivalently, the local zeta function is sometimes defined as follows:
In other words, the local zeta function with coefficients in the finite field is defined as a function whose logarithmic derivative generates the number of solutions of the equation defining in the degree extension
Formulation
Given a finite field F, there is, up to isomorphism, only one field Fk with
,
for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number
of solutions in Fk and create the generating function
.
The correct definition for Z(t) is to set log Z equal to G, so
and Z(0) = 1, since G(0) = 0, and Z(t) is a priori a formal power series.
The logarithmic derivative
equals the generating function
.
Examples
For example, assume all the Nk are 1; this happens for example if we start with an equation like X = 0, so that geometrically we are taking V to be a point. Then
is the expansion of a logarithm (for |t| < 1). In this case we have
To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including the one point at infinity. Therefore, we have
and
for |t| small enough, and therefore
The first study of these functions was in the 1923 dissertation of Emil Artin. He obtained results for the case of a hyperelliptic curve, and conjectured the further main points of the theory as applied to curves. The theory was then developed by F. K. Schmidt and Helmut Hasse. The earliest known nontrivial cases of local zeta functions were implicit in Carl Friedrich Gauss's Disquisitiones Arithmeticae, article 358. There, certain particular examples of elliptic curves over finite fields having complex multiplication have their points counted by means of cyclotomy.
For the definition and some examples, see also.
Motivations
The relationship between the definitions of G and Z can be explained in a number of ways. (See for example the infinite product formula for Z below.) In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over finite field.
The local Z zeta functions are multiplied to get global zeta functions,
These generally involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers).
In these fields, the variable t is substituted by p−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.)
The global products of Z in the two cases used as exampl
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https://en.wikipedia.org/wiki/Contact%20%28mathematics%29
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In mathematics, two functions have a contact of order if, at a point , they have the same value and equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation.
One speaks also of curves and geometric objects having -th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tangent angle and curvature), etc.
Applications
Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds; see contact geometry. Contact transformations are related changes of coordinates, of importance in classical mechanics. See also Legendre transformation.
Contact between manifolds is often studied in singularity theory, where the type of contact are classified, these include the A series (A0: crossing, A1: tangent, A2: osculating, ...) and the umbilic or D-series where there is a high degree of contact with the sphere.
Contact between curves
Two curves in the plane intersecting at a point p are said to have:
0th-order contact if the curves have a simple crossing (not tangent).
1st-order contact if the two curves are tangent.
2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating.
3rd-order contact if the derivatives of the curvature are equal.
4th-order contact if the second derivatives of the curvature are equal.
Contact between a curve and a circle
For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal of κ(t), the curvature of S at t. Where curvature is zero (at an inflection point on the curve), the osculating circle is a straight line. The locus of the centers of all the osculating circles (also called "centers of curvature") is the evolute of the curve.
If the derivative of curvature κ'(t) is zero, then the osculating circle will have 3rd-order contact and the curve is said to have a vertex. The evolute will have a cusp at the center of the circle. The sign of the second derivative of curvature determines whether the curve has a local minimum or maximum of curvature. All closed curves will have at least four vertices, two minima and two maxima (the four-vertex theorem).
In general a curve will not have 4th-order contact with any circle. However, 4th-order contact can occur generically in a 1-parameter family of curves, at a curve in the family
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https://en.wikipedia.org/wiki/Iwasawa%20theory
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In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
Formulation
Iwasawa worked with so-called -extensions - infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. (These were called -extensions in early papers.) Every closed subgroup of is of the form so by Galois theory, a -extension is the same thing as a tower of fields
such that Iwasawa studied classical Galois modules over by asking questions about the structure of modules over
More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
Example
Let be a prime number and let be the field generated over by the th roots of unity. Iwasawa considered the following tower of number fields:
where is the field generated by adjoining to the pn+1-st roots of unity and
The fact that implies, by infinite Galois theory, that In order to get an interesting Galois module, Iwasawa took the ideal class group of , and let be its p-torsion part. There are norm maps whenever , and this gives us the data of an inverse system. If we set
then it is not hard to see from the inverse limit construction that is a module over In fact, is a module over the Iwasawa algebra . This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description it is possible to recover information about the p-part of the class group of
The motivation here is that the p-torsion in the ideal class group of had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem.
Connections with p-adic analysis
From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes.
Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by for and for all totally real number fields by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the so-called Herbrand–Ribet theorem).
Karl Rubin found a more elementary proof of
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https://en.wikipedia.org/wiki/Oblique
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Oblique may refer to:
an alternative name for the character usually called a slash (punctuation) ( / )
Oblique angle, in geometry
Oblique triangle, in geometry
Oblique lattice, in geometry
Oblique leaf base, a characteristic shape of the base of a leaf
Oblique angle, a synonym for Dutch angle, a cinematographic technique
Oblique (album), by jazz vibraphonist Bobby Hutcherson
Oblique (film), a 2008 Norwegian film
Oblique (Vasarely), a 1966 collage, by Victor Vasarely
Oblique banded rattail, a fish also known as a rough-head whiptail
Oblique case, in linguistics
Oblique argument, in linguistics
Oblique correction, in particle physics
Oblique motion, in music
Oblique order, a military formation
Oblique projection, in geometry and drawing, including cavalier and cabinet projection
Oblique reflection, in Euclidean geometry
Oblique shock, in gas dynamics
Oblique type, in typography
Oblique wing, in aircraft design
Oblique icebreaker, a special type of icegoing ship
Anatomy
Oblique arytenoid muscle, in the neck
Oblique cord, near the elbow point
Oblique fissure, separating the inferior and superior lobes of the lungs
Oblique muscle (disambiguation), any of several in the human body
Abdominal muscles
Abdominal external oblique muscle
Abdominal internal oblique muscle
Oblique strain, an injury of either of these muscles, common in baseball
Eye muscles
Inferior oblique muscle
Superior oblique muscle
See also
Obliq, a computer programming language
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https://en.wikipedia.org/wiki/%C3%89tale%20cohomology
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In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
History
Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed-point theorem in this context.
Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in Zermelo–Fraenkel set theory) led to some speculation that étale cohomology and its applications (such as the proof of Fermat's Last Theorem) require axioms beyond ZFC. However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories.
Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.
Motivation
For complex algebraic varieties, invariants from algebraic topology such as the fundamental group and cohomology groups are very useful, and one would like to have analogues of these for varieties over other fields, such as finite fields. (One reason for this is that Weil suggested that the Weil conjectures could be proved using suc
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https://en.wikipedia.org/wiki/Solvable
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In mathematics, solvable may refer to:
Solvable group, a group that can be constructed by compositions of abelian groups, or equivalently a group whose derived series reaches the trivial group in finitely many steps
Solvable extension, a field extension whose Galois group is a solvable group
Solvable equation, a polynomial equation whose Galois group is solvable, or equivalently, one whose solutions may be expressed by nested radicals
Solvable Lie algebra, a Lie algebra whose derived series reaches the zero algebra in finitely many steps
Solvable problem, a computational problem that can be solved by a Turing machine
Exactly solvable model in statistical mechanics, a system whose solution can be expressed in closed form, or alternatively, another name for completely integrable systems
See also
solved game
solubility
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https://en.wikipedia.org/wiki/F%CF%83%20set
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{{DISPLAYTITLE:Fσ set}}
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for (French: closed) and σ for (French: sum, union).
The complement of an Fσ set is a Gδ set.
Fσ is the same as in the Borel hierarchy.
Examples
Each closed set is an Fσ set.
The set of rationals is an Fσ set in . More generally, any countable set in a T1 space is an Fσ set, because every singleton is closed.
The set of irrationals is not a Fσ set.
In metrizable spaces, every open set is an Fσ set.
The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.
The set of all points in the Cartesian plane such that is rational is an Fσ set because it can be expressed as the union of all the lines passing through the origin with rational slope:
where , is the set of rational numbers, which is a countable set.
See also
Gδ set — the dual notion.
Borel hierarchy
P-space, any space having the property that every Fσ set is closed
References
Topology
Descriptive set theory
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https://en.wikipedia.org/wiki/Descriptive%20set%20theory
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In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.
Polish spaces
Descriptive set theory begins with the study of Polish spaces and their Borel sets.
A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line , the Baire space , the Cantor space , and the Hilbert cube .
Universality properties
The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.
Every Polish space is homeomorphic to a Gδ subspace of the Hilbert cube, and every Gδ subspace of the Hilbert cube is Polish.
Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.
Because of these universality properties, and because the Baire space has the convenient property that it is homeomorphic to , many results in descriptive set theory are proved in the context of Baire space alone.
Borel sets
The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that:
Every open subset of X is a Borel set.
If A is a Borel set, so is . That is, the class of Borel sets are closed under complementation.
If An is a Borel set for each natural number n, then the union is a Borel set. That is, the Borel sets are closed under countable unions.
A fundamental result shows that any two uncountable Polish spaces X and Y are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.
Borel hierarchy
Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes , , and .
Every open set is declared to be .
A set is declared to be if and only if its complement is .
A set A is declared to be , δ > 1, if there is a sequence 〈
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https://en.wikipedia.org/wiki/Polish%20space
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In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval is Polish.
Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the cardinality of the continuum.
Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.
Properties
Every Polish space is second countable (by virtue of being separable metrizable).
(Alexandrov's theorem) If is Polish then so is any -subset of .
A subspace of a Polish space is Polish if and only if is the intersection of a sequence of open subsets of . (This is the converse to Alexandrov's theorem.)
(Cantor–Bendixson theorem) If is Polish then any closed subset of can be written as the disjoint union of a perfect set and a countable set. Further, if the Polish space is uncountable, it can be written as the disjoint union of a perfect set and a countable open set.
Every Polish space is homeomorphic to a -subset of the Hilbert cube (that is, of , where is the unit interval and is the set of natural numbers).
The following spaces are Polish:
closed subsets of a Polish space,
open subsets of a Polish space,
products and disjoint unions of countable families of Polish spaces,
locally compact spaces that are metrizable and countable at infinity,
countable intersections of Polish subspaces of a Hausdorff topological space,
the set of irrational numbers with the topology induced by the standard topology of the real line.
Characterization
There are numerous characterizations that tell when a second-countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology.
There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game. A separable metric space is completely metrizable if and only if the second player has a winning strategy in this
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https://en.wikipedia.org/wiki/Ben%20Roy%20Mottelson
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Ben Roy Mottelson (9 July 1926 – 13 May 2022) was an American-Danish nuclear physicist. He won the 1975 Nobel Prize in Physics for his work on the non-spherical geometry of atomic nuclei.
Early life
Mottelson was born in Chicago, Illinois, on 9 July 1926, the son of Georgia (Blum) and Goodman Mottelson, an engineer. His family was Jewish. After graduating from Lyons Township High School in La Grange, Illinois, he joined the United States Navy and was sent to attend officers training at Purdue University, where he received a bachelor's degree in 1947. He then earned a PhD in nuclear physics from Harvard University in 1950. His thesis adviser was Julian Schwinger, the theoretical physicist who later won the Nobel Prize in 1965 for his work on quantum electrodynamics.
Career
He moved to Institute for Theoretical Physics (later the Niels Bohr Institute) at the University of Copenhagen on the Sheldon Traveling Fellowship from Harvard, and remained in Denmark. In 1953 he was appointed staff member in CERN's Theoretical Study Group, which was based in Copenhagen, a position he held until he became professor at the newly formed Nordic Institute for Theoretical Physics (Nordita) in 1957. He was a visiting professor at the University of California, Berkeley in Spring 1959. In 1971 he became a naturalized Danish citizen.
In 1950–1951, James Rainwater and Aage Bohr had developed models of the atomic nucleus which began to take into account the behaviour of the individual nucleons. These models, which moved beyond the simpler liquid drop treatment of the nucleus as having effectively no internal structure, were the first models which could explain a number of nuclear properties, including the non-spherical distribution of charge in certain nuclei. Mottelson worked with Aage Bohr to compare the theoretical models with experimental data. In 1952–1953, Bohr and Mottelson published a series of papers demonstrating close agreement between theory and experiment, for example showing that the energy levels of certain nuclei could be described by a rotation spectrum. This work stimulated new theoretical and experimental studies.
In the summer of 1957, David Pines visited Copenhagen, and introduced Bohr and Mottelson to the pairing effect developed in theories of superconductivity, which inspired them to introduce a similar pairing effect to explain the differences in the energy levels between even and odd atomic nuclei.
Nobel Prize (1975)
Rainwater, Bohr and Mottelson were jointly awarded the 1975 Nobel Prize in Physics "for the discovery of the connection between collective motion and particle motion in atomic nuclei and the development of the theory of the structure of the atomic nucleus based on this connection".
Post–Nobel Prize work
Bohr and Mottelson continued to work together, publishing a two-volume monograph, Nuclear Structure. The first volume, Single-Particle Motion, appeared in 1969, and the second volume, Nuclear Deformations, in 1975.
Professor Mo
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https://en.wikipedia.org/wiki/Sheldon%20Glashow
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Sheldon Lee Glashow (, ; born December 5, 1932) is a Nobel Prize-winning American theoretical physicist. He is the Metcalf Professor of Mathematics and Physics at Boston University and Eugene Higgins Professor of Physics, emeritus, at Harvard University, and is a member of the board of sponsors for the Bulletin of the Atomic Scientists.
Birth and education
Sheldon Glashow was born on December 5, 1932 in New York City, to Jewish immigrants from Russia, Bella (née Rubin) and Lewis Gluchovsky, a plumber. He graduated from Bronx High School of Science in 1950. Glashow was in the same graduating class as Steven Weinberg, whose own research, independent of Glashow's, would result in Glashow, Weinberg, and Abdus Salam sharing the 1979 Nobel Prize in Physics (see below). Glashow received a Bachelor of Arts degree from Cornell University in 1954 and a PhD degree in physics from Harvard University in 1959 under Nobel-laureate physicist Julian Schwinger. Afterwards, Glashow became a NSF fellow at NORDITA and met Murray Gell-Mann, who convinced him to become a research fellow at the California Institute of Technology. Glashow then became an assistant professor at Stanford University before joining the University of California, Berkeley where he was an associate professor from 1962 to 1966. He joined the Harvard physics department as a professor in 1966, and was named Eugene Higgins Professor of Physics in 1979; he became emeritus in 2000. Glashow has been a visiting scientist at CERN, and professor at Aix-Marseille University, MIT, Brookhaven Laboratory, Texas A&M, the University of Houston, and Boston University.
Research
In 1961, Glashow extended electroweak unification models due to Schwinger by including a short range neutral current, the Z0. The resulting symmetry structure that Glashow proposed, SU(2) × U(1), forms the basis of the accepted theory of the electroweak interactions. For this discovery, Glashow along with Steven Weinberg and Abdus Salam, was awarded the 1979 Nobel Prize in Physics.
In collaboration with James Bjorken, Glashow was the first to predict a fourth quark, the charm quark, in 1964. This was at a time when 4 leptons had been discovered but only 3 quarks proposed. The development of their work in 1970, the GIM mechanism showed that the two quark pairs: (d.s), (u,c), would largely cancel out flavor changing neutral currents, which had been observed experimentally at far lower levels than theoretically predicted on the basis of 3 quarks only. The prediction of the charm quark also removed a technical disaster for any quantum field theory with unequal numbers of quarks and leptons — an anomaly — where classical field theory symmetries fail to carry over into the quantum theory.
In 1973, Glashow and Howard Georgi proposed the first grand unified theory. They discovered how to fit the gauge forces in the standard model into an SU(5) group, and the quarks and leptons into two simple representations. Their theory qualitatively pre
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https://en.wikipedia.org/wiki/Numeracy
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Numeracy is the ability to understand, reason with, and to apply simple numerical concepts. The charity National Numeracy states: "Numeracy means understanding how mathematics is used in the real world and being able to apply it to make the best possible decisions...It’s as much about thinking and reasoning as about 'doing sums'". Basic numeracy skills consist of comprehending fundamental arithmetical operations like addition, subtraction, multiplication, and division. For example, if one can understand simple mathematical equations such as 2 + 2 = 4, then one would be considered to possess at least basic numeric knowledge. Substantial aspects of numeracy also include number sense, operation sense, computation, measurement, geometry, probability and statistics. A numerically literate person can manage and respond to the mathematical demands of life.
By contrast, innumeracy (the lack of numeracy) can have a negative impact. Numeracy has an influence on healthy behaviors, financial literacy, and career decisions. Therefore, innumeracy may negatively affect economic choices, financial outcomes, health outcomes, and life satisfaction. It also may distort risk perception in health decisions. Greater numeracy has been associated with reduced susceptibility to framing effects, less influence of nonnumerical information such as mood states, and greater sensitivity to different levels of numerical risk. Ellen Peters and her colleagues argue that achieving the benefits of numeric literacy, however, may depend on one's numeric self-efficacy or confidence in one's skills.
Representation of numbers
Humans have evolved to mentally represent numbers in two major ways from observation (not formal math). These representations are often thought to be innate (see Numerical cognition), to be shared across human cultures, to be common to multiple species, and not to be the result of individual learning or cultural transmission. They are:
Approximate representation of numerical magnitude, and
Precise representation of the quantity of individual items.
Approximate representations of numerical magnitude imply that one can relatively estimate and comprehend an amount if the number is large (see Approximate number system). For example, one experiment showed children and adults arrays of many dots. After briefly observing them, both groups could accurately estimate the approximate number of dots. However, distinguishing differences between large numbers of dots proved to be more challenging.
Precise representations of distinct items demonstrate that people are more accurate in estimating amounts and distinguishing differences when the numbers are relatively small (see Subitizing). For example, in one experiment, an experimenter presented an infant with two piles of crackers, one with two crackers the other with three. The experimenter then covered each pile with a cup. When allowed to choose a cup, the infant always chose the cup with more crackers because the infan
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https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin%20theorem
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In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index . In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.
Theorem
Let be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that is isomorphic to a product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index .
There is also a version of the Wedderburn–Artin theorem for algebras over a field . If is a finite-dimensional semisimple -algebra, then each in the above statement is a finite-dimensional division algebra over . The center of each need not be ; it could be a finite extension of .
Note that if is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.
Proof
There are various proofs of the Wedderburn–Artin theorem. A common modern one takes the following approach.
Suppose the ring is semisimple. Then the right -module is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of ). Write this direct sum as
where the are mutually nonisomorphic simple right -modules, the th one appearing with multiplicity . This gives an isomorphism of endomorphism rings
and we can identify with a ring of matrices
where the endomorphism ring of is a division ring by Schur's lemma, because is simple. Since we conclude
Here we used right modules because ; if we used left modules would be isomorphic to the opposite algebra of , but the proof would still go through. To see this proof in a larger context, see Decomposition of a module. For the proof of an important special case, see Simple Artinian ring.
Consequences
Since a finite-dimensional algebra over a field is Artinian, the Wedderburn–Artin theorem implies that every finite-dimensional simple algebra over a field is isomorphic to an n-by-n matrix ring over some finite-dimensional division algebra D over , where both n and D are uniquely determined. This was shown by Joseph Wedderburn. Emil Artin later generalized this result to the case of simple left or right Artinian rings.
Since the only finite-dimensional division algebra over an algebraically closed field is the field itself, the Wedderburn–Artin theorem has strong consequences in this case. Let be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field . Then is a finite product where the are positive integers and is the algebra
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https://en.wikipedia.org/wiki/Normalizing%20constant
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In probability theory, a normalizing constant or normalizing factor is used to reduce any probability function to a probability density function with total probability of one.
For example, a Gaussian function can be normalized into a probability density function, which gives the standard normal distribution. In Bayes' theorem, a normalizing constant is used to ensure that the sum of all possible hypotheses equals 1. Other uses of normalizing constants include making the value of a Legendre polynomial at 1 and in the orthogonality of orthonormal functions.
A similar concept has been used in areas other than probability, such as for polynomials.
Definition
In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.
Examples
If we start from the simple Gaussian function
we have the corresponding Gaussian integral
Now if we use the latter's reciprocal value as a normalizing constant for the former, defining a function as
so that its integral is unit
then the function is a probability density function. This is the density of the standard normal distribution. (Standard, in this case, means the expected value is 0 and the variance is 1.)
And constant is the normalizing constant of function .
Similarly,
and consequently
is a probability mass function on the set of all nonnegative integers. This is the probability mass function of the Poisson distribution with expected value λ.
Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. In that context, the normalizing constant is called the partition function.
Bayes' theorem
Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function. Proportional to implies that one must multiply or divide by a normalizing constant to assign measure 1 to the whole space, i.e., to get a probability measure. In a simple discrete case we have
where P(H0) is the prior probability that the hypothesis is true; P(D|H0) is the conditional probability of the data given that the hypothesis is true, but given that the data are known it is the likelihood of the hypothesis (or its parameters) given the data; P(H0|D) is the posterior probability that the hypothesis is true given the data. P(D) should be the probability of producing the data, but on its own is difficult to calculate, so an alternative way to describe this relationship is as one of proportionality:
Since P(H|D) is a probability, the sum over all possible (mutually exclusive) hypotheses should be 1, leading to the conclusion that
In this case, the reciprocal of the value
is the normalizing constant.
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https://en.wikipedia.org/wiki/Option%20time%20value
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In finance, the time value (TV) (extrinsic or instrumental value) of an option is the premium a rational investor would pay over its current exercise value (intrinsic value), based on the probability it will increase in value before expiry. For an American option this value is always greater than zero in a fair market, thus an option is always worth more than its current exercise value. As an option can be thought of as 'price insurance' (e.g., an airline insuring against unexpected soaring fuel costs caused by a hurricane), TV can be thought of as the risk premium the option seller charges the buyer—the higher the expected risk (volatility time), the higher the premium. Conversely, TV can be thought of as the price an investor is willing to pay for potential upside.
Time value decays to zero at expiration, with a general rule that it will lose of its value during the first half of its life and in the second half. As an option moves closer to expiry, moving its price requires an increasingly larger move in the price of the underlying security.
Intrinsic value
The intrinsic value (IV) of an option is the value of exercising it now. If the price of the underlying stock is above a call option strike price, the option has a positive monetary value, and is referred to as being in-the-money. If the underlying stock is priced cheaper than the call option's strike price, the call option is referred to as being out-of-the-money. If an option is out-of-the-money at expiration, its holder simply abandons the option and it expires worthless. Hence, a purchased option can never have a negative value. This is because a rational investor would choose to buy the underlying stock at the market price rather than exercise an out-of-the-money call option to buy the same stock at a higher-than-market price.
For the same reasons, a put option is in-the-money if it allows the purchase of the underlying at a market price below the strike price of the put option. A put option is out-of-the-money if the underlying's spot price is higher than the strike price.
As shown in the below equations and graph, the intrinsic value (IV) of a call option is positive when the underlying asset's spot price S exceeds the option's strike price K.
Value of a call option: , or
Value of a put option: , or
Option value
Option value (i.e.,. price) is estimated via a predictive formula such as Black-Scholes or using a numerical method such as the Binomial model. This price incorporates the expected probability of the option finishing "in-the-money". For an out-of-the-money option, the further in the future the expiration date—i.e. the longer the time to exercise—the higher the chance of this occurring, and thus the higher the option price; for an in-the-money option the chance of being in the money decreases; however the fact that the option cannot have negative value also works in the owner's favor. The sensitivity of the option value to the amount of time to expiry is known as th
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https://en.wikipedia.org/wiki/Euler%20product
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In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.
Definition
In general, if is a bounded multiplicative function, then the Dirichlet series
is equal to
where the product is taken over prime numbers , and is the sum
In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes .
An important special case is that in which is totally multiplicative, so that is a geometric series. Then
as is the case for the Riemann zeta function, where , and more generally for Dirichlet characters.
Convergence
In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region
that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.
In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here. The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree , and the representation theory for .
Examples
The following examples will use the notation for the set of all primes, that is:
The Euler product attached to the Riemann zeta function , also using the sum of the geometric series, is
while for the Liouville function , it is
Using their reciprocals, two Euler products for the Möbius function are
and
Taking the ratio of these two gives
Since for even values of the Riemann zeta function has an analytic expression in terms of a rational multiple of , then for even exponents, this infinite product evaluates to a rational number. For example, since , , and , then
and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to
where counts the number of distinct prime factors of , and is the number of square-free divisors.
If is a Dirichlet character of conductor , so that is totally multiplicative and only depends on , and if is not coprime to , then
Here it is convenient to omit the primes dividing the conductor from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as
for where is the polylogarithm. For the product above is just .
Notable constants
Many well known constants have Euler product expansions.
The Leibniz formula for
can be in
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https://en.wikipedia.org/wiki/Projective%20line
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In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case).
There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field K, commonly denoted P1(K), as the set of one-dimensional subspaces of a two-dimensional K-vector space. This definition is a special instance of the general definition of a projective space.
The projective line over the reals is a manifold; see real projective line for details.
Homogeneous coordinates
An arbitrary point in the projective line P1(K) may be represented by an equivalence class of homogeneous coordinates, which take the form of a pair
of elements of K that are not both zero. Two such pairs are equivalent if they differ by an overall nonzero factor λ:
Line extended by a point at infinity
The projective line may be identified with the line K extended by a point at infinity. More precisely,
the line K may be identified with the subset of P1(K) given by
This subset covers all points in P1(K) except one, which is called the point at infinity:
This allows to extend the arithmetic on K to P1(K) by the formulas
Translating this arithmetic in terms of homogeneous coordinates gives, when does not occur:
Examples
Real projective line
The projective line over the real numbers is called the real projective line. It may also be thought of as the line K together with an idealised point at infinity ∞ ; the point connects to both ends of K creating a closed loop or topological circle.
An example is obtained by projecting points in R2 onto the unit circle and then identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup
Compare the extended real number line, which distinguishes ∞ and −∞.
Complex projective line: the Riemann sphere
Adding a point at infinity to the complex plane results in a space that is topologically a sphere. Hence the complex projective line is also known as the Riemann sphere (or sometimes the Gauss sphere). It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.
For a finite field
The projective line over a finite field Fq of q elements has points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates , q of these points have the form:
for each in ,
and the remaining point at infinity may be represented as [1 : 0].
Symmetry group
Quite generally, the group of homographies with coefficients in K acts on the projective line P1(K). This group action is transitive, so that P1(K) is a homogeneous
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https://en.wikipedia.org/wiki/Georges%20de%20Rham
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Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
Biography
Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in Switzerland. He was the fifth born of the six children in the family of Léon de Rham, a constructions engineer. Georges de Rham grew up in Roche but went to school in nearby Aigle, the main town of the district, travelling daily by train. By his own account, he was not an extraordinary student in school, where he mainly enjoyed painting and dreamed of becoming a painter. In 1919 he moved with his family to Lausanne in a rented apartment in Beaulieu Castle, where he would live for the rest of his life. Georges de Rham started the Gymnasium in Lausanne with a focus on humanities, following his passion for literature and philosophy but learning little mathematics. On graduating from the Gymnasium in 1921 however, he decided not to continue with the Faculty of Letters in order to avoid Latin. He opted instead for the Faculty of Sciences of the University of Lausanne. At the faculty he started out studying biology, physics and chemistry and no mathematics initially. While trying to learn some mathematics by himself as a tool for physics, his interest was raised and by the third year he abandoned biology to focus decisively on mathematics.
At the University he was mainly influenced by two professors, Gustave Dumas and Dmitry Mirimanoff, who guided him in studying the works of Émile Borel, René-Louis Baire, Henri Lebesgue, and Joseph Serret. After graduating in 1925, de Rham remained at the University of Lausanne as an assistant to Dumas. Starting work towards completing his doctorate, he read the works of Henri Poincaré on topology on the advice of Dumas. Although he found inspiration for a thesis subject in Poincaré, progress was slow as topology was a relatively new topic and access to the relevant literature was difficult in Lausanne. With the recommendation of Dumas, de Rham contacted Lebesgue and went to Paris for a few months in 1926 and, again, for a few months in 1928. Both trips were financed by his own savings and he spent his time in Paris taking classes and studying at the University of Paris and the Collège de France. Lebesgue provided de Rham with a lot of help in this period, both with his studies and supporting his first research publications. When he finished his thesis Lebesgue advised him to send it to Élie Cartan and, in 1931, De Rham received his doctorate from the University of Paris before a commission led by Cartan and including Paul Montel and Gaston Julia as examiners.
In 1932 de Rham returned to the University of Lausanne as an extraordinary professor. In 1936 he also became a professor at the University of Geneva and continued to hold both positions in parallel until his retirement in 1971.
de Rham was also one of the best mountaineers in Switzerland. As a member of the Independent
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https://en.wikipedia.org/wiki/Associated%20bundle
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In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . For a fiber bundle F with structure group G, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ. One may then construct a fiber bundle F′ as a new fiber bundle having the same transition functions, but possibly a different fiber.
An example
A simple case comes with the Möbius strip, for which is the cyclic group of order 2, . We can take as any of: the real number line , the interval , the real number line less the point 0, or the two-point set . The action of on these (the non-identity element acting as in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles and together: what we really need is the data to identify to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for as for .
Construction
In general it is enough to explain the transition from a bundle with fiber , on which acts, to the associated principal bundle (namely the bundle where the fiber is , considered to act by translation on itself). For then we can go from to , via the principal bundle. Details in terms of data for an open covering are given as a case of descent.
This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.
Associated bundles in general
Let be a fiber bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective.
There is a local trivialization of the bundle E consisting of an open cover Ui of X, and a collection of fibre mapssuch that the transition maps are given by elements of G. More precisely, there are continuous functions gij : (Ui ∩ Uj) → G such that
Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated with E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover Ui whose transition functions are given bywhere the G-valued functions gij(u) are t
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https://en.wikipedia.org/wiki/Lebesgue%20covering%20dimension
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In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by open sets.
In general, a topological space X can be covered by open sets, in that one can find a collection of open sets such that X lies inside of their union. The covering dimension is the smallest number n such that for every cover, there is a refinement in which every point in X lies in the intersection of no more than n + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an integer) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under homeomorphisms.
The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.
Formal definition
The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.
A modern definition is as follows. An open cover of a topological space is a family of open sets such that their union is the whole space, = . The order or ply of an open cover = {} is the smallest number (if it exists) for which each point of the space belongs to at most open sets in the cover: in other words 1 ∩ ⋅⋅⋅ ∩ +1 = for 1, ..., +1 distinct. A refinement of an open cover = {} is another open cover = {}, such that each is contained in some . The covering dimension of a topological space is defined to be the minimum value of such that every finite open cover of X has an open refinement with order + 1. Thus, if is finite, 1 ∩ ⋅⋅⋅ ∩ +2 = for 1, ..., +2 distinct. If no such minimal exists, the space is said to have infinite covering dimension.
As a special case, a non-empty topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement.
The empty set has covering dimension -1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.
Examples
Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs
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https://en.wikipedia.org/wiki/Measure-preserving%20dynamical%20system
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In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium.
Definition
A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system
with the following structure:
is a set,
is a σ-algebra over ,
is a probability measure, so that , and ,
is a measurable transformation which preserves the measure , i.e., .
Discussion
One may ask why the measure preserving transformation is defined in terms of the inverse instead of the forward transformation . This can be understood in a fairly easy fashion. Consider a mapping of power sets:
Consider now the special case of maps which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends to (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map by writing . Of course, one could also define , but this is not enough to specify all such possible maps . That is, conservative, Borel-preserving maps cannot, in general, be written in the form one might consider, for example, the map of the unit interval given by this is the Bernoulli map.
has the form of a pushforward, whereas is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map ; the measure can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)
There are two classification problems of interest. One, discussed below, fixes and asks about the isomorphism classes of a transformation map . The other, discussed in transfer operator, fixes and , and asks about maps that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into dissipative systems and the route to equilibrium.
In terms of physics, the measure-preserving dynamical system often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map describes this stirring,
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https://en.wikipedia.org/wiki/43%20%28number%29
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43 (forty-three) is the natural number following 42 and preceding 44.
In mathematics
Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is 47. 43 is the smallest prime that is not a Chen prime. It is also the third Wagstaff prime.
43 is the fourth term of Sylvester's sequence, one more than the product of the previous terms (2 × 3 × 7).
43 is a centered heptagonal number.
Let a = a = 1, and thenceforth a = (a + a + ... + a). This sequence continues 1, 1, 2, 3, 5, 10, 28, 154... . a is the first term of this sequence that is not an integer.
43 is a Heegner number.
43 is the largest prime which divides the order of the Janko group J4.
43 is a repdigit in base 6 (111).
43 is the number of triangles inside the Sri Yantra.
43 is the largest natural number that is not an (original) McNugget number.
43 is the smallest prime number expressible as the sum of 2, 3, 4, or 5 different primes:
43 = 41 + 2
43 = 11 + 13 + 19
43 = 2 + 11 + 13 + 17
43 = 3 + 5 + 7 + 11 + 17.
43 is the smallest number with the property 43 = 4*prime(4) + 3*prime(3). Where prime(n) is the n-th prime number. There are only two numbers with that property, the other one is 127.
When taking the first six terms of the Taylor series for computing e, one obtains
which is also five minus the fifth harmonic number.
Every solvable configuration of the Fifteen puzzle can be solved in no more than 43 multi-tile moves (i.e. when moving two or three tiles at once is counted as one move).
In science
The chemical element with the atomic number 43 is technetium. It has the lowest atomic number of any element that does not possess stable isotopes.
Astronomy
Messier object M43, a magnitude 7.0 H II region in the constellation of Orion, a part of the Orion Nebula, and also sometimes known as de Mairan's Nebula
The New General Catalogue object NGC 43, a barred spiral galaxy in the constellation Andromeda
In sports
In auto racing:
The number for Richard Petty's race car when he won his seven Winston Cup Championships. He also won 200 races in his career, 95% of them in the famous #43.
The maximum number of cars participating in a NASCAR race in the Cup Series until 2016, and, through the 2012 season, the Nationwide Series.
In American football:
Strong safety Troy Polamalu wore #43 for the Pittsburgh Steelers and played his entire NFL career with the team for eleven seasons (2003–2014).
Arts, entertainment, and media
Music
The number of notes in Harry Partch's 43-tone scale of just intonation.
Popular culture
Movie 43 (2013) is a film consisting of a series of interconnected short stories, featuring some of the biggest stars in Hollywood, which make up the insane storylines a washed-up producer is pitching to a movie company.
In The Big Bang Theory episode "The 43 Peculiarity", Howard and Raj try to solve the mystery of Sheldon disappearing every afternoon to a room with a chalkboard that has the number 43 w
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https://en.wikipedia.org/wiki/44%20%28number%29
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44 (forty-four) is the natural number following 43 and preceding 45.
In mathematics
Forty-four X is a composite number; a square-prime, of the form (p2,q) and fourth of this form and of the form (22.q), where q is a higher prime.
44 is a repdigit and palindromic number in decimal. It is the tenth 10-happy number, and the fourth octahedral number.
It is the first member of the first cluster of two square-primes; of the form (p2,q), specifically, {(22.11)=44, (32.5)=45}. The next such cluster of two square-primes comprises {(22.29)=116, (32.13)=117}.
44 has an aliquot sum of 40, within an aliquot sequence of three composite numbers (44,40,50,43,1,0) to the prime in the 43-aliquot tree.
Since the greatest prime factor of 442 + 1 = 1937 is 149 and thus more than 44 twice, 44 is a Størmer number. Given Euler's totient function, φ(44) = 20 and φ(69) = 44.
44 is a tribonacci number, preceded by 7, 13, and 24, whose sum it equals.
44 is the number of derangements of 5 items.
There are only 44 kinds of Schwarz triangles, aside from the infinite dihedral family of triangles (p 2 2) with p = {2, 3, 4, ...}.
There are 44 distinct stellations of the truncated cube and truncated octahedron, per Miller's rules.
44 four-dimensional crystallographic point groups of a total 227 contain dual enantiomorphs, or mirror images.
There are forty-four classes of finite simple groups that arise from four general families of such groups:
Two general groups stem from cyclic groups and alternating groups.
Sixteen families of groups stem from simple groups of Lie type.
Twenty-six groups are sporadic.
Sometimes the Tits group is considered a 17th non-strict simple group of Lie type, or a 27th sporadic group, which would yield a total of 45 finite simple groups.
In science
The atomic number of ruthenium
Astronomy
Messier object M44, a magnitude 4.0 open cluster in the constellation Cancer, also known as the Beehive Cluster
The New General Catalogue object NGC 44, a double star in the constellation Andromeda
In technology
+44 is the ITU country code for international direct dial telephone calls to the United Kingdom
The .44 Magnum or .44 Special revolver cartridges
In other fields
Forty-four is:
The total number of candles lit on the menorah during the Jewish holiday of Chanukah, which starts on the 25th day of the Hebrew month of Kislev and ends on the 2nd or the 3rd day of Tevet.
The number of candles in a box of Hanukkah candles.
The name of a mysterious savior of Poland prophesied by the Polish national poet Adam Mickiewicz in his masterpiece dramatic poem Dziady (Forefathers): In scene 5 of act 3, the priest Piotr announces a "reviver of the nation" who is to bring back the lost freedom of Poland, and describes him with these words:
Born from a foreign mother, his blood of ancient heroes,
And his name will be forty and four.
A poker game in which each player is dealt four cards down, and four cards are dealt face down on the table in
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https://en.wikipedia.org/wiki/46%20%28number%29
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46 (forty-six) is the natural number following 45 and preceding 47.
In mathematics
Forty-six is
thirteenth discrete semiprime () and the eighth of the form (2.q), where q is a higher prime.
with an aliquot sum of 26; itself a semiprime, within an aliquot sequence of six composite numbers (46,26,16,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
a Wedderburn-Etherington number,
an enneagonal number
a centered triangular number.
the number of parallelogram polyominoes with 6 cells.
It is the sum of the totient function for the first twelve integers. 46 is the largest even integer that cannot be expressed as a sum of two abundant numbers. It is also the sixteenth semiprime.
Since it is possible to find sequences of 46+1 consecutive integers such that each inner member shares a factor with either the first or the last member, 46 is an Erdős–Woods number.
In science
The atomic number of palladium.
The number of human chromosomes.
The approximate molar mass of ethanol (46.07 g mol)
Astronomy
Messier object M46, a magnitude 6.5 open cluster in the constellation Puppis.
The New General Catalogue object NGC 46, a star in the constellation Pisces.
In music
Japanese idol group franchise Sakamichi Series, which consists of Nogizaka46, Keyakizaka46, Hinatazaka46, and Yoshimotozaka46
In sports
Valentino Rossi used 46 as his number in the MotoGP world motorcycle championship.
The number of mountains in the 46 peaks of the Adirondack mountain range. People who have climbed all of them are called "forty-sixers"; there is also an unofficial 47th peak.
The name of a defensive scheme used in American football; see 46 defense.
In religion
The total of books in the Old Testament, Catholic version, if the Book of Lamentations is counted as a book separate from the Book of Jeremiah
In other fields
Forty-six is also:
The code for international direct dial phone calls to Sweden.
The number of samurai, out of 47, who carried out the attack in the historical Ako vendetta; sometimes referred to as the 46 Ronins to discount the one samurai forced to turn back.
In the title of the movie Code 46, starring Tim Robbins and Samantha Morton.
Several routes numbered 46 exist throughout the world.
Because 46 in Japanese can be pronounced as "yon roku", and "yoroshiku" (よろしく) means "my best regards" in Japanese, people sometimes use 46 for greeting.
46 is the number of the City Chevrolet and Superflo cars driven by Cole Trickle in the movie Days of Thunder.
The number of the French department Lot.
46 is the number that unlocks the Destiny spaceship on the Sci-Fi TV show Stargate Universe. Dr. Rush discovers that the number 46 relates to the number of human chromosomes and begins sequencing different genetic codes to finally gain control of the ship's operating system. The episode was called "Humans".
The number depicted in the first flag of Oklahoma (replaced in 1925), signifying the fact that Oklahoma was the 46th state to join the
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https://en.wikipedia.org/wiki/48%20%28number%29
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48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens.
In mathematics
Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number.
48 is the smallest number with exactly ten divisors, and the first multiple of 12 not to be a sum of twin primes.
The Sum of Odd Anti-Factors of 48 = number * (n/2) where n is an Odd number. So, 48 is an Odd Anti-Factor Hemiperfect Number.
Other such numbers include 6048, 38688, 82132, 975312, etc.
Odd Anti-Factors of 48 = 5, 19
Sum of Odd Anti-Factors = 5 + 19 = 24 = 48 * 1/2
There are 11 solutions to the equation φ(x) = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number.
Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number.
48 is a Harshad number in base 10. It has 24, 2, 12, and 4 as factors.
In science
The atomic number of cadmium.
The number of Ptolemaic constellations.
The number of symmetries of a cube.
Astronomy
Messier object M48, a magnitude 5.5 open cluster in the constellation Hydra.
The New General Catalogue object NGC 48, a barred spiral galaxy in the constellation Andromeda.
In religion
The prophecies of 48 Jewish prophets and 7 prophetesses were recorded in the Tanakh for posterity.
According to the Mishnah, Torah wisdom is acquired via 48 ways (Pirkei Avoth 6:6).
In Buddhism, Amitabha Buddha had made 48 great vows and promises to provide ultimate salvation to countless beings through countless eons, with benefits said to be available merely by thinking about his name with Nianfo practice. He is thus hailed as "King of Buddhas" through such skillful compassion and became a popular and formal refuge figure in Pureland Buddhism.
In music
Johann Sebastian Bach's Well-Tempered Clavier is informally known as because it consists of a prelude and a fugue in each major and minor key, for a total of forty-eight pieces.
"48" is a song by Sunny Day Real Estate.
"48" is a song by Tyler, The Creator.
"Forty eight" is a song by Truckfighters on their 2007 album, Phi.
"48 Hour Parole" is a song by the Hollies.
"48 Crash" is a song by Suzi Quatro.
Familiar 48 is an alternative pop/rock band formerly known as Bonehead.
On Tool's album Ænima, there is a song named "Forty-Six & 2", the sum of which is 48.
AKB48 Group is a Japanese female idol group.
In sports
48 is the total number of minutes in a full NBA game.
In other fields
Forty-eight may also refer to:
the code for international direct dial phone calls to Poland.
the model number of the HP-48 S/SX/G/GX/G+/GII.
the 48 Hour Film Project.
The First 48, an American crime program, 2004-present.
48 Hours is a television news program on CBS.
48 Hrs., a 1982 film starring Nick Nolte and Eddie Murphy, followed by Another 48 H
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https://en.wikipedia.org/wiki/49%20%28number%29
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49 (forty-nine) is the natural number following 48 and preceding 50.
In mathematics
Forty-nine is the square of the prime number seven and hence the fourth non-unitary square prime of the form p2
47 has an aliquot sum of 8; itself a prime power, and hence an aliquot sequence of two composite members (49, 8, 7,1,0).
It appears in the Padovan sequence, preceded by the terms 21, 28, 37 (it is the sum of the first two of these).
Along with the number that immediately derives from it, 77, the only number under 100 not having its home prime known ().
Decimal representation
The sum of the digits of the square of 49 (2401) is the square root of 49.
49 is the first square where the digits are squares. In this case, 4 and 9 are squares.
Reciprocal
The fraction is a repeating decimal with a period of 42:
= (42 digits repeat)
There are 42 (note that this number is the period) positive integers that are less than 49 and coprime to 49. Multiplying 020408163265306122448979591836734693877551 by each of these integers results in a cyclic permutation of the original number:
020408163265306122448979591836734693877551 × 2 = 040816326530612244897959183673469387755102
020408163265306122448979591836734693877551 × 3 = 061224489795918367346938775510204081632653
020408163265306122448979591836734693877551 × 4 = 081632653061224489795918367346938775510204
...
The repeating number can be obtained from 02 and repetition of doubles placed at two places to the right:
02
04
08
16
32
64
128
256
512
1024
2048
+ ...
----------------------
020408163265306122448979591836734693877551...0204081632...
because satisfies:
In chemistry
The atomic number of indium.
During the Manhattan Project, plutonium was also often referred to simply as "49". Number 4 was for the last digit in 94 (atomic number of plutonium) and 9 for the last digit in Pu-239, the weapon-grade fissile isotope used in nuclear bombs.
In astronomy
Messier object M49, a magnitude 10.0 galaxy in the constellation Virgo.
The New General Catalogue object NGC 49, a spiral galaxy in the constellation Cetus.
In religion
In Judaism: the number of days of the Counting of the Omer and the number of years in a Jubilee (biblical) cycle.
The number of days and night Siddhartha Gautama spent meditating as a holy man
In Buddhism, 49 days is one of the lengths of the intermediate state (bardo)
In sports
49er, a member of the San Francisco 49ers team of the National Football League (United States football).
Arsenal had a 49-game unbeaten run between May 2003 and October 2004 until they lost to Manchester United, which is a national record in English football.
Rocky Marciano ended his boxing career as the only heavyweight champion with a perfect record—49 wins in 49 professional bouts, with 43 knockouts.
Indian Premier League cricket team Royal Challengers Bangalore holds
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https://en.wikipedia.org/wiki/51%20%28number%29
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51 (fifty-one) is the natural number following 50 and preceding 52.
In mathematics
Fifty-one is
a pentagonal number as well as a centered pentagonal number and an 18-gonal number
the 6th Motzkin number, telling the number of ways to draw non-intersecting chords between any six points on a circle's boundary, no matter where the points may be located on the boundary.
a Perrin number, coming after 22, 29, 39 in the sequence (and the sum of the first two)
a Størmer number, since the greatest prime factor of 512 + 1 = 2602 is 1301, which is substantially more than 51 twice.
There are 51 different cyclic Gilbreath permutations on 10 elements, and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set.
Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass and straightedge, the angle is constructible, and the number cos is expressible in terms of square roots.
In other fields
51 is:
The atomic number of antimony
The code for international direct dial phone calls to Peru
The last possible television channel number in the UHF bandplan for American terrestrial television from December 31, 2011, when channels 52–69 were withdrawn, to July 3, 2020, when channels 38–51 were removed from the bandplan.
The number of the laps of the Azerbaijan Grand Prix.
In the 2006 film Cars, 51 was Doc Hudson's number.
The Area 51.
The fire station number in the television series Emergency!.
The number of essays Alexander Hamilton wrote as part of The Federalist Papers defending the US constitution
See also
AD 51, a year in the Julian calendar
List of highways numbered 51
The model number of the P-51 Mustang World War II fighter aircraft
Area 51, a parcel of U.S. military-controlled land in southern Nevada, apparently containing a secret aircraft testing facility
Photo 51, an X-ray image of key importance in elucidating the structure of DNA in the 1950s
Fifty-One Tales, part of the title of a collection of stories by Lord Dunsany
"51st State", any future US state, usually referring to Washington, D.C., or Puerto Rico. Occasionally used in commentary to refer to non-US entities (such as Alberta) adopting US-like policies or seeking to become part of the US.
51 (film), a 2011 American horror film
"Fifty-One", an episode of Breaking Bad
Pastis 51, often just called "51", is a brand of pastis owned by Pernod Ricard.
Greg Murphy used "#51" for most of the seasons he raced in V8 Supercars
Clubhouse Games: 51 Worldwide Classics, a Nintendo Switch video game with 51 activities
References
Integers
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https://en.wikipedia.org/wiki/52%20%28number%29
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52 (fifty-two) is the natural number following 51 and preceding 53.
In mathematics
Fifty-two is
a composite number; a square-prime, of the form (p2, q) where q is a higher prime. It is the sixth of this form and the fifth of the form (22.q).
the 5th Bell number, the number of ways to partition a set of 5 objects.
a decagonal number.
with an aliquot sum of 46; within an aliquot sequence of seven composite numbers (52,46,26,16,15, 9,4,3,1,0) to the prime in the 3-aliquot tree. This sequence does not extend above 52 because it is,
an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is not equal to x − φ(x) for any x.
a vertically symmetrical number.
In science
The atomic number of tellurium
Astronomy
Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, also known as NGC 7654.
The New General Catalogue object NGC 52, a spiral galaxy in the constellation Pegasus.
In other fields
Fifty-two is:
The approximate number of weeks in a year. 52 weeks is 364 days, while the tropical year is 365.24 days long. According to ISO 8601, most years have 52 weeks while some have 53.
A significant number in the Maya calendar
On the modern piano, the number of white keys (notes in the C major scale)
The number of cards in a standard deck of playing cards, not counting Jokers or advertisement cards
The name of a practical joke card game 52 Pickup
52 Pick-Up is a film starring Roy Scheider and Ann Margaret
The code for international direct dial phone calls to Mexico
A weekly comic series from DC Comics entitled 52 has 52 issues, with a plot spanning one full year.
The New 52 is a 2011 revamp and relaunch by DC Comics of its entire line of ongoing monthly superhero books.
The number of letters in the English alphabet, if majuscules are distinguished from minuscules
The number of the French department Haute-Marne
52nd Street (disambiguation)
52 Hand Blocks, a variant of the martial art jailhouse rock.
52 is the car number of retired NASCAR driver Jimmy Means
52 American hostages were held in the Iran hostage crisis
The number of the laps of the British Grand Prix since 2010.
Historical years
52 BC, AD 52, 1052, 1952 etc.
See also
B52 (disambiguation)
List of highways numbered 52
References
Integers
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https://en.wikipedia.org/wiki/53%20%28number%29
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53 (fifty-three) is the natural number following 52 and preceding 54. It is the 16th prime number.
In mathematics
Fifty-three is the 16th prime number. It is also an Eisenstein prime, an isolated prime, a balanced prime and a Sophie Germain prime.
The sum of the first 53 primes is 5830, which is divisible by 53, a property shared by only a few other numbers.
53 cannot be expressed as the sum of any integer and its decimal digits, making 53 a self number.
53 is the smallest prime number that does not divide the order of any sporadic group.
In science
The atomic number of iodine
Astronomy
Messier object M53, a magnitude 8.5 globular cluster in the constellation Coma Berenices
The New General Catalogue object NGC 53, a magnitude 12.6 barred spiral galaxy in the constellation Tucana
In other fields
Fifty-three is:
The racing number of Herbie, a fictional Volkswagen Beetle with a mind of his own, first appearing in the 1968 film The Love Bug
The code for international direct dial phone calls to Cuba
53 Days is a northeastern USA rock band
53 Days a novel by Georges Perec
In How the Grinch Stole Christmas!, and its animated TV special adaptation the Grinch says he's put up with the Whos' Christmas cheer for 53 years.
Fictional 53rd Precinct in the Bronx was found in the TV comedy "Car 54, Where Are You?"
"53rd & 3rd" a song by the Ramones
The number of Hail Mary beads on a standard, five decade Catholic Rosary (the Dominican Rosary).
The number of bytes in an Asynchronous Transfer Mode packet.
UDP and TCP port number for the Domain Name System protocol.
53-TET (53 tone, equal temperament) is a musical temperament that has a fifth that is closer to pure than our current system.
53 More Things To Do In Zero Gravity is a book mentioned in The Hitchhiker's Guide to the Galaxy
Sports
The maximum number of players on a National Football League roster
Most points by a rookie in an NBA playoff game, by Philadelphia's Wilt Chamberlain, 1960
Most field goals (three-game series, NBA playoffs), by Michael Jordan, 1992
See also
List of highways numbered 53
References
Integers
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https://en.wikipedia.org/wiki/54%20%28number%29
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54 (fifty-four) is the natural number following 53 and preceding 55.
In mathematics
54 is an abundant number and a semiperfect number, like all other multiples of 6.
It is twice the third power of three, 3 + 3 = 54, and hence is a Leyland number.
54 is the smallest number that can be written as the sum of three positive squares in more than two different ways: = = = 54.
It is a 19-gonal number,
In base 10, 54 is a Harshad number.
The Holt graph has 54 edges.
The sine of an angle of 54 degrees is half the golden ratio.
The number of primes ≤ 28.
A Lehmer-Comtet number.
54 is the only non-trivial Neon Number in Power 9: 549 = 3,904,305,912,313,344; 3 + 9 + 0 + 4 + 3 + 0 + 5 + 9 + 1 + 2 + 3 + 1 + 3 + 3 + 4 + 4 = 54
In science
The atomic number of xenon is 54.
Astronomy
Messier object M54, a magnitude 8.5 globular cluster in the constellation Sagittarius
The New General Catalogue object NGC 54, a spiral galaxy in the constellation Cetus
The number of years in three Saros cycles of eclipses of the sun and moon is known as a Triple Saros or exeligmos (Greek: "turn of the wheel").
In sports
Fewest points in an NBA playoff game: Chicago (96), Utah (54), June 7, 1998
The New York Rangers won the Stanley Cup in 1994, 54 years after their previous Cup win. It is the longest drought in the trophy's history.
For years car number 54 was driven by NASCAR's Lennie Pond. More recently, it is known as the Nationwide Series car number for Kyle Busch.
A score of 54 on a par 72 course in golf is colloquially referred to as a perfect round. This score has never been achieved in competition.
The number used when a player is defeated 3 games in a row in racquetball.
In other fields
54 is also:
+54 The code for international direct dial phone calls to Argentina
A broadcast television channel number
54, a 1998 film about Studio 54 starring Ryan Phillippe, Mike Myers, and Salma Hayek
54, a novel by the Wu Ming collective of authors
In the title of a 1960s television show Car 54, Where Are You?
The number of the French department Meurthe-et-Moselle
New York's Warwick New York Hotel is on West 54th Street
The number of cards in a deck of playing cards, if two jokers are included
The number of countries in Africa
Year identifier used on motor vehicles registered in the UK between 1 September 2004 and 28 February 2005
Six by nine, the incorrect Answer to the Ultimate Question of Life, the Universe, and Everything
See also
List of highways numbered 54
References
Integers
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https://en.wikipedia.org/wiki/55%20%28number%29
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55 (fifty-five) is the natural number following 54 and preceding 56.
Mathematics
55 is
the fifteenth discrete semiprime () and the second with 5 as the lowest non-unitary factor thus of the form (5.q), where q is a higher prime.
with an aliquot sum of 17; a prime, within an aliquot sequence of one composite number (55, 17, 1,0) to the Prime 17 in the 17-aliquot tree.
a triangular number (the sum of the consecutive numbers 1 to 10), and a doubly triangular number.
the 10th Fibonacci number. It is the largest Fibonacci number to also be a triangular number.
a square pyramidal number (the sum of the squares of the integers 1 to 5) as well as a heptagonal number, and a centered nonagonal number.
in base 10, a Kaprekar number.
the product of 5 and 11, 5 being the prime index of 11.
the first number to be a sum of more than one pair of numbers which mirror each other (23 + 32 and 14 + 41).
Science
The atomic number of caesium.
Astronomy
Messier object M55, a magnitude 7.0 globular cluster in the constellation Sagittarius
The New General Catalogue object NGC 55, a magnitude 7.9 barred spiral galaxy in the constellation Sculptor
Music
The name of a song by Kasabian. The song was released as a B side to Club Foot and was recorded live when the band performed at London's Brixton Academy.
"55", a song by Mac Miller
"I Can't Drive 55", a song by Sammy Hagar
"Ol' '55", a song by Tom Waits
Ol' 55 (band), an Australian rock band.
Primer 55 an American band
Station 55, an album released in 2005 by Cristian Vogel
55 Cadillac, an album by Andrew W.K.
Transportation
In the United States, the National Maximum Speed Law prohibited speed limits higher than from 1974 to 1987
Film
55 Days at Peking a film starring Charlton Heston and David Niven
Years
AD 55
55 BC
1755
1855
1955
Other uses
Gazeta 55, an Albanian newspaper
Agitation and Propaganda against the State, also known as Constitution law 55, a law during Communist Albania.
The code for international direct dial phone calls to Brazil
A 55-gallon drum for containing oil, etc.
The Élysée, the official residency of the French Republic president, which address is 55 rue du Faubourg-Saint-Honoré in Paris.
See also
55th Regiment of Foot (disambiguation)
Channel 55 (disambiguation)
Type 55 (disambiguation)
Class 55 (disambiguation)
List of highways numbered 55
References
Integers
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https://en.wikipedia.org/wiki/56%20%28number%29
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56 (fifty-six) is the natural number following 55 and preceding 57.
Mathematics
56 is:
The sum of the first six triangular numbers (making it a tetrahedral number).
The number of ways to choose 3 out of 8 objects or 5 out of 8 objects, if order does not matter.
The sum of six consecutive primes (3 + 5 + 7 + 11 + 13 + 17)
a tetranacci number and as a multiple of 7 and 8, a pronic number. Interestingly it is one of a few pronic numbers whose digits in decimal also are successive (5 and 6).
a refactorable number, since 8 is one of its 8 divisors.
The sum of the sums of the divisors of the first 8 positive integers.
A semiperfect number, since 56 is twice a perfect number.
A partition number – the number of distinct ways 11 can be represented as the sum of natural numbers.
An Erdős–Woods number, since it is possible to find sequences of 56 consecutive integers such that each inner member shares a factor with either the first or the last member.
The only known number n such that , where φ(m) is Euler's totient function and σ(n) is the sum of the divisor function, see .
The maximum determinant in an 8 by 8 matrix of zeroes and ones.
The number of polygons formed by connecting all the 8 points on the perimeter of a two-times-two-square by straight lines.
Plutarch states that the Pythagoreans associated a polygon of 56 sides with Typhon and that they associated certain polygons of smaller numbers of sides with other figures in Greek mythology. While it is impossible to construct a perfect regular 56-sided polygon using a compass and straightedge, a close approximation has recently been discovered which it is claimed might have been used at Stonehenge, and it is constructible if the use of an angle trisector is allowed since 56 = 23 × 7.
Science, technology, and biology
The atomic number of barium.
In humans, olfactory receptors are categorized in 56 families.
The maximum speed of analog data transmission over a POTS in the 20th century was 56 kbit/s.
The number of bits in a key used in the Data Encryption Standard.
Astronomy
Messier object M56, a magnitude 9.5 globular cluster in the constellation Lyra
The New General Catalogue object NGC 56, an unverified object in the constellation Pisces, which does not appear to be a real object
Music
"56 Minutes", a 2007 David Woodard composition for piano, violin, cello and electronics
Flatfoot 56, a Christian punk rock band
"Along For The Ride ('56 T-bird)" sung by Danny O'Keefe
This song was covered by John Denver
"Five Feet of Lovin '56" sung by Gene Vincent
Elvis '56, an Elvis Presley CD
The name of a Plexi song
Xperimento56, a Spanish Funk/Rock band
56 Nights, a mixtape by Future
Television and film
Nasser 56, a documentary
Sports
Joe DiMaggio's 56-game hitting streak, which DiMaggio accomplished in 1941 with the New York Yankees. This remains a record today.
Hack Wilson hit 56 home runs in 1930, a National League record until 1998
56 people died in a fire at Valley
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https://en.wikipedia.org/wiki/57%20%28number%29
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57 (fifty-seven) is the natural number following 56 and preceding 58.
In mathematics
Fifty-seven is the sixteenth discrete semiprime (specifically, the sixth distinct semiprime of the form , where is a higher prime). It also forms the fourth discrete semiprime pair with 58.
57 is the third Blum integer since its two prime factors (3 and 19) are both Gaussian primes. 57 has an aliquot sum of 23, which makes it the tenth number to contain a prime aliquot sum. This also makes 57 the first composite member of the 23-aliquot tree (..., 57, 23, 1, 0). The only other numbers to generate an aliquot sum of 57 are 99, 159, 343, 559, and 703; where 343 is the cube of 7, and 703 the sum of the first thirty-seven nonzero integers. Fifty seven is also a repdigit in base-7 (111).
57 is the fifth Leyland number, as it can be written in the form:
57 is the number of compositions of 10 into distinct parts.
57 is the seventh fine number, equivalently the number of ordered rooted trees with seven nodes having root of even degree.
57 is also the number of nodes in a regular octagon when all of its diagonals are drawn, and the first non-trivial icosagonal (20-gonal) number.
In geometry, there are:
57 non-prismatic uniform star polyhedra in 3-space, including four Kepler-Poinsot star polyhedra that are regular.
57 vertices and hemi-dodecahedral facets in the 57-cell, a 4-dimensional abstract regular polytope.
57 uniform prismatic 5-polytopes in the fifth dimension based on four different finite prismatic families, and inclusive of one special non-Wythoffian figure: the grand antiprism prism.
57 uniform prismatic 6-polytopes in the sixth dimension, as prisms of all non-prismatic uniform 5-polytopes.
The split Lie algebra E has a 57-dimensional Heisenberg algebra as its nilradical, and the smallest possible homogeneous space for E8 is also 57-dimensional.
57 lies between prime numbers 53 and 61, which are the only two prime numbers less than 71 that do not divide the order of any sporadic group, inclusive of the six pariahs. 71, the twentieth prime number, is the largest supersingular prime that divides the largest of these groups while 57, on the other hand, is the fortieth composite number whose sum of divisors σ(57) is 80 and averages 20.
Although fifty-seven is not prime, it is jokingly known as the "Grothendieck prime" after a story in which mathematician Alexander Grothendieck supposedly gave it as an example of a particular prime number. This story is repeated in Part 2 of a biographical article on Grothendieck in
Notices of the American Mathematical Society.
In science
The atomic number of lanthanum (La), the first of the lanthanides
Astronomy
Messier object M57, a magnitude 9.5 planetary nebula in the constellation Lyra, also known as the Ring Nebula
The New General Catalogue object NGC 57, an elliptical galaxy in the constellation Pisces.
In fiction and media
In films
Passenger 57, a film starring Wesley Snipes
In the movie Contagion
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https://en.wikipedia.org/wiki/58%20%28number%29
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58 (fifty-eight) is the natural number following 57 and preceding 59.
In mathematics
Fifty-eight is the 17th discrete semiprime and the 9th with 2 as the lowest non-unitary factor; thus of the form (2.q), where q is a higher prime.
Fifty-eight is the first member of a cluster of two semiprimes (57, 58), the next such cluster is (118, 119).
Fifty-eight has an aliquot sum of 32 within an aliquot sequence of two composite numbers (58, 32, 13, 1, 0) in the 13-aliquot tree.
Fifty-eight is an 11-gonal number, after 30 (and 11). It is also a Smith number,
and given 58, the Mertens function returns zero.
58 is the smallest integer whose square root has a continued fraction with period 7.
58 is equal to the sum of the first seven consecutive prime numbers: This is a difference of 1 from the 17th prime number and 7th super-prime, 59.
There is no solution to the equation x – φ(x) = 58, making 58 a noncototient. However, the sum of the totient function for the first thirteen integers is 58.
The regular icosahedron produces 58 distinct stellations, the most of any other Platonic solid, which collectively produce 62 stellations.
Coxeter groups
With regard to Coxeter groups and uniform polytopes in higher dimensional spaces, there are:
58 distinct uniform polytopes in the fifth dimension that are generated from symmetries of three Coxeter groups, they are the A5 simplex group, B5 cubic group, and the D5 demihypercubic group;
58 fundamental Coxeter groups that generate uniform polytopes in the seventh dimension, with only four of these generating uniform non-prismatic figures.
There exist 58 total paracompact Coxeter groups of ranks four through ten, with realizations in dimensions three through nine. These solutions all contain infinite facets and vertex figures, in contrast from compact hyperbolic groups that contain finite elements; there are no other such groups with higher or lower ranks.
In science
The atomic number of cerium, a lanthanide.
Astronomy
Messier object M58, a magnitude 11.0 galaxy in the constellation Virgo.
The New General Catalogue object NGC 58, a barred spiral galaxy in the constellation Cetus. It is also the object designated as NGC 47.
In music
John Cage composition Fifty-Eight.
58 was the name of a side project involving Nikki Sixx of Mötley Crüe. They covered the song "Alone Again (Naturally)".
"58 Poems" by Chicago.
In sports
In the NBA, the most points ever scored in a fourth quarter was 58 by the Buffalo Braves (at Boston Celtics), October 20, 1972. The most points in a game by a rookie player: Wilt Chamberlain, 58: Philadelphia vs. Detroit, January 25, 1960, and Philadelphia vs. New York Knicks, February 21, 1960.
In MotoGP, 58 was the number of Marco Simoncelli who died in an accident at the Malaysian Round of the 2011 MotoGP season. MotoGP's governing body, the FIM, are considering to retire number 58 from use in MotoGP as they did before with the numbers 74 and 48 of Daijiro Kato and Shoya Tomizawa, resp
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https://en.wikipedia.org/wiki/59%20%28number%29
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59 (fifty-nine) is the natural number following 58 and preceding 60.
In mathematics
Fifty-nine is the 17th prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary part and real part of the form . Since is divisible by 59 but 59 is not one more than a multiple of 15, 59 is a Pillai prime.
It is also a highly cototient number.
There are 59 stellations of the regular icosahedron, inclusive of the icosahedron.
59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13# + 1 = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509 = 30031.
59 is the highest integer a single symbol may represent in the Sexagesimal system.
As 17 is prime, 59 is a super-prime.
The number 59 takes 3 iterations of the "reverse and add" process to form the palindrome 1111. All smaller integers (1 through 58) take either one or two iterations to form a palindrome through this process.
In science
The atomic number of praseodymium, a lanthanide.
Astronomy
Messier object M59, a magnitude 11.5 galaxy in the constellation Virgo.
The New General Catalogue object NGC 59, a magnitude 12.4 spiral galaxy in the constellation Cetus.
In music
Beethoven's Opus 59 consists of the three so-called Razumovsky Quartets
59, an album by Puffy AmiYumi
The 1960s song "The 59th Street Bridge Song (Feelin' Groovy)" was popularized by Simon & Garfunkel and Harpers Bizarre
The '59 Sound, an album by The Gaslight Anthem; includes the song of the same name
The album 14:59 by Sugar Ray
"11:59", a song by Blondie from Parallel Lines
.59 is a song by from beatmania IIDX 2nd Style and Dance Dance Revolution 4thMIX
'59 is the sixth track on the album Ignition! by Brian Setzer
59 is an area code of Andheri, Mumbai. Used by Vivian Divine in various songs with Gully Gang.
In sports
Satchel Paige became the oldest Major League Baseball player at age 59.
59 is the lowest golf score in a single round on the LPGA Tour by Annika Sörenstam, and on the Champions Tour by Kevin Sutherland.
In other fields
Fifty-nine is:
The number corresponding to the last minute in a given hour, and the last second in a given minute
The number of beads on a Roman Catholic rosary (Dominican).
Approximately the number of days in two lunar months
The Queensboro Bridge in New York City is also known as the 59th Street Bridge
The number on a button commonly worn by feminist activists in the 1970s; this was based on the claim that a woman earned 59 cents to an equally qualified man's dollar
Art Project 59's "59 Seconds Video Festival" at 59 Franklin Street, showed 59 videos to 59 different audiences, each 59 seconds long and incorporating the number 59
In amateur radio, a perfect signal report
Five Nine, an amateur radio magazine published in Japan
The number of the French department Nord
The "59-minute rule" is an informal r
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https://en.wikipedia.org/wiki/Dividing%20a%20circle%20into%20areas
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In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem, has a solution by an inductive method. The greatest possible number of regions, , giving the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (). Though the first five terms match the geometric progression , it deviates at , showing the risk of generalising from only a few observations.
Lemma
If there are n points on the circle and one more point is added, n lines can be drawn from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact.
Lemma. The new point A can be chosen so that case b occurs for each of the new lines.
Proof. For the case a, three points must be on one line: the new point A, the old point O to which the line is drawn, and the point I where two of the old lines intersect. There are n old points O, and hence finitely many points I where two of the old lines intersect. For each O and I, the line OI crosses the circle in one point other than O. Since the circle has infinitely many points, it has a point A which will be on none of the lines OI. Then, for this point A and all of the old points O, case b will be true.
This lemma means that, if there are k lines crossing AO, then each of them crosses AO at a different point and k + 1 new areas are created by the line AO.
Solution
Inductive method
The lemma establishes an important property for solving the problem. By employing an inductive proof, one can arrive at a formula for f(n) in terms of f(n − 1).
In the figure the dark lines are connecting points 1 through 4 dividing the circle into 8 total regions (i.e., f(4) = 8). This figure illustrates the inductive step from n = 4 to n = 5 with the dashed lines. When the fifth point is added (i.e., when computing f(5) using f(4)), this results in four new lines (the dashed lines in the diagram) being added, numbered 1 through 4, one for each point that they connect to. The number of new regions introduced by the fifth point can therefore be determined by considering the number of regions added by each of the 4 lines. Set i to count the lines being added. Each new line can cross a number of existing lines, depending on which point it is to (the value of i). The new lines will never cross each other, except at the new point.
The number of lines that each new line intersects can be determined by considering the number of points on the "left" of the line and the number of points on the "right" of the line. Since all existing points already have lines between them, the number of points on the left multiplied by the number of p
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https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell%20theorem
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In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.
The Rao–Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator g(X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal.
The theorem is named after C.R. Rao and David Blackwell. The process of transforming an estimator using the Rao–Blackwell theorem can be referred to as Rao–Blackwellization. The transformed estimator is called the Rao–Blackwell estimator.
Definitions
An estimator δ(X) is an observable random variable (i.e. a statistic) used for estimating some unobservable quantity. For example, one may be unable to observe the average height of all male students at the University of X, but one may observe the heights of a random sample of 40 of them. The average height of those 40—the "sample average"—may be used as an estimator of the unobservable "population average".
A sufficient statistic T(X) is a statistic calculated from data X to estimate some parameter θ for which no other statistic which can be calculated from data X provides any additional information about θ. It is defined as an observable random variable such that the conditional probability distribution of all observable data X given T(X) does not depend on the unobservable parameter θ, such as the mean or standard deviation of the whole population from which the data X was taken. In the most frequently cited examples, the "unobservable" quantities are parameters that parametrize a known family of probability distributions according to which the data are distributed.
In other words, a sufficient statistic T(X) for a parameter θ is a statistic such that the conditional probability of the data X, given T(X), does not depend on the parameter θ.
A Rao–Blackwell estimator δ1(X) of an unobservable quantity θ is the conditional expected value E(δ(X) | T(X)) of some estimator δ(X) given a sufficient statistic T(X). Call δ(X) the "original estimator" and δ1(X) the "improved estimator". It is important that the improved estimator be observable, i.e. that it does not depend on θ. Generally, the conditional expected value of one function of these data given another function of these data does depend on θ, but the very definition of sufficiency given above entails that this one does not.
The mean squared error of an estimator is the expected value of the square of its deviation from the unobservable quantity being estimated of θ.
The theorem
Mean-squared-error version
One case of Rao–Blackwell theorem st
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https://en.wikipedia.org/wiki/Dirichlet%20L-function
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In mathematics, a Dirichlet L-series is a function of the form
where is a Dirichlet character and s a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).
These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire.
Euler product
Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:
where the product is over all prime numbers.
Primitive characters
Results about L-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications. This is because of the relationship between a imprimitive character and the primitive character which induces it:
(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:
(This formula holds for all s, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.) The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors.
As a special case, the L-function of the principal character modulo q can be expressed in terms of the Riemann zeta function:
Functional equation
Dirichlet L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of to the value of . Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:
In this equation, Γ denotes the gamma function; a is 0 if χ(−1) = 1, or 1 if χ(−1) = −1; and
where τ ( χ) is a Gauss sum:
It is a property of Gauss sums that |τ ( χ) | = q1/2, so |ɛ ( χ) | = 1.
Another way to state the functional equation is in terms of
The functional equation can be expressed as:
The functional equation implies that (and ) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then has a pole at s = 1.)
For generalizations, see: Functional equation (L-function).
Zeros
Let χ be a primitive character modulo q, with q > 1.
There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative integers s:
If χ(−1) = 1, the only zeros of L(s, χ) with Re(s) < 0 are
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https://en.wikipedia.org/wiki/Additive%20Schwarz%20method
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In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results.
Overview
Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down.
(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem:
fxx(x,y) + fyy(x,y) = 0
f(0,y) = 1; f(x,0) = f(x,1) = f(1,y) = 0
where f is the unknown function, fxx and fyy denote the second partial derivatives with respect to x and y, respectively.
Here, the domain is the square [0,1] × [0,1].
This particular problem can be solved exactly on paper, so there is no need for a computer. However, this is an exceptional case, and most BVPs cannot be solved exactly. The only possibility is to use a computer to find an approximate solution.
Solving on a computer
A typical way of doing this is to sample f at regular intervals in the square [0,1] × [0,1]. For instance, we could take 8 samples in the x direction at x = 0.1, 0.2, ..., 0.8 and 0.9, and 8 samples in the y direction at similar coordinates. We would then have 64 samples of the square, at places like (0.2,0.8) and (0.6,0.6). The goal of the computer program would be to calculate the value of f at those 64 points, which seems easier than finding an abstract function of the square.
There are some difficulties, for instance it is not possible to calculate fxx(0.5,0.5) knowing f at only 64 points in the square. To overcome this, one uses some sort of numerical approximation of the derivatives, see for instance the finite element method or finite differences. We ignore these difficulties and concentrate on another aspect of the problem.
Solving linear problems
Whichever method we choose to solve this problem, we will need to solve a large linear system of equations. The reader may recall linear systems of equations from high school, they look like this:
2a + 5b = 12 (*)
6a − 3b = −3
This is a system of 2 equations in 2 unknowns (a and b). If we solve the BVP above in the manner suggested, we will need to solve a system of 64 equations in 64 unknowns. This is not a hard problem for modern computers, but if we use a larger number of samples, even modern computers cannot solve the BVP very efficiently.
Domain decomposition
Which brings us to domain decomposition methods. If we split the domain [0,1] × [0,1] into two subdomains [0,0.5] × [0,1] and [0.5,1] × [0,1], each has only half of the sample points. So we can try to solve a version of
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https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov%20derivative
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In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
Constructing the Grünwald–Letnikov derivative
The formula
for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:
Assuming that the h 's converge synchronously, this simplifies to:
which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient):
Removing the restriction that n be a positive integer, it is reasonable to define:
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
So the Grünwald–Letnikov derivative may be succinctly written as:
An alternative definition
In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as
or removing the restriction that n must be a positive integer:
This equation is called the reverse Grünwald–Letnikov derivative. If the substitution h → −h is made, the resulting equation is called the direct Grünwald–Letnikov derivative:
References
Further reading
The Fractional Calculus, by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974.
Fractional calculus
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https://en.wikipedia.org/wiki/Shadow%20volume
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Shadow volume is a technique used in 3D computer graphics to add shadows to a rendered scene. They were first proposed by Frank Crow in 1977 as the geometry describing the 3D shape of the region occluded from a light source. A shadow volume divides the virtual world in two: areas that are in shadow and areas that are not.
The stencil buffer implementation of shadow volumes is generally considered among the most practical general purpose real-time shadowing techniques for use on modern 3D graphics hardware. It has been popularized by the video game Doom 3, and a particular variation of the technique used in this game has become known as Carmack's Reverse.
Shadow volumes have become a popular tool for real-time shadowing, alongside the more venerable shadow mapping. The main advantage of shadow volumes is that they are accurate to the pixel (though many implementations have a minor self-shadowing problem along the silhouette edge, see construction below), whereas the accuracy of a shadow map depends on the texture memory allotted to it as well as the angle at which the shadows are cast (at some angles, the accuracy of a shadow map unavoidably suffers). However, the technique requires the creation of shadow geometry, which can be CPU intensive (depending on the implementation). The advantage of shadow mapping is that it is often faster, because shadow volume polygons are often very large in terms of screen space and require a lot of fill time (especially for convex objects), whereas shadow maps do not have this limitation.
Construction
In order to construct a shadow volume, project a ray from the light source through each vertex in the shadow casting object to some point (generally at infinity). These projections will together form a volume; any point inside that volume is in shadow, everything outside is lit by the light.
For a polygonal model, the volume is usually formed by classifying each face in the model as either facing toward the light source or facing away from the light source. The set of all edges that connect a toward-face to an away-face form the silhouette with respect to the light source. The edges forming the silhouette are extruded away from the light to construct the faces of the shadow volume. This volume must extend over the range of the entire visible scene; often the dimensions of the shadow volume are extended to infinity to accomplish this (see optimization below.) To form a closed volume, the front and back end of this extrusion must be covered. These coverings are called "caps". Depending on the method used for the shadow volume, the front end may be covered by the object itself, and the rear end may sometimes be omitted (see depth pass below).
There is also a problem with the shadow where the faces along the silhouette edge are relatively shallow. In this case, the shadow an object casts on itself will be sharp, revealing its polygonal facets, whereas the usual lighting model will have a gradual change in the light
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https://en.wikipedia.org/wiki/Cross-ratio
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In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defined as
where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)
The point is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is , called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio.
The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry.
The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.
Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.
In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
Terminology and history
Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII. Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.
Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position. Chasles coined the French term [anharmonic ratio] in 1837. German geometers call it [double ratio].
Carl von Staudt was unsatisfied with past definitions of the cross-ratio relying on algebraic manipulation of Euclidean distances rather than being based purely on synthetic projective geometry concepts. In 1847, von Staudt demonstrated that the algebraic structure is implicit in projective geometry, by creating an algebra based on construction of the projective harmonic conjugate, which he called a throw (German: Wurf): given three points on a line, the harmonic conjugate is a fourth point that makes the cross ratio equal to . His algebra of throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.
The English term "cross-ratio" was introduced in 1878 by William Kingdon Clifford.
Definition
If , , , and are four points on an oriented affine line, their cross ratio is:
with the notation defined to mean the signed ratio of the displacement from to to th
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https://en.wikipedia.org/wiki/Linear%20fractional%20transformation
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In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form
The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear.
In the most basic setting, , and are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then . Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line.
When are integer (or, more generally, belong to an integral domain), is supposed to be a rational number (or to belong to the field of fractions of the integral domain. In this case, the invertibility condition is that must be a unit of the domain (that is or in the case of integers).
In the most general setting, the and are elements of a ring, such as square matrices. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the real matrix ring.
Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical geometry, number theory (they are used, for example, in Wiles's proof of Fermat's Last Theorem), group theory, control theory.
General definition
In general, a linear fractional transformation is a homography of , the projective line over a ring . When is a commutative ring, then a linear fractional transformation has the familiar form
where are elements of such that is a unit of (that is has a multiplicative inverse in )
In a non-commutative ring , with in , the units determine an equivalence relation An equivalence class in the projective line over A is written , where the brackets denote projective coordinates. Then linear fractional transformations act on the right of an element of :
The ring is embedded in its projective line by , so recovers the usual expression. This linear fractional transformation is well-defined since does not depend on which element is selected from its equivalence class for the operation.
The linear fractional transformations over form a group, denoted
The group of the linear fractional transformations is called the modular group. It has been widely studied because of its numerous applications to number theory, which include, in particular, Wiles's proof of Fermat's Last Theorem.
Use in hyperbolic geometry
In the complex plane a generalized circle is either a line or a circle. When completed with the point at infinity, the generalized circles in the plane correspond to circles on the surface of the Riemann sphere, an expression of the complex projective line. Linear fractional transformations permute these circles on the sphere, and the corresponding finite points of the generalized circl
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https://en.wikipedia.org/wiki/Monopole
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Monopole may refer to:
Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole
Monopole (mathematics), a connection over a principal bundle G with a section (the Higgs field) of the associated adjoint bundle
Monopole, the first term in a multipole expansion
Monopole (wine), an appellation owned by only one winery
Monopole (album), a 2011 album by White Town
Monopole antenna, a radio antenna that replaces half of a dipole antenna with a ground plane at right-angles to the remaining half
Monopole, a tubular self-supporting telecommunications mast
The Monopole, a bar in Plattsburgh, NY
Établissements Monopole, a French auto parts manufacturer, racing car builder and racing team.
See also
Dipole, a particle with a north and south pole
Dyon, a particle with electric and magnetic charge
Instanton, a class of field solutions that includes monopoles
Monomial, a polynomial which has only one term
Monopoly (disambiguation)
Seiberg-Witten monopole, a solution of the Seiberg-Witten equations
't Hooft–Polyakov monopole, analogous to Dirac monopole, but without singularities
Wu–Yang monopole, a monopole solution of Yang-Mills equations
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https://en.wikipedia.org/wiki/Ray%20Solomonoff
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Ray Solomonoff (July 25, 1926 – December 7, 2009) was the inventor of algorithmic probability, his General Theory of Inductive Inference (also known as Universal Inductive Inference), and was a founder of algorithmic information theory. He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability. He circulated the first report on non-semantic machine learning in 1956.
Solomonoff first described algorithmic probability in 1960, publishing the theorem that launched Kolmogorov complexity and algorithmic information theory. He first described these results at a conference at Caltech in 1960, and in a report, Feb. 1960, "A Preliminary Report on a General Theory of Inductive Inference." He clarified these ideas more fully in his 1964 publications, "A Formal Theory of Inductive Inference," Part I and Part II.
Algorithmic probability is a mathematically formalized combination of Occam's razor, and the Principle of Multiple Explanations.
It is a machine independent method of assigning a probability value to each hypothesis (algorithm/program) that explains a given observation, with the simplest hypothesis (the shortest program) having the highest probability and the increasingly complex hypotheses receiving increasingly small probabilities.
Solomonoff founded the theory of universal inductive inference, which is based on solid philosophical foundations and has its root in Kolmogorov complexity and algorithmic information theory. The theory uses algorithmic probability in a Bayesian framework. The universal prior is taken over the class of all computable measures; no hypothesis will have a zero probability. This enables Bayes' rule (of causation) to be used to predict the most likely next event in a series of events, and how likely it will be.
Although he is best known for algorithmic probability and his general theory of inductive inference, he made many other important discoveries throughout his life, most of them directed toward his goal in artificial intelligence: to develop a machine that could solve hard problems using probabilistic methods.
Life history through 1964
Ray Solomonoff was born on July 25, 1926, in Cleveland, Ohio, son of Jewish Russian immigrants Phillip Julius and Sarah Mashman Solomonoff. He attended Glenville High School, graduating in 1944. In 1944 he joined the United States Navy as Instructor in Electronics. From 1947–1951 he attended the University of Chicago, studying under Professors such as Rudolf Carnap and Enrico Fermi, and graduated with an M.S. in Physics in 1951.
From his earliest years he was motivated by the pure joy of mathematical discovery and by the desire to explore where no one had gone before. At the age of 16, in 1942, he began to search for a general method to solve mathematical problems.
In 1952 he met Marvin Minsky, John McCarthy and others interested in machine intelligence. In 1956 Minsky and McCarthy and others organized the Dartmouth Su
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https://en.wikipedia.org/wiki/Algorithmic%20probability
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In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s.
It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the method together with Bayes' rule to obtain probabilities of prediction for an algorithm's future outputs.
In the mathematical formalism used, the observations have the form of finite binary strings viewed as outputs of Turing machines, and the universal prior is a probability distribution over the set of finite binary strings calculated from a probability distribution over programs (that is, inputs to a universal Turing machine). The prior is universal in the
Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated.
Overview
Algorithmic probability is the main ingredient of Solomonoff's theory of inductive inference, the theory of prediction based on observations; it was invented with the goal of using it for machine learning; given a sequence of symbols, which one will come next? Solomonoff's theory provides an answer that is optimal in a certain sense, although it is incomputable. Unlike, for example, Karl Popper's informal inductive inference theory, Solomonoff's is mathematically rigorous.
Four principal inspirations for Solomonoff's algorithmic probability were: Occam's razor, Epicurus' principle of multiple explanations, modern computing theory (e.g. use of a universal Turing machine) and Bayes’ rule for prediction.
Occam's razor and Epicurus' principle are essentially two different non-mathematical approximations of the universal prior.
Occam's razor: among the theories that are consistent with the observed phenomena, one should select the simplest theory.
Epicurus' principle of multiple explanations: if more than one theory is consistent with the observations, keep all such theories.
At the heart of the universal prior is an abstract model of a computer, such as a universal Turing machine. Any abstract computer will do, as long as it is Turing-complete, i.e. every computable function has at least one program that will compute its application on the abstract computer.
The abstract computer is used to give precise meaning to the phrase "simple explanation". In the formalism used, explanations, or theories of phenomena, are computer programs that generate observation strings when run on the abstract computer. Each computer program is assigned a weight corresponding to its length. The universal probability distribution is the probability distribution on all possible output strings with random input, assigning for each finite output prefix q the sum of the probabilities of the programs that compute something starting with q. Thus, a simple explanation is a short computer program. A complex explanation is a long
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https://en.wikipedia.org/wiki/List%20of%20equations
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This is a list of equations, by Wikipedia page under appropriate bands of their field.
Eponymous equations
The following equations are named after researchers who discovered them.
Mathematics
Cauchy–Riemann equations
Chapman–Kolmogorov equation
Maurer–Cartan equation
Pell's equation
Poisson's equation
Riccati equation
sine-Gordon equation
Verhulst equation
Physics
Ampère's circuital law
Bernoulli's equation
Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
Bessel's differential equation
Boltzmann equation
Borda–Carnot equation
Burgers' equation
Darcy–Weisbach equation
Dirac equation
Dirac equation in the algebra of physical space
Dirac–Kähler equation
Doppler equations
Drake equation (aka Green Bank equation)
Einstein's field equations
Euler equations (fluid dynamics)
Euler's equations (rigid body dynamics)
Relativistic Euler equations
Euler–Lagrange equation
Faraday's law of induction
Fokker–Planck equation
Fresnel equations
Friedmann equations
Gauss's law for electricity
Gauss's law for gravity
Gauss's law for magnetism
Gibbs–Helmholtz equation
Gross–Pitaevskii equation
Hamilton–Jacobi–Bellman equation
Helmholtz equation
Karplus equation
Kepler's equation
Kepler's laws of planetary motion
Kirchhoff's diffraction formula
Klein–Gordon equation
Korteweg–de Vries equation
Landau–Lifshitz–Gilbert equation
Lane–Emden equation
Langevin equation
Levy–Mises equations
Lindblad equation
Lorentz equation
Maxwell's equations
Maxwell's relations
Newton's laws of motion
Navier–Stokes equations
Reynolds-averaged Navier–Stokes equations
Prandtl–Reuss equations
Prony equation
Rankine–Hugoniot equation
Roothaan equations
Saha ionization equation
Sackur–Tetrode equation
Samik Hazra equation
Schrödinger equation
screened Poisson equation
Schwinger–Dyson equation
Sellmeier equation
Stokes–Einstein relation
Tsiolkovsky rocket equation
Van der Waals equation
Vlasov equation
Wiener equation
Chemistry
Arrhenius equation
Butler–Volmer equation
Eyring equation
Henderson–Hasselbalch equation
Michaelis–Menten equation
Nernst equation
Schrödinger equation
Urey-Bigeleisen-Mayer equation
Biology
Breeder's equation
Hardy–Weinberg principle
Hill equation
Lotka–Volterra equation
Michaelis–Menten equation
Poiseuille equation
Price equation
Economics
Black–Scholes equation
Fisher equation
Technology
Mansour's equation
Other equations
Mathematics
Polynomial equation
Linear equation
Quadratic equation
Cubic equation
Biquadratic equation
Quartic equation
Quintic equation
Sextic equation
Characteristic equation
Class equation
Comparametric equation
Difference equation
Matrix difference equation
Differential equation
Matrix differential equation
Ordinary differential equation
Partial differential equation
Total differential equation
Diophantine equation
Equation
Modular equation
Parametric equation
Replicator equation
Physics
Advection equation
Bar
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https://en.wikipedia.org/wiki/Point%20at%20infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring.
In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).
In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric.
Affine geometry
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
As a projective space over a field is a smooth algebraic variety, the same is true for the set of points at infinity. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a manifold.
Perspective
In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point.
Hyperbolic geometry
In hyperbolic geometry, points at infinity are typically named ideal points. Unlike Euclidean and elliptic geometries, each line has two points at infinity: given a line l and a point P not on l, the right- and left-limiting parallels converge asymptotically to different points at infinity.
All points at infinity together form the Cayley absolute or boundary of a hyperbolic plane.
Projective geometry
A symmetry of points and lines arises in a projective plane: just as a pair of points determine a line, so a pair of lines determine a point. The existence of parallel lines leads to establishing a point at infinity which represents the intersection of these parallels. This axiomatic symmetry grew out of a study of graphical perspective where a parallel projection arises as a central projection where the center C is a point at infinity, or figurative point. The axiomatic symmetry of points and lines is called duality.
Though a point at infinity is considered on a par with any other point of a projective range, in the representation of p
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https://en.wikipedia.org/wiki/Line%20at%20infinity
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In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.
Geometric formulation
In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel.
Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept.
In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are closed curves, i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.
Topological perspective
The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the affine plane and the line at infinity makes the real projective plane, .
A hyperbola can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two asymptotes of the hyperbola. Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
The analogue for the complex projective plane is a 'line' at infinity that is (naturally) a complex projective line. Topologically this is quite different, in that it is a Riemann sphere, which is therefore a 2-sphere, being added to a complex affine space of two dimensions over C (so four real dimensions), resulting in a four-dimensional compact manifold. The result is orientable, while the real projective plane is not.
History
The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a conic constrained to pass through two points at infinity, the solutions of
X2 + Y2 = 0.
This equation is the form taken by that of any ci
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https://en.wikipedia.org/wiki/Plane%20at%20infinity
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In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case.
Definition
There are two approaches to defining the plane at infinity which depend on whether one starts with a projective 3-space or an affine 3-space.
If a projective 3-space is given, the plane at infinity is any distinguished projective plane of the space. This point of view emphasizes the fact that this plane is not geometrically different than any other plane. On the other hand, given an affine 3-space, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. Meaning that the points of the plane at infinity are the points where parallel lines of the affine 3-space will meet, and the lines are the lines where parallel planes of the affine 3-space will meet. The result of the addition is the projective 3-space, . This point of view emphasizes the internal structure of the plane at infinity, but does make it look "special" in comparison to the other planes of the space.
If the affine 3-space is real, , then the addition of a real projective plane at infinity produces the real projective 3-space .
Analytic representation
Since any two projective planes in a projective 3-space are equivalent, we can choose a homogeneous coordinate system so that any point on the plane at infinity is represented as (X:Y:Z:0).
Any point in the affine 3-space will then be represented as (X:Y:Z:1). The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to any rescaling:
,
so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely a projective plane).
Proposition: Any line which passes through the origin (0:0:0:1) and through a point (X:Y:Z:1) will intersect the plane at infinity at the point (X:Y:Z:0).
Proof: A line which passes through points (0:0:0:1) and (X:Y:Z:1) will consist of points which are linear combinations of the two given points:
For such a point to lie on the plane at infinity we must have, . So, by choosing , we obtain the point
, as required. Q.E.D.
Any pair of parallel lines in 3-space will intersect each other at a point on the plane at infinity. Also, every line in 3-space intersects the plane at infinity at a unique point. This point is determined by the direction—and only by the direction—of the line. To determine this point, consider a line parallel to the given line, but passing through the origin, if the line does not already pass through the origin. Then choose any point, other than the origin, on this second line. If the homogeneous coordinates of this point are (X:Y:Z:1), then the homogeneous coordinates of the point at infinity through
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https://en.wikipedia.org/wiki/Suzuki%20sporadic%20group
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In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order
213 · 37 · 52 · 7 · 11 · 13 = 448345497600 ≈ 4.
History
Suz is one of the 26 Sporadic groups and was discovered by as a rank 3 permutation group on 1782 points with point stabilizer G2(4). It is not related to the Suzuki groups of Lie type. The Schur multiplier has order 6 and the outer automorphism group has order 2.
Complex Leech lattice
The 24-dimensional Leech lattice has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers, called the complex Leech lattice. The automorphism group of the complex Leech lattice is the universal cover 6 · Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice, and shows that it has two complex irreducible representations of dimension 12. The group 6 · Suz acting on the complex Leech lattice is analogous to the group 2 · Co1 acting on the Leech lattice.
Suzuki chain
The Suzuki chain or Suzuki tower is the following tower of rank 3 permutation groups from , each of which is the point stabilizer of the next.
G2(2) = U(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
J2 · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G2(2)
G2(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J2 · 2
Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G2(4) · 2
Maximal subgroups
found the 17 conjugacy classes of maximal subgroups of Suz as follows:
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
External links
MathWorld: Suzuki group
Atlas of Finite Group Representations: Suzuki group
Sporadic groups
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https://en.wikipedia.org/wiki/Higman%E2%80%93Sims%20group
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In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order
29⋅32⋅53⋅7⋅11 = 44352000
≈ 4.
The Schur multiplier has order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group.
History
HS is one of the 26 sporadic groups and was found by . They were attending a presentation by Marshall Hall on the Hall–Janko group J2. It happens that J2 acts as a permutation group on the Hall–Janko graph of 100 points, the stabilizer of one point being a subgroup with two other orbits of lengths 36 and 63. Inspired by this they decided to check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system has 77 blocks.) By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M22.
HS is the simple subgroup of index two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes, so the Higman–Sims group HS is a transitive group of permutations of a 100 element set. The smallest faithful complex representation of HS has dimension 22.
independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points.
Construction
GAP code to build the Higman-Sims group is presented as an example in the GAP documentation itself.
The Higman-Sims group can be constructed with the following two generators:
and
Relationship to Conway groups
identified the Higman–Sims group as a subgroup of the Conway group Co0. In Co0 HS arises as a pointwise stabilizer of a 2-3-3 triangle, one whose edges (differences of vertices) are type 2 and 3 vectors. HS thus is a subgroup of each of the Conway groups Co0, Co2 and Co3.
(p. 208) shows that the group HS is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of Co3. Count the type 2 points w such that the inner product v·w = 2 (and thus v-w is type 3). He shows that their number is and that this Co3 is transitive on these w.
|HS| = |Co3|/11,178 = 44,352,000.
In fact, and there are instances of HS including a permutation matrix representation of the Mathieu group M22.
If an instance of HS in Co0 fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3 that this copy of HS permutes in orbits of 176 and 100. This fact leads to Graham Higman's construction as well as to the Higman–Sims graph. HS is doubly transitive on the 176 and rank 3 on the 100.
A 2-3-3 triangle defines a 2-dimensional subspace fixed pointwise by HS. The standard representation of HS can thus be reduced to a 22-dimensional one.
A Higman-Sims graph
(p. 210) gives an example of a Higman-Sims graph within the Leech lattice, permuted by the representation of M22
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https://en.wikipedia.org/wiki/McLaughlin%20group
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McLaughlin group may refer to:
McLaughlin group (mathematics), a sporadic finite simple group
The McLaughlin Group, a weekly public affairs program broadcast in the United States
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https://en.wikipedia.org/wiki/61%20%28number%29
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61 (sixty-one) is the natural number following 60 and preceding 62.
In mathematics
61 is the 18th prime number, and a twin prime with 59. It is the sum of two consecutive squares, It is also a centered decagonal number, a centered hexagonal number, and a centered square number.
61 is the fourth cuban prime of the form where , and the forth Pillai prime since is divisible by 61, but 61 is not one more than a multiple of 8. It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ...
61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number.
61 is the smallest proper prime, a prime which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, times).
In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760,350,154,212,639,131).
61 is the exponent of the ninth Mersenne prime, and the next candidate exponent for a potential fifth double Mersenne prime:
The exotic sphere is the last odd-dimensional sphere to contain a unique smooth structure; , and are the only other such spheres.
In science
The chemical element with the atomic number 61 is promethium.
Astronomy
Messier object M61, a magnitude 10.5 galaxy in the constellation Virgo
The New General Catalogue object NGC 61, a double spiral galaxy in the constellation Cetus
61 Ursae Majoris is located about 31.1 light-years from the Sun.
61 Cygni was christened the "Flying Star" in 1792 by Giuseppe Piazzi (1746–1826) for its unusually large proper motion.
In other fields
Sixty-one is:
The number of the French department Orne
The code for international direct dial phone calls to Australia
61*, a 2001 baseball movie directed by Billy Crystal
Highway 61 Revisited is a Bob Dylan album
The Highway 61 Blues Festival occurs annually in Leland, Mississippi
Highway 61 is a 1991 film set on U.S. Route 61
U.S. Route 61 is the highway that inspired so much attention on "Highway 61"
Part 61 is a law created by the FAA regarding medical exams. This law has often come under attack by AOPA.
The P-61 is the Northrop-designed fighter first designated as the XP-61. It first flew on May 26, 1942. It is also known as the Black Widow as it was the first fighter aircraft designed to be a night fighter
Sixty 1 is a brand tobacco produced by Nationwide Tobacco
61A is the London address of Margot Wendice (Grace Kelly) and Tony Wendice (Ray Milland) in the movie Dial M for Murder
1 Liberty Place is one of Philadelphia's tallest buildings at 61 stories
The number of cadets on The Summerall Guards
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https://en.wikipedia.org/wiki/62%20%28number%29
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62 (sixty-two) is the natural number following 61 and preceding 63.
In mathematics
62 is:
the eighteenth discrete semiprime () and tenth of the form (2.q), where q is a higher prime.
with an aliquot sum of 34; itself a semiprime, within an aliquot sequence of seven composite numbers (62,34,20,22,14,10,8,7,1,0) to the Prime in the 7-aliquot tree. This is the longest aliquot sequence for a semiprime up to 118 which has one more sequence member. 62 is the tenth member of the 7-aliquot tree (7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, etc).
a nontotient.
palindromic and a repdigit in bases 5 (2225) and 30 (2230)
the sum of the number of faces, edges and vertices of icosahedron or dodecahedron.
the number of faces of two of the Archimedean solids, the rhombicosidodecahedron and truncated icosidodecahedron.
the smallest number that is the sum of three distinct positive squares in two (or more) ways,
the only number whose cube in base 10 (238328) consists of 3 digits each occurring 2 times.
The 20th & 21st, 72nd & 73rd, 75th & 76th digits of pi.
In science
Sixty-two is the atomic number of samarium, a lanthanide.
In other fields
62 is the code for international direct dial calls to Indonesia.
In the 1998 Home Run Race, Mark McGwire hit his 62nd home run on September 8, breaking the single-season record. Sammy Sosa hit his 62nd home run just days later on September 13.
Under Social Security (United States), the earliest age at which a person may begin receiving retirement benefits (other than disability).
References
Integers
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https://en.wikipedia.org/wiki/63%20%28number%29
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63 (sixty-three) is the natural number following 62 and preceding 64.
Mathematics
63 is the sum of the first six powers of 2 (20 + 21 + ... 25). It is the eighth highly cototient number, and the fourth centered octahedral number; after 7 and 25. For five unlabeled elements, there are 63 posets.
Sixty-three is the seventh square-prime of the form and the second of the form . It contains a prime aliquot sum of 41, the thirteenth indexed prime; and part of the aliquot sequence (63, 41, 1, 0) within the 41-aliquot tree.
Zsigmondy's theorem states that where are coprime integers for any integer , there exists a primitive prime divisor that divides and does not divide for any positive integer , except for when
, with having no prime divisors,
, a power of two, where any odd prime factors of are contained in , which is even;
and for a special case where with and , which yields .
63 is a Mersenne number of the form with an of , however this does not yield a Mersenne prime, as 63 is the forty-fourth composite number. It is the only number in the Mersenne sequence whose prime factors are each factors of at least one previous element of the sequence (3 and 7, respectively the first and second Mersenne primes). In the list of Mersenne numbers, 63 lies between Mersenne primes 31 and 127, with 127 the thirty-first prime number. The thirty-first odd number, of the simplest form , is 63. It is also the fourth Woodall number of the form with , with the previous members being 1, 7 and 23 (they add to 31, the third Mersenne prime).
In the integer positive definite quadratic matrix representative of all (even and odd) integers, the sum of all nine terms is equal to 63.
63 is the third Delannoy number, which represents the number of pathways in a grid from a southwest corner to a northeast corner, using only single steps northward, eastward, or northeasterly.
Finite simple groups
63 holds thirty-six integers that are relatively prime with itself (and up to), equivalently its Euler totient. In the classification of finite simple groups of Lie type, 63 and 36 are both exponents that figure in the orders of three exceptional groups of Lie type. The orders of these groups are equivalent to the product between the quotient of (with prime and a positive integer) by the GCD of , and a (in capital pi notation, product over a set of terms):
the order of exceptional Chevalley finite simple group of Lie type,
the order of exceptional Chevalley finite simple group of Lie type,
the order of one of two exceptional Steinberg groups,
Lie algebra holds 36 positive roots in sixth-dimensional space, while holds 63 positive root vectors in the seven-dimensional space (with 126 total root vectors, twice 63).
There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic Coxeter group (sometimes, the demicube is also included in this family), that is associated with classical Chevalley Lie algebra via
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https://en.wikipedia.org/wiki/64%20%28number%29
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64 (sixty-four) is the natural number following 63 and preceding 65.
In mathematics
Sixty-four is the square of 8, the cube of 4, and the sixth-power of 2. It is the smallest number with exactly seven divisors. 64 is the first non-unitary sixth-power prime of the form p6 where p is a prime number.
The aliquot sum of a 2-power (2n) is always one less than the 2-power itself therefore the aliquot sum of 64 is 63, within an aliquot sequence of two composite members ( 64,63,41,1,0) to the prime 41 in the 41-aliquot tree.
It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number (greater than 1) that is both a perfect square and a perfect cube.
Since it is possible to find sequences of 65 consecutive integers (intervals of length 64) such that each inner member shares a factor with either the first or the last member, 64 is an Erdős–Woods number.
In base 10, no integer added to the sum of its own digits yields 64; hence 64 is a self number.
64 is a superperfect number—a number such that σ(σ(n)) = 2n.
64 is the index of Graham's number in the rapidly growing sequence 3↑↑↑↑3, 3 ↑ 3,…
In the fourth dimension, there are 64 uniform polychora aside from two infinite families of duoprisms and antiprismatic prisms, and 64 Bravais lattices.
In science
The atomic number of gadolinium, a lanthanide
In astronomy
Messier object M64, a magnitude 9.0 galaxy in the constellation Coma Berenices, also known as the Black Eye Galaxy.
The New General Catalogue object NGC 64, a barred spiral galaxy in the constellation Cetus.
In technology
In some computer programming languages, the size in bits of certain data types
64-bit computing
A 64-bit integer can represent up to 18,446,744,073,709,551,616 values.
Base 64 is used in with Base64 encoding and other data compression formats.
In 8-bit home computers, a common shorthand for the Commodore 64
The ASCII code 64 is for the @ symbol
The Nintendo 64 video game console and (historically) the Commodore 64.
Since 1996, the number 64 has been an abbreviation or slang for Nintendo 64 (though N64 is more common) along with the games Super Mario 64, Mario Kart 64 and more.
64 is used in the term Smash 64 which is used to distinguish Super Smash Bros. (the video game) from the name of the series Super Smash Bros.
64 is the maximum size of most items and blocks held in the inventory of the player in the video game Minecraft. In the game, it is also known as a "stack".
In other fields
Sixty-four is:
Up to the year 1957, the Indian currency Rupee had 64 paise. Likewise, the Pakistani Rupee also had 64 paise up to 1961.
In chess or draughts, the total number of black (dark) and white (light) squares on the 8 by 8 game board
The total number of gems in a standard Bejeweled game board
64 was
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https://en.wikipedia.org/wiki/65%20%28number%29
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65 (sixty-five) is the natural number following 64 and preceding 66.
In mathematics
65 is the nineteenth distinct semiprime, (5.13); and the third of the form (5.q), where q is a higher prime.
65 has a prime aliquot sum of 19 within an aliquot sequence of one composite numbers (65,19,1,0) to the prime; as the first member' of the 19-aliquot tree.
It is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0.
This number is the magic constant of a 5x5 normal magic square:
This number is also the magic constant of n-Queens Problem for n = 5.
65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two (or more) ways, 65 = 82 + 12 = 72 + 42.
It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).
65 is a Stirling number of the second kind, the number of ways of dividing a set of six objects into four non-empty subsets.
65 = 15 + 24 + 33 + 42 + 51.
65 is the length of the hypotenuse of 4 different Pythagorean triangles, the lowest number to have more than 2: 652 = 162 + 632 = 332 + 562 = 392 + 522 = 252 + 602. The first two are "primitive", and 65 is the lowest number to be the largest side of more than one such triple.
65 is the number of compositions of 11 into distinct parts.
In science
The atomic number of terbium, a lanthanide
Astronomy
Messier object M65, a galaxy of magnitude 10.5 in the constellation Leo
The New General Catalogue object NGC 65, a spiral galaxy in the constellation Cetus
In music
"65 Love Affair", singer Paul Davis' hit song in 1982
Sammy Hagar re-recorded his hit "I Can't Drive 55", with the 55 changed to 65, in 2001 for NBC's NASCAR broadcasts to reflect higher speed limits; the song was used from 2001 until 2004 to introduce Budweiser Pole Award winners on NBC and TNT broadcasts
65 is a commonly used abbreviation for the Sheffield, UK, post-rock band 65daysofstatic
Referenced in "Heroes and Villains" by the Beach Boys: "At 60 and 5 / I'm very much alive / I've still got the jive / to survive with the Heroes and Villains"
Odd Future group MellowHype has performed a song entitled "65"
In other fields
65 miles per hour is a common speed limit on expressways in many U.S. states, primarily in the eastern and central United States. (In the western United States, a common speed limit is 70 m.p.h., and in some places it is 75 m.p.h.).
+65 is the code for international direct dial telephone calls to Singapore.
the traditional age for retirement in the United Kingdom, Germany, the United States, Canada, and several other countries.
in the U.S., the age at which a person is eligible to obtain Medicare.
CVN-65 is the designation of the U.S. Navy's first nuclear-powered aircraft carrier, the USS Enterprise (CVN-65).
65 is the minimum grade or average required to pass an exam, or a class, in some schools.
The setting of the American classic TV series Naked City (1958–1963) was the 65th Preci
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https://en.wikipedia.org/wiki/66%20%28number%29
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66 (sixty-six) is the natural number following 65 and preceding 67.
Usages of this number include:
In mathematics
66 is:
a sphenic number.
a triangular number.
a hexagonal number.
a semi-meandric number.
a semiperfect number, being a multiple of a perfect number.
an Erdős–Woods number, since it is possible to find sequences of 66 consecutive integers such that each inner member shares a factor with either the first or the last member.
palindromic and a repdigit in bases 10 (6610), 21 (3321) and 32 (2232)
In science
Astronomy
Messier object Spiral Galaxy M66, a magnitude 10.0 galaxy in the constellation Leo.
The New General Catalogue object NGC 66, a peculiar barred spiral galaxy in the constellation Cetus.
66 Maja, a carbonaceous background asteroid from the central regions of the asteroid belt.
Physics
The atomic number of dysprosium, a lanthanide.
In computing
66 (more specifically 66.667) megahertz (MHz) is a common divisor for the front side bus (FSB) speed, overall central processing unit (CPU) speed, and base bus speed. On a Core 2 CPU, and a Core 2 motherboard, the FSB is 1066 MHz (~16 × 66 MHz), the memory speed is usually 666.67 MHz (~10 × 66 MHz), and the processor speed ranges from 1.86 gigahertz (GHz) (~66 MHz × 28) to 2.93 GHz (~66 MHz × 44), in 266 MHz (~66 MHz × 4) increments.
In motor vehicle transportation
The designation of the historic U.S. Route 66, dubbed the "Mother Road" by novelist John Steinbeck, and other roads.
Phillips 66, a brand of gasoline and service station in the United States.
In religion
The total number of chapters in the Bible Book of Isaiah.
The number of verses in Chapter 3 of the Book of Lamentations in the Old Testament.
The total number of books in the Protestant edition of the Bible (Old Testament and New Testament) combined.
In Abjad numerals, The Name Of Allah (الله) numeric value is 66.
In sports
The number of the laps of the Spanish Grand Prix.
The longest field goal made in NFL history was 66 yards and kicked by Justin Tucker.
In entertainment
Cinema
Sixty Six is a 2006 British movie about a bar mitzvah in London on the day of the 1966 World Cup final.
In the Star Wars movie series, Order 66 is a prepared order to the clone troopers to kill the Jedi commanding them.
Television
Route 66 was a popular US television series on CBS from 1960 to 1964.
Video games
In the video game Fullmetal Alchemist, elusive villain Barry the Chopper is imprisoned in cell number 66, which later becomes his alias when battling the brothers at Laboratory Five.
In other fields
The international direct dialing (IDD) code for Thailand.
The number of the French department Pyrénées-Orientales.
The name of a German card game, translated from sechsundsechzig (see Sixty-six (game)).
In telecommunications a 66 block is used to organize telephone lines.
Sergio Mendes and Brasil '66 was a 1960s group.
66 WNBC radio was a popular New York radio station, which became WFAN on 1 July 1987.
Lil B has a song entitled "OMG
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https://en.wikipedia.org/wiki/67%20%28number%29
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67 (sixty-seven) is the natural number following 66 and preceding 68. It is an odd number.
In mathematics
67 is:
the 19th prime number (the next is 71).
a Chen prime.
an irregular prime.
a lucky prime.
the sum of five consecutive primes (7 + 11 + 13 + 17 + 19).
a Heegner number.
a Pillai prime since 18! + 1 is divisible by 67, but 67 is not one more than a multiple of 18.
palindromic in quinary (2325) and senary (1516).
a super-prime. (19 is prime)
an isolated prime. (65 and 69 are not prime)
In science
The atomic number of holmium, a lanthanide.
Astronomy
Messier object M67, a magnitude 7.5 open cluster in the constellation Cancer.
The New General Catalogue object NGC 67, an elliptical galaxy in the constellation Andromeda.
In music
"Car 67", a song by the band Driver 67
Chicago's song "Questions 67 and 68"
Elton John's song "Old '67" on The Captain & The Kid CD, (2006)
British rap group called 67
Rapper Drake released the song named "Star67" off his album If You're Reading This It's Too Late
In other fields
Sixty-seven is:
The registry of the U.S. Navy's aircraft carrier , named after U.S. President John F. Kennedy.
The number of the French department Bas-Rhin.
The number of counties in Alabama, Florida, and Pennsylvania.
The province/traffic code of Zonguldak Province in Turkey.
In the U.S., *67 is a common prefix-code for blocking caller ID info on the subsequent call.
In sports
Buddy Arrington's best-known NASCAR car number.
The Ottawa 67's, founded in 1967.
Pekka Koskela skated the 1000 metres in 1:07:00 (67 seconds) on 10 November 2007, a world record at the time.
The number of the laps of the German Grand Prix since 2002 if the race was held at Hockenheimring.
External links
References
Integers
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https://en.wikipedia.org/wiki/68%20%28number%29
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68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number.
In mathematics
68 is a composite number; a square-prime, of the form (p2, q) where q is a higher prime. It is the eighth of this form and the sixth of the form (22.q).
68 is a Perrin number.
It has an aliquot sum of 58 within an aliquot sequence of two composite numbers (68, 58,32,31,1,0) to the Prime in the 31-aliquot tree.
It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37. All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the Goldbach conjecture and, like it, remains unproven.
Because of the factorization of 68 as , a 68-sided regular polygon may be constructed with compass and straightedge.
There are exactly 68 10-bit binary numbers in which each bit has an adjacent bit with the same value, exactly 68 combinatorially distinct triangulations of a given triangle with four points interior to it, and exactly 68 intervals in the Tamari lattice describing the ways of parenthesizing five items. The largest graceful graph on 14 nodes has exactly 68 edges. There are 68 different undirected graphs with six edges and no isolated nodes, 68 different minimally 2-connected graphs on seven unlabeled nodes, 68 different degree sequences of four-node connected graphs, and 68 matroids on four labeled elements.
Størmer's theorem proves that, for every number p, there are a finite number of pairs of consecutive numbers that are both p-smooth (having no prime factor larger than p). For p = 13 this finite number is exactly 68. On an infinite chessboard, there are 68 squares three knight's moves away from any cell.
As a decimal number, 68 is the last two-digit number to appear for the first time in the digits of pi. It is a happy number, meaning that repeatedly summing the squares of its digits eventually leads to 1:
68 → 6 + 8 = 100 → 1 + 0 + 0 = 1.
Other uses
68 is the atomic number of erbium, a lanthanide.
In the restaurant industry, 68 may be used as a code meaning "put back on the menu", being the opposite of 86 which means "remove from the menu".
68 may also be used as slang for oral sex, based on a play on words involving the number 69.
The NCAA Division I men's basketball tournament has involved 68 teams in each edition since 2011, when the First Four round was introduced.
The NCAA Division I women's basketball tournament expanded to 68 teams in 2022, matching the men's tournament.
See also
68 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/Algebraic%20equation
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In mathematics, an algebraic equation or polynomial equation is an equation of the form , where P is a polynomial with coefficients in some field, often the field of the rational numbers.
For example, is an algebraic equation with integer coefficients and
is a multivariate polynomial equation over the rationals.
For many authors, the term algebraic equation refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the multivariate case), in which case the term polynomial equation is usually preferred.
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).
Terminology
The term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory.
Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve th roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old problem. So the term polynomial equation is generally preferred when this ambiguity may occur, specially when considering multivariate equations.
History
The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets).
Univariate algebraic equations over the rationals (i.e., with rational coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like for the positive solution of . The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant. During the Renaissance in 1545, Gerolamo Cardano
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https://en.wikipedia.org/wiki/List%20of%20mathematics-based%20methods
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This is a list of mathematics-based methods.
Adams' method (differential equations)
Akra–Bazzi method (asymptotic analysis)
Bisection method (root finding)
Brent's method (root finding)
Condorcet method (voting systems)
Coombs' method (voting systems)
Copeland's method (voting systems)
Crank–Nicolson method (numerical analysis)
D'Hondt method (voting systems)
D21 – Janeček method (voting system)
Discrete element method (numerical analysis)
Domain decomposition method (numerical analysis)
Epidemiological methods
Euler's forward method
Explicit and implicit methods (numerical analysis)
Finite difference method (numerical analysis)
Finite element method (numerical analysis)
Finite volume method (numerical analysis)
Highest averages method (voting systems)
Method of exhaustion
Method of infinite descent (number theory)
Information bottleneck method
Inverse chain rule method (calculus)
Inverse transform sampling method (probability)
Iterative method (numerical analysis)
Jacobi method (linear algebra)
Largest remainder method (voting systems)
Level-set method
Linear combination of atomic orbitals molecular orbital method (molecular orbitals)
Method of characteristics
Least squares method (optimization, statistics)
Maximum likelihood method (statistics)
Method of complements (arithmetic)
Method of moving frames (differential geometry)
Method of successive substitution (number theory)
Monte Carlo method (computational physics, simulation)
Newton's method (numerical analysis)
Pemdas method (order of operation)
Perturbation methods (functional analysis, quantum theory)
Probabilistic method (combinatorics)
Romberg's method (numerical analysis)
Runge–Kutta method (numerical analysis)
Sainte-Laguë method (voting systems)
Schulze method (voting systems)
Sequential Monte Carlo method
Simplex method
Spectral method (numerical analysis)
Variational methods (mathematical analysis, differential equations)
Welch's method
See also
Automatic basis function construction
List of graphical methods
Scientific method
Methods
Scientific method
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https://en.wikipedia.org/wiki/Circular%20segment
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In geometry, a circular segment (symbol: ⌓), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by a circular arc (of less than π radians by convention) and by the circular chord connecting the endpoints of the arc.
Formulae
Let R be the radius of the arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta (height) of the segment, d the apothem of the segment, and a the area of the segment.
Usually, chord length and height are given or measured, and sometimes the arc length as part of the perimeter, and the unknowns are area and sometimes arc length. These can't be calculated simply from chord length and height, so two intermediate quantities, the radius and central angle are usually calculated first.
Radius and central angle
The radius is:
The central angle is
Chord length and height
The chord length and height can be back-computed from radius and central angle by:
The chord length is
The sagitta is
The apothem is
Arc length and area
The arc length, from the familiar geometry of a circle, is
The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):
In terms of and ,
In terms of and ,
What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area a rapidly and asymptotically approaches . If , is a substantially good approximation.
If is held constant, and the radius is allowed to vary, then we have
As the central angle approaches π, the area of the segment is converging to the area of a semicircle, , so a good approximation is a delta offset from the latter area:
for h>.75R
As an example, the area is one quarter the circle when θ ~ 2.31 radians (132.3°) corresponding to a height of ~59.6% and a chord length of ~183% of the radius.
Etc.
The perimeter p is the arclength plus the chord length,
As a proportion of the whole area of the disc, , you have
Applications
The area formula can be used in calculating the volume of a partially-filled cylindrical tank laying horizontally.
In the design of windows or doors with rounded tops, c and h may be the only known values and can be used to calculate R for the draftsman's compass setting.
One can reconstruct the full dimensions of a complete circular object from fragments by measuring the arc length and the chord length of the fragment.
To check hole positions on a circular pattern. Especially useful for quality checking on machined products.
For calculating the area or centroid of a planar shape that contains circular segments.
See also
Chord (geometry)
Spherical cap
Circular sector
References
External links
Definition of a circula
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https://en.wikipedia.org/wiki/Brauer%20group
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In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.
The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
Construction
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A such that A is a simple ring and the center of A is equal to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras.
For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – or – is a CSA over the reals, but not a division algebra (if n > 1).
We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n, D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying with for all positive integers m and n, we get the Brauer equivalence relation on CSAs over K. The elements of the Brauer group are the Brauer equivalence classes of CSAs over K.
Given central simple algebras A and B, one can look at their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterization: a central simple algebra A over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K. This result also shows that the dimension of a central simple algebra A as a K-vector space is always a square. The degree of A is defined to be the square root of its dimension.
As a result, the isomorphism classes of CSAs over K form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse of an algebra A is given by its opposite algebra Aop (the opposite ring with the same action by K since the image of is in the center of A). Explicitly, for a CSA A we have , where n is the degree of A over K.
The Brauer group of any field is a torsion group. In more detail, define the period of a central simple algebra A over K to be its order as an element of the Brauer group. Define the index of A to be the degree of the division algebra that is Brauer equivalent to A. Then the period of A divides the index of A (and hence is finite).
Examples
In the following cases, every finite-di
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https://en.wikipedia.org/wiki/Central%20simple%20algebra
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In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.)
For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below).
Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.
Properties
According to the Artin–Wedderburn theorem a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Hence, there is a unique division algebra in each Brauer equivalence class.
Every automorphism of a central simple algebra is an inner automorphism (this follows from the Skolem–Noether theorem).
The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension. The Schur index of a central simple algebra is the degree of the equivalent division algebra: it depends only on the Brauer class of the algebra.
The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index, and the two numbers are composed of the same prime factors.
If S is a simple subalgebra of a central simple algebra A then dimF S divides dimF A.
Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra.
If D is a central division algebra over K for which the index has prime factorisation
then D has a tensor product decomposition
where each component Di is a central division algebra of index , and the components are uniquely determined up to isomorphism.
Splitting field
We call a field E a splitting field for A over K if A⊗E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed, in the case when A is a division algebra, then a maximal subfield of A is a splitting field. In general by theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of K
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https://en.wikipedia.org/wiki/Piecewise
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In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself.
A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example.
Notation and interpretation
Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pairwise disjoint, i.e. form a partition of the domain. In order for the overall function to be called "piecewise", the subdomains are usually required to be intervals (some may be degenerated intervals, i.e. single points or unbounded intervals). For bounded intervals, the number of subdomains is required to be finite, for unbounded intervals it is often only required to be locally finite. For example, consider the piecewise definition of the absolute value function:
For all values of less than zero, the first sub-function () is used, which negates the sign of the input value, making negative numbers positive. For all values of greater than or equal to zero, the second sub-function is used, which evaluates trivially to the input value itself.
The following table documents the absolute value function at certain values of :
In order to evaluate a piecewise-defined function at a given input value, the appropriate subdomain needs to be chosen in order to select the correct sub-function—and produce the correct output value.
Continuity and differentiability of piecewise-defined functions
A piecewise-defined function is continuous on a given interval in its domain if the following conditions are met:
its sub-functions are continuous on the corresponding intervals (subdomains),
there is no discontinuity at an endpoint of any subdomain within that interval.
The pictured function, for example, is piecewise-continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at . The filled circle indicates that the value of the right sub-function is used in this position.
For a piecewise-defined function to be differentiable on a given interval in its domain, the following conditions have to fulfilled in addition to those for continuity above:
its sub-functions are differentiable on the corresponding open intervals,
the one-sided derivatives exist at all intervals' endpoints,
at the points where two subintervals touch, the
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https://en.wikipedia.org/wiki/Closed%20and%20exact%20differential%20forms
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In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
For an exact form α, for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α. Since the exterior derivative of a closed form is zero, β is not unique, but can be modified by the addition of any closed form of degree one less than that of α.
Because , every exact form is necessarily closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.
Examples
A simple example of a form that is closed but not exact is the 1-form given by the derivative of argument on the punctured plane Since is not actually a function (see the next paragraph) is not an exact form. Still, has vanishing derivative and is therefore closed.
Note that the argument is only defined up to an integer multiple of since a single point can be assigned different arguments etc. We can assign arguments in a locally consistent manner around but not in a globally consistent manner. This is because if we trace a loop from counterclockwise around the origin and back to the argument increases by Generally, the argument changes by
over a counter-clockwise oriented loop
Even though the argument is not technically a function, the different local definitions of at a point differ from one another by constants. Since the derivative at only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative
The upshot is that is a one-form on that is not actually the derivative of any well-defined function We say that is not exact. Explicitly, is given as:
which by inspection has derivative zero. Because has vanishing derivative, we say that it is closed.
This form generates the de Rham cohomology group meaning that any closed form is the sum of an exact form and a multiple of where accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a potential function) being the derivative of a globally defined function.
Examples in low dimensions
Differential forms in and were well known in the mathematical physics of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element , so that it is the 1-form
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https://en.wikipedia.org/wiki/Normal%20function
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In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the case that f(γ) = sup {f(ν) : ν < γ}.
For all ordinals α < β, it is the case that f(α) < f(β).
Examples
A simple normal function is given by (see ordinal arithmetic). But is not normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when λ is a limit ordinal. If β is a fixed ordinal, then the functions , (for ), and (for ) are all normal.
More important examples of normal functions are given by the aleph numbers , which connect ordinal and cardinal numbers, and by the beth numbers .
Properties
If f is normal, then for any ordinal α,
f(α) ≥ α.
Proof: If not, choose γ minimal such that f(γ) < γ. Since f is strictly monotonically increasing, f(f(γ)) < f(γ), contradicting minimality of γ.
Furthermore, for any non-empty set S of ordinals, we have
f(sup S) = sup f(S).
Proof: "≥" follows from the monotonicity of f and the definition of the supremum. For "≤", set δ = sup S and consider three cases:
if δ = 0, then S = {0} and sup f(S) = f(0);
if δ = ν + 1 is a successor, then there exists s in S with ν < s, so that δ ≤ s. Therefore, f(δ) ≤ f(s), which implies f(δ) ≤ sup f(S);
if δ is a nonzero limit, pick any ν < δ, and an s in S such that ν < s (possible since δ = sup S). Therefore, f(ν) < f(s) so that f(ν) < sup f(S), yielding f(δ) = sup {f(ν) : ν < δ} ≤ sup f(S), as desired.
Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function f' : Ord → Ord, called the derivative of f, such that f' (α) is the α-th fixed point of f. For a hierarchy of normal functions, see Veblen functions.
Notes
References
.
Set theory
Ordinal numbers
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https://en.wikipedia.org/wiki/Fixed-point%20lemma%20for%20normal%20functions
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The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908.
Background and formal statement
A normal function is a class function from the class Ord of ordinal numbers to itself such that:
is strictly increasing: whenever .
is continuous: for every limit ordinal (i.e. is neither zero nor a successor), .
It can be shown that if is normal then commutes with suprema; for any nonempty set of ordinals,
.
Indeed, if is a successor ordinal then is an element of and the equality follows from the increasing property of . If is a limit ordinal then the equality follows from the continuous property of .
A fixed point of a normal function is an ordinal such that .
The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal , there exists an ordinal such that and .
The continuity of the normal function implies the class of fixed points is closed (the supremum of any subset of the class of fixed points is again a fixed point). Thus the fixed point lemma is equivalent to the statement that the fixed points of a normal function form a closed and unbounded class.
Proof
The first step of the proof is to verify that for all ordinals and that commutes with suprema. Given these results, inductively define an increasing sequence by setting , and for . Let , so . Moreover, because commutes with suprema,
The last equality follows from the fact that the sequence increases.
As an aside, it can be demonstrated that the found in this way is the smallest fixed point greater than or equal to .
Example application
The function f : Ord → Ord, f(α) = ωα is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ωθ. In fact, the lemma shows that there is a closed, unbounded class of such θ.
References
Ordinal numbers
Normal Functions
Lemmas in set theory
Articles containing proofs
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https://en.wikipedia.org/wiki/Alternating
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Alternating may refer to:
Mathematics
Alternating algebra, an algebra in which odd-grade elements square to zero
Alternating form, a function formula in algebra
Alternating group, the group of even permutations of a finite set
Alternating knot, a knot or link diagram for which the crossings alternate under, over, under, over, as one travels along each component of the link
Alternating map, a multilinear map that is zero whenever any two of its arguments are equal
Alternating operator, a multilinear map in algebra
Alternating permutation, a type of permutation studied in combinatorics
Alternating series, an infinite series in which the signs of the general terms alternate between positive and negative
Electronics
Alternating current, a flow of electric charge that periodically reverses direction
Other
Alternating turns, the process by which people in a conversation decide who is to speak next
See also
Alternate bass
Alternative (disambiguation)
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https://en.wikipedia.org/wiki/Bayesian%20statistics
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Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation that views probability as the limit of the relative frequency of an event after many trials.
Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability of an event based on data as well as prior information or beliefs about the event or conditions related to the event. For example, in Bayesian inference, Bayes' theorem can be used to estimate the parameters of a probability distribution or statistical model. Since Bayesian statistics treats probability as a degree of belief, Bayes' theorem can directly assign a probability distribution that quantifies the belief to the parameter or set of parameters.
Bayesian statistics is named after Thomas Bayes, who formulated a specific case of Bayes' theorem in a paper published in 1763. In several papers spanning from the late 18th to the early 19th centuries, Pierre-Simon Laplace developed the Bayesian interpretation of probability. Laplace used methods that would now be considered Bayesian to solve a number of statistical problems. Many Bayesian methods were developed by later authors, but the term was not commonly used to describe such methods until the 1950s. During much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many Bayesian methods required much computation to complete, and most methods that were widely used during the century were based on the frequentist interpretation. However, with the advent of powerful computers and new algorithms like Markov chain Monte Carlo, Bayesian methods have seen increasing use within statistics in the 21st century.
Bayes' theorem
Bayes' theorem is used in Bayesian methods to update probabilities, which are degrees of belief, after obtaining new data. Given two events and , the conditional probability of given that is true is expressed as follows:
where . Although Bayes' theorem is a fundamental result of probability theory, it has a specific interpretation in Bayesian statistics. In the above equation, usually represents a proposition (such as the statement that a coin lands on heads fifty percent of the time) and represents the evidence, or new data that is to be taken into account (such as the result of a series of coin flips). is the prior probability of which expresses one's beliefs about before evidence is taken into account. The prior probability may also quantify prior knowledge or information about . is the likelihood fu
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https://en.wikipedia.org/wiki/Finitary
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In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values.
In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of infinitary logic.
Finitary argument
A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper.
By contrast, infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary disjunction.
History
Logicians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language without semantics. In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes."
The stress on finiteness came from the idea that human mathematical thought is based on a finite number of principles and all the reasonings follow essentially one rule: the modus ponens. The project was to fix a finite number of symbols (essentially the numerals 1, 2, 3, ... the letters of alphabet and some special symbols like "+", "⇒", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some rules of inference which would model the way humans make conclusions. From these, regardless of the semantic interpretation of the symbols the remaining theorems should follow formally using only the stated rules (which make mathematics look like a game with symbols more than a science) without the need to rely on ingenuity. The hope was to prove that from these axioms and rules all the theorems of mathematics could be deduced. That aim is known as logicism.
Notes
External links
Stanford Encyclopedia of Philosophy entry on Infinitary Logic
Mathematical logic
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https://en.wikipedia.org/wiki/Trefoil%20knot
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In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
The trefoil knot is named after the three-leaf clover (or trefoil) plant.
Descriptions
The trefoil knot can be defined as the curve obtained from the following parametric equations:
The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus :
Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation.
In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic).
If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot.
Symmetry
The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.)
Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve.
Nontriviality
The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil.
Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants.
Classification
In knot theory, the trefoil is the first nontrivial knot, and is the only knot with crossing number three. It is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 4 6 2, and the Conway notation is [3].
The trefoil can be described as the (2,3)-torus knot. It is also the knot obtained by closing the braid σ13.
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https://en.wikipedia.org/wiki/New%20Math
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New Mathematics or New Math was a dramatic but temporary change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries and elsewhere, during the 1950s1970s.
Overview
In 1957, the U.S. National Science Foundation funded the development of several new curricula in the sciences, such as the Physical Science Study Committee high school physics curriculum, Biological Sciences Curriculum Study in biology, and CHEM Study in chemistry. Several mathematics curriculum development efforts were also funded as part of the same initiative, such as the Madison Project, School Mathematics Study Group, and University of Illinois Committee on School Mathematics.
These curricula were quite diverse, yet shared the idea that children's learning of arithmetic algorithms would last past the exam only if memorization and practice were paired with teaching for comprehension. More specifically, elementary school arithmetic beyond single digits makes sense only on the basis of understanding place value. This goal was the reason for teaching arithmetic in bases other than ten in the New Math, despite critics' derision: In that unfamiliar context, students couldn't just mindlessly follow an algorithm, but had to think why the place value of the "hundreds" digit in base seven is 49. Keeping track of non-decimal notation also explains the need to distinguish numbers (values) from the numerals that represent them.
Topics introduced in the New Math include set theory, modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, and abstract algebra.
All of the New Math projects emphasized some form of discovery learning. Students worked in groups to invent theories about problems posed in the textbooks. Materials for teachers described the classroom as "noisy." Part of the job of the teacher was to move from table to table assessing the theory that each group of students had developed and "torpedo" wrong theories by providing counterexamples. For that style of teaching to be tolerable for students, they had to experience the teacher as a colleague rather than as an adversary or as someone concerned mainly with grading. New Math workshops for teachers, therefore, spent as much effort on the pedagogy as on the mathematics.
Criticism
Parents and teachers who opposed the New Math in the U.S. complained that the new curriculum was too far outside of students' ordinary experience and was not worth taking time away from more traditional topics, such as arithmetic. The material also put new demands on teachers, many of whom were required to teach material they did not fully understand. Parents were concerned that they did not understand what their children were learning and could not help them with their studies. In an effort to learn the material, many parents attended their children's classes. In the end, it was concluded that the experiment was not working, and New Math f
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https://en.wikipedia.org/wiki/Italian%20National%20Institute%20of%20Statistics
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The Italian National Institute of Statistics (; Istat) is the primary source of official statistics in Italy. The institute conducts a variety of activities, including the census of population, economic censuses, and numerous social, economic, and environmental surveys and analyses. Istat is the largest producer of statistical information in Italy and is actively involved in the European Statistical System, which is overseen by Eurostat.
History
The Italian National Institute of Statistics () was established by Legislative decree no. 1162 on 9 July, 1926, as the Central Institute of Statistics () in order to replace the General Statistics Division of the Ministry of Agriculture. Corrado Gini was established as the first director of the institute, under the authority of the head of state.
The institute, with a staff of about 170 workers, was charged with publishing the data of the 6th general population census, generated by updating the figures from previous censuses carried out by the General Statistics Division up until 1921. After ramping up activities in the early 1930s, national statistics operations in Italy suffered serious setbacks due to economic sanctions imposed as a result of the Second Italo-Ethiopian War, which essentially halted any publication of economic or financial data. The figures that had been already collected but not reported during this period were eventually published in 1937, although this activity was ceased only two years afterwards.
After the outbreak of the Second World War, publications decreased due to the lack of personnel, most of whom had been called up for military service. This led to a postponement of the 9th population census, which was instead held in 1951. Due to the Armistice of Cassibile in 1943, the institute headquarters were relocated within the territory of the Italian Social Republic.
During the late 1940s, the archives were recovered and transferred back to Rome, allowing the institute to fully resume its activities. With post-war reconstruction underway, the institute mainly focused on collecting new data concerning national development that eventually lead to the publication of the volume " (Studies on National Revenue) in 1950.
Legislative decree no. 322, published on 6 September, 1989, established the National Statistics System () and changed the name of the institution to the National Institute of Statistics (), without changing its acronym, which remained Istat.
Institute publications are released under a Creative Commons "Attribution" (CC BY) license.
Organization
The administration of the institute is as follows:
President, appointed by the President of Italy upon the proposal of the President of the Council of Ministers (Prime Minister) after the approval of the Council of Ministers. Their term lasts for four years and can be renewed only once. They are responsible for the performance of the institute and its technical and scientific coordination.
Policy-making and Statistics In
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https://en.wikipedia.org/wiki/Almost%20periodic%20function
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In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.
Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in.
Motivation
There are several inequivalent definitions of almost periodic functions. The first was given by Harald Bohr. His interest was initially in finite Dirichlet series. In fact by truncating the series for the Riemann zeta function ζ(s) to make it finite, one gets finite sums of terms of the type
with s written as (σ + it) – the sum of its real part σ and imaginary part it. Fixing σ, so restricting attention to a single vertical line in the complex plane, we can see this also as
Taking a finite sum of such terms avoids difficulties of analytic continuation to the region σ < 1. Here the 'frequencies' log n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n are multiplicatively independent – which comes down to their prime factorizations).
With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematical analysis was applied to discuss the closure of this set of basic functions, in various norms.
The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner and others in the 1920s and 1930s.
Uniform or Bohr or Bochner almost periodic functions
Bohr (1925) defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm
(on bounded functions f on R). In other words, a function f is uniformly almost periodic if for every ε > 0 there is a finite linear combination of sine and cosine waves that is of distance less than ε from f with respect to the uniform norm. Bohr proved that this definition was equivalent to the existence of a relatively dense set of ε almost-periods, for all ε > 0: that is, translations T(ε) = T of the variable t making
An alternative definition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to
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https://en.wikipedia.org/wiki/Solomonoff%27s%20theory%20of%20inductive%20inference
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Solomonoff's theory of inductive inference is a mathematical theory of induction introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory.
Solomonoff's induction naturally formalizes Occam's razor by assigning larger prior credences to theories that require a shorter algorithmic description.
Origin
Philosophical
The theory is based in philosophical foundations, and was founded by Ray Solomonoff around 1960. It is a mathematically formalized combination of Occam's razor and the Principle of Multiple Explanations. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.
Principle
Solomonoff's induction has been argued to be the computational formalization of pure Bayesianism. To understand, recall that Bayesianism derives the posterior probability of a theory given data by applying Bayes rule, which yields , where theories are alternatives to theory . For this equation to make sense, the quantities and must be well-defined for all theories and . In other words, any theory must define a probability distribution over observable data . Solomonoff's induction essentially boils down to demanding that all such probability distributions be computable.
Interestingly, the set of computable probability distributions is a subset of the set of all programs, which is countable. Similarly, the sets of observable data considered by Solomonoff were finite. Without loss of generality, we can thus consider that any observable data is a finite bit string. As a result, Solomonoff's induction can be defined by only invoking discrete probability distributions.
Solomonoff's induction then allows to make probabilistic predictions of future data , by simply obeying the laws of probability. Namely, we have . This quantity can be interpreted as the average predictions of all theories given past data , weighted by their posterior credences .
Mathematical
The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to one (as per the definition of probability) thus the probabilities must roughly decrease as we enumerate the infinite set of all programs, otherwise S will be strictly greater than one. To be more precise, for every > 0, there is some length l such that the probabili
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https://en.wikipedia.org/wiki/Exact%20differential
|
In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).
An exact differential is sometimes also called a total differential, or a full differential, or, in the study of differential geometry, it is termed an exact form.
The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics.
Overview
Definition
Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type
is called a differential form. This form is called exact on an open domain in space if there exists some differentiable scalar function defined on such that
throughout , where are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some open domain of a space, a differential form is an exact differential if it is equal to the general differential of a differentiable function in an orthogonal coordinate system.
Note: In this mathematical expression, the subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the partial derivative, these subscripts are not required, but they are explicitly shown here as reminders.
Integral path independence
The exact differential for a differentiable scalar function defined in an open domain is equal to , where is the gradient of , represents the scalar product, and is the general differential displacement vector, if an orthogonal coordinate system is used. If is of differentiability class (continuously differentiable), then is a conservative vector field for the corresponding potential by the definition. For three dimensional spaces, expressions such as and can be made.
The gradient theorem states
that does not depend on which integral path between the given path endpoints and is chosen. So it is concluded that the integral of an exact differential is independent of the choice of an integral path between given path endpoints (path independence).
For three dimensional spaces, if defined on an open domain is of differentiability class (equivalently is of ), then this integral path independence can also be proved by using the vector calculus identity and the Stokes' theorem.
for a simply closed loop with the smooth oriented surface in it. If the open domain is simply connected open space (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a vector f
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https://en.wikipedia.org/wiki/Teacher%20in%20Space%20Project
|
The Teacher in Space Project (TISP) was a NASA program announced by Ronald Reagan in 1984 designed to inspire students, honor teachers, and spur interest in mathematics, science, and space exploration. The project would carry teachers into space as Payload Specialists (non-astronaut civilians), who would return to their classrooms to share the experience with their students.
NASA cancelled the program in 1990, following the death of its first participant, Christa McAuliffe, in the Space Shuttle Challenger disaster (STS-51-L) on January 28, 1986. NASA replaced Teachers in Space in 1998 with the Educator Astronaut Project, which required its participants to become astronaut Mission Specialists. The first Educator Astronauts were selected as part of NASA Astronaut Group 19 in 2004.
Barbara Morgan, who was selected as a mission specialist as part of NASA Astronaut Group 17 in 1998, has often been incorrectly referred to as an Educator Astronaut. However, she was selected as a mission specialist before the Educator Astronaut Project.
NASA programs
TISP was announced by President Ronald Reagan on August 27, 1984. Not members of NASA's Astronaut Corps, the teachers would fly as Payload Specialists and return to their classrooms after flight. More than 40,000 applications were mailed to interested teachers while 11,000 teachers sent completed applications to NASA. Each application included a potential lesson that would be taught from space while on the Space Shuttle. The applications were sorted and then sent to the various State Departments of Education, who were then responsible for narrowing down their state applicants to a final set of two each. These 114 applicants were notified of their selections and were gathered together for further selection processes down to ten finalists. These were then trained for a time, and in 1985 NASA selected Christa McAuliffe to be the first teacher in space, with Barbara Morgan as her backup. McAuliffe was a high school social studies teacher from Concord, New Hampshire. She planned to teach two 15-minute lessons from the Space Shuttle.
McAuliffe died in the Space Shuttle Challenger disaster (STS-51-L) on January 28, 1986. After the accident, Reagan spoke on national television and assured the nation that the Teacher in Space program would continue. "We'll continue our quest in space", he said. "There will be more shuttle flights and more shuttle crews and, yes, more volunteers, more civilians, more teachers in space. Nothing ends here; our hopes and our journeys continue." However, NASA decided in 1990 that spaceflight was still too dangerous to risk the lives of civilian teachers, and eliminated the Teacher in Space project. Morgan returned to teaching in Idaho and later became a mission specialist on STS-118.
Educator Astronaut Project
In January 1998, NASA replaced the Teacher In Space project with the Educator Astronaut Project. Instead of training teachers for five months as Payload Specialists who wou
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https://en.wikipedia.org/wiki/Time%20series
|
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series.
Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility).
Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).
Methods for analysis
Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.
Additionally, t
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https://en.wikipedia.org/wiki/Clausen%20function
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In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.
The Clausen function of order 2 – often referred to as the Clausen function, despite being but one of a class of many – is given by the integral:
In the range the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation:
The Clausen functions, as a class of functions, feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet L-series.
Basic properties
The Clausen function (of order 2) has simple zeros at all (integer) multiples of since if is an integer, then
It has maxima at
and minima at
The following properties are immediate consequences of the series definition:
See .
General definition
More generally, one defines the two generalized Clausen functions:
which are valid for complex z with Re z >1. The definition may be extended to all of the complex plane through analytic continuation.
When z is replaced with a non-negative integer, the standard Clausen functions are defined by the following Fourier series:
N.B. The SL-type Clausen functions have the alternative notation and are sometimes referred to as the Glaisher–Clausen functions (after James Whitbread Lee Glaisher, hence the GL-notation).
Relation to the Bernoulli polynomials
The SL-type Clausen function are polynomials in , and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli polynomials:
Setting in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:
where the Bernoulli polynomials are defined in terms of the Bernoulli numbers by the relation:
Explicit evaluations derived from the above include:
Duplication formula
For , the duplication formula can be proven directly from the integral definition (see also for the result – although no proof is given):
Denoting Catalan's constant by , immediate consequences of the duplication formula include the relations:
For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace with the dummy variable , and integrate over the interval Applying the same process repeatedly yi
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https://en.wikipedia.org/wiki/Dawson%20function
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In mathematics, the Dawson function or Dawson integral
(named after H. G. Dawson)
is the one-sided Fourier–Laplace sine transform of the Gaussian function.
Definition
The Dawson function is defined as either:
also denoted as or or alternatively
The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function,
It is closely related to the error function erf, as
where erfi is the imaginary error function, Similarly,
in terms of the real error function, erf.
In terms of either erfi or the Faddeeva function the Dawson function can be extended to the entire complex plane:
which simplifies to
for real
For near zero, For large, More specifically, near the origin it has the series expansion
while for large it has the asymptotic expansion
More precisely
where is the double factorial.
satisfies the differential equation
with the initial condition Consequently, it has extrema for
resulting in x = ±0.92413887... (), F(x) = ±0.54104422... ().
Inflection points follow for
resulting in x = ±1.50197526... (), F(x) = ±0.42768661... (). (Apart from the trivial inflection point at )
Relation to Hilbert transform of Gaussian
The Hilbert transform of the Gaussian is defined as
P.V. denotes the Cauchy principal value, and we restrict ourselves to real can be related to the Dawson function as follows. Inside a principal value integral, we can treat as a generalized function or distribution, and use the Fourier representation
With we use the exponential representation of and complete the square with respect to to find
We can shift the integral over to the real axis, and it gives Thus
We complete the square with respect to and obtain
We change variables to
The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives
where is the Dawson function as defined above.
The Hilbert transform of is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let
Introduce
The th derivative is
We thus find
The derivatives are performed first, then the result evaluated at A change of variable also gives Since we can write where and are polynomials. For example, Alternatively, can be calculated using the recurrence relation (for )
See also
References
External links
gsl_sf_dawson in the GNU Scientific Library
libcerf, numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision. It is based on the Faddeeva function as implemented in the MIT Faddeeva Package
Dawson's Integral (at Mathworld)
Error functions
Gaussian function
Special functions
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https://en.wikipedia.org/wiki/Debye%20function
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In mathematics, the family of Debye functions is defined by
The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model.
Mathematical properties
Relation to other functions
The Debye functions are closely related to the polylogarithm.
Series expansion
They have the series expansion
where is the n-th Bernoulli number.
Limiting values
If is the gamma function and is the Riemann zeta function, then, for ,
Derivative
The derivative obeys the relation
where is the Bernoulli function.
Applications in solid-state physics
The Debye model
The Debye model has a density of vibrational states
for
with the Debye frequency ωD.
Internal energy and heat capacity
Inserting g into the internal energy
with the Bose–Einstein distribution
.
one obtains
.
The heat capacity is the derivative thereof.
Mean squared displacement
The intensity of X-ray diffraction or neutron diffraction at wavenumber q is given by
the Debye-Waller factor or the Lamb-Mössbauer factor.
For isotropic systems it takes the form
).
In this expression, the mean squared displacement refers to just once Cartesian component
ux of the vector u that describes the displacement of atoms from their equilibrium positions.
Assuming harmonicity and developing into normal modes,
one obtains
Inserting the density of states from the Debye model, one obtains
.
From the above power series expansion of follows that the mean square displacement at high temperatures is linear in temperature
.
The absence of indicates that this is a classical result. Because goes to zero for it follows that for
(zero-point motion).
References
Further reading
"Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xn
Implementations
Fortran 77 code
Fortran 90 version
C version of the GNU Scientific Library
Special functions
Peter Debye
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https://en.wikipedia.org/wiki/Legendre%20form
|
In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity (the ellipse being defined parametrically by , ).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
Definition
The incomplete elliptic integral of the first kind is defined as,
the second kind as
and the third kind as
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik as well as Numerical Recipes. The choice of sign is that of Abramowitz and Stegun as well as Gradshteyn and Ryzhik, but corresponds to the of Numerical Recipes.
The respective complete elliptic integrals are obtained by setting the amplitude, , the upper limit of the integrals, to .
The Legendre form of an elliptic curve is given by
Numerical evaluation
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus towards zero, while increasing the amplitude . Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of approaching zero or one, the integral is readily evaluated.
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.
References
See also
Carlson symmetric form
Special functions
|
https://en.wikipedia.org/wiki/Carlson%20symmetric%20form
|
In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.
The Carlson elliptic integrals are:
Since and are special cases of and , all elliptic integrals can ultimately be evaluated in terms of just and .
The term symmetric refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of is the same for any permutation of its arguments, and the value of is the same for any permutation of its first three arguments.
The Carlson elliptic integrals are named after Bille C. Carlson (1924-2013).
Relation to the Legendre forms
Incomplete elliptic integrals
Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms:
(Note: the above are only valid for and )
Complete elliptic integrals
Complete elliptic integrals can be calculated by substituting φ = π:
Special cases
When any two, or all three of the arguments of are the same, then a substitution of renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions.
Similarly, when at least two of the first three arguments of are the same,
Properties
Homogeneity
By substituting in the integral definitions for any constant , it is found that
Duplication theorem
where .
where and
Series Expansion
In obtaining a Taylor series expansion for or it proves convenient to expand about the mean value of the several arguments. So for , letting the mean value of the arguments be , and using homogeneity, define , and by
that is etc. The differences , and are defined with this sign (such that they are subtracted), in order to be in agreement with Carlson's papers. Since is symmetric under permutation of , and , it is also symmetric in the quantities , and . It follows that both the integrand of and its integral can be expressed as functions of the elementary symmetric polynomials in , and which are
Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term...
The advantage of expanding about the mean value of the arguments is now apparent; it reduces identically to zero, and so eliminates all terms involving - which otherwise would be the most numerous.
An ascending series for may be found in a similar way. There is a slight difficulty because is not fully symmetric; its dependence on its fourth argument, , is different from its dependence on , and . This is overcome by treating as a fully symmetric function of five arguments, two of which happen to have the same value . The mean value of the arguments is therefore taken to be
and the differences , and defined by
The elementary symmetric polynomials in
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https://en.wikipedia.org/wiki/Complete%20Fermi%E2%80%93Dirac%20integral
|
In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by
This equals
where is the polylogarithm.
Its derivative is
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for appears in the literature, for instance some authors omit the factor . The definition used here matches that in the NIST DLMF.
Special values
The closed form of the function exists for j = 0:
For x = 0, the result reduces to
where is the Dirichlet eta function.
See also
Incomplete Fermi–Dirac integral
Gamma function
Polylogarithm
References
External links
GNU Scientific Library - Reference Manual
Fermi-Dirac integral calculator for iPhone/iPad
Notes on Fermi-Dirac Integrals
Section in NIST Digital Library of Mathematical Functions
npplus: Python package that provides (among others) Fermi-Dirac integrals and inverses for several common orders.
Wolfram's MathWorld: Definition given by Wolfram's MathWorld.
Special functions
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https://en.wikipedia.org/wiki/Incomplete%20Fermi%E2%80%93Dirac%20integral
|
In mathematics, the incomplete Fermi–Dirac integral for an index j is given by
This is an alternate definition of the incomplete polylogarithm.
See also
Complete Fermi–Dirac integral
External links
GNU Scientific Library - Reference Manual
Special functions
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https://en.wikipedia.org/wiki/Polygamma%20function
|
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers defined as the th derivative of the logarithm of the gamma function:
Thus
holds where is the digamma function and is the gamma function. They are holomorphic on . At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function.
Integral representation
When and , the polygamma function equals
where is the Hurwitz zeta function.
This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function.
Setting in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the case above but which has an extra term .
Recurrence relation
It satisfies the recurrence relation
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
and
for all , where is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their recurrence relation and one given function-value, say , except in the case where the additional condition of strict monotonicity on is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case must be treated differently because is not normalizable at infinity (the sum of the reciprocals doesn't converge).
Reflection relation
where is alternately an odd or even polynomial of degree with integer coefficients and leading coefficient . They obey the recursion equation
Multiplication theorem
The multiplication theorem gives
and
for the digamma function.
Series representation
The polygamma function has the series representation
which holds for integer values of and any complex not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as
This relation can for example be used to compute the special values
Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.
One more series may be permitted for the polygamma functions. As given by Schlömilch,
This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:
Now, the natural logarithm of the gamma function is easily representable:
Finally, we arrive at a summation representation for the polygamma function:
Where is the Kronecker delta.
Also the Lerch transcendent
can be denoted in terms of polygamma function
Taylor series
The Taylor series at is
and
which converges for . Here, is the Riema
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https://en.wikipedia.org/wiki/Digamma%20function
|
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , and it asymptotically behaves as
for large arguments () in the sector with some infinitesimally small positive constant .
The digamma function is often denoted as or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
Relation to harmonic numbers
The gamma function obeys the equation
Taking the derivative with respect to gives:
Dividing by or the equivalent gives:
or:
Since the harmonic numbers are defined for positive integers as
the digamma function is related to them by
where and is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values
Integral representations
If the real part of is positive then the digamma function has the following integral representation due to Gauss:
Combining this expression with an integral identity for the Euler–Mascheroni constant gives:
The integral is Euler's harmonic number , so the previous formula may also be written
A consequence is the following generalization of the recurrence relation:
An integral representation due to Dirichlet is:
Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of .
This formula is also a consequence of Binet's first integral for the gamma function. The integral may be recognized as a Laplace transform.
Binet's second integral for the gamma function gives a different formula for which also gives the first few terms of the asymptotic expansion:
From the definition of and the integral representation of the gamma function, one obtains
with .
Infinite product representation
The function is an entire function, and it can be represented by the infinite product
Here is the kth zero of (see below), and is the Euler–Mascheroni constant.
Note: This is also equal to due to the definition of the digamma function: .
Series representation
Series formula
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16):
Equivalently,
Evaluation of sums of rational functions
The above identity can be used to evaluate sums of the form
where and are polynomials of .
Performing partial fraction on in the complex field, in the case when all roots of are simple roots,
For the series to converge,
otherwise the series will be greater than the harmonic series and thus diverge. Hence
and
With the series expansion of higher rank polygamma function a generalized formula can be given as
provided the series on the left converges.
Taylor series
The digamma has a rational zeta series, given by the Taylor series at . This is
|
https://en.wikipedia.org/wiki/Transport%20function
|
In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by
Note that
See also
Incomplete gamma function
Special functions
Transportation theory
|
https://en.wikipedia.org/wiki/Synchrotron%20function
|
In mathematics the synchrotron functions are defined as follows (for x ≥ 0):
First synchrotron function
Second synchrotron function
where Kj is the modified Bessel function of the second kind.
Use in astrophysics
In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.
References
Further reading
Special functions
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