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https://en.wikipedia.org/wiki/Hurwitz%20zeta%20function
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In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by
This series is absolutely convergent for the given values of and and can be extended to a meromorphic function defined for all . The Riemann zeta function is . The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.
Integral representation
The Hurwitz zeta function has an integral representation
for and (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing
and then interchanging the sum and integral.
The integral representation above can be converted to a contour integral representation
where is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation . Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.
The contour integral representation provides an analytic continuation of to all . At , it has a simple pole with residue .
Hurwitz's formula
The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:
valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:
Hurwitz's formula can also be expressed as
(for Re(s) < 0 and 0 < a ≤ 1).
Hurwitz's formula has a variety of different proofs. One proof uses the contour integration representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral
(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.
Functional equation for rational a
When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers ,
holds for all values of s.
This functional equation can be written as another equivalent form:
.
Some finite sums
Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form
where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.
Series representation
A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:
This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of ; that is,
where Δ is the forward difference operator. Thus, one may write:
Taylor series
The partial derivative of the zeta in the second argument is a shift:
Thus, the Taylor series can be written as:
Alternatively,
with .
Closely related is the Stark–Keiper formula
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https://en.wikipedia.org/wiki/Eta%20function
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In mathematics, eta function may refer to:
The Dirichlet eta function η(s), a Dirichlet series
The Dedekind eta function η(τ), a modular form
The Weierstrass eta function η(w) of a lattice vector
The eta function η(s) used to define the eta invariant
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https://en.wikipedia.org/wiki/Functor%20category
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In category theory, a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons:
many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;
every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
Definition
Suppose is a small category (i.e. the objects and morphisms form a set rather than a proper class) and is an arbitrary category.
The category of functors from to , written as Fun(, ), Funct(,), , or , has as objects the covariant functors from to ,
and as morphisms the natural transformations between such functors. Note that natural transformations can be composed:
if is a natural transformation from the functor to the functor , and
is a natural transformation from the functor to the functor , then the composition defines a natural transformation
from to . With this composition of natural transformations (known as vertical composition, see natural transformation),
satisfies the axioms of a category.
In a completely analogous way, one can also consider the category of all contravariant functors from to ; we write this as Funct().
If and are both preadditive categories (i.e. their morphism sets are abelian groups and the composition of morphisms is bilinear),
then we can consider the category of all additive functors from to , denoted by Add(,).
Examples
If is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from to essentially consists of a family of objects of , indexed by ; the functor category can be identified with the corresponding product category: its elements are families of objects in and its morphisms are families of morphisms in .
An arrow category (whose objects are the morphisms of , and whose morphisms are commuting squares in ) is just , where 2 is the category with two objects and their identity morphisms as well as an arrow from one object to the other (but not another arrow back the other way).
A directed graph consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex. The category of all directed graphs is thus nothing but the functor category , where is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the category of sets.
Any group can be considered as a one-object category in which every morphism is invertible. The category of all -sets is the same as the functor category Set. Natural transformations are -maps.
Similar to the previous e
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https://en.wikipedia.org/wiki/Morera%27s%20theorem
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In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies
for every closed piecewise C1 curve in D must be holomorphic on D.
The assumption of Morera's theorem is equivalent to f locally having an antiderivative on D.
The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.
The standard counterexample is the function , which is holomorphic on C − {0}. On any simply connected neighborhood U in C − {0}, 1/z has an antiderivative defined by , where . Because of the ambiguity of θ up to the addition of any integer multiple of 2, any continuous choice of θ on U will suffice to define an antiderivative of 1/z on U. (It is the fact that θ cannot be defined continuously on a simple closed curve containing the origin in its interior that is the root of why 1/z has no antiderivative on its entire domain C − {0}.) And because the derivative of an additive constant is 0, any constant may be added to the antiderivative and the result will still be an antiderivative of 1/z.
In a certain sense, the 1/z counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/z itself does not have an antiderivative on C − {0}.
Proof
There is a relatively elementary proof of the theorem. One constructs an anti-derivative for f explicitly.
Without loss of generality, it can be assumed that D is connected. Fix a point z0 in D, and for any , let be a piecewise C1 curve such that and . Then define the function F to be
To see that the function is well-defined, suppose is another piecewise C1 curve such that and . The curve (i.e. the curve combining with in reverse) is a closed piecewise C1 curve in D. Then,
And it follows that
Then using the continuity of f to estimate difference quotients, we get that F′(z) = f(z). Had we chosen a different z0 in D, F would change by a constant: namely, the result of integrating f along any piecewise regular curve between the new z0 and the old, and this does not change the derivative.
Since f is the derivative of the holomorphic function F, it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent power series, and the fact that power series may be differentiated term by term. This completes the proof.
Applications
Morera's theorem is a standard tool in complex analysis. It is used in
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https://en.wikipedia.org/wiki/Divisor%20function
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In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
Definition
The sum of positive divisors function σz(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as
where is shorthand for "d divides n".
The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) ().
The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, ), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.
Example
For example, σ0(12) is the number of the divisors of 12:
while σ1(12) is the sum of all the divisors:
and the aliquot sum s(12) of proper divisors is:
σ-1(n) is sometimes called the abundancy index of n, and we have:
Table of values
The cases x = 2 to 5 are listed in through , x = 6 to 24 are listed in through .
Properties
Formulas at prime powers
For a prime number p,
because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the primorial,
since n prime factors allow a sequence of binary selection ( or 1) from n terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a power of two; instead, the smallest such number may be obtained by multiplying together the first n Fermi–Dirac primes, prime powers whose exponent is a power of two.
Clearly, for all , and for all , .
The divisor function is multiplicative (since each divisor c of the product mn with distinctively correspond to a divisor a of m and a divisor b of n), but not completely multiplicative:
The consequence of this is that, if we write
where r = ω(n) is the number of distinct prime factors of n, pi is the ith prime factor, and ai is the maximum power of pi by which n is divisible, then we have:
which, when x ≠ 0, is equivalent to the useful formula:
When x = 0, is:
This result can be directly deduced from the fact that all divisors of are uniquely determined by the distinct tuples of integers with (i.e. independent choices for e
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https://en.wikipedia.org/wiki/91%20%28number%29
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91 (ninety-one) is the natural number following 90 and preceding 92.
In mathematics
91 is:
the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime.
the aliquot sum of 91 is 21 33; itself a semiprime, within an aliquot sequence of two composite numbers (91,21,11, 1,0) to the prime in the 11-aliquot tree. 91 is the fourth composite number in the 11-aliquot tree.(91,51,21,18).
a triangular number.
a hexagonal number, one of the few such numbers to also be a centered hexagonal number.
a centered nonagonal number.
a centered cube number.
a square pyramidal number, being the sum of the squares of the first six integers.
the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): . (See 1729 for more details). This implies that 91 is the second cabtaxi number.
the smallest positive integer expressible as a sum of six distinct squares: .
The only other ways to write 91 as a sum of distinct squares are: and
.
the smallest pseudoprime satisfying the congruence .
a repdigit in base 9 (1119).
palindromic in bases 3 (101013), 9 (1119), and 12 (7712).
a Riordan number.
The decimal equivalent of the fraction can be obtained by using powers of 9.
In science
91 is the atomic number of protactinium, an actinide.
McCarthy 91 function, a recursive function in discrete mathematics
Messier object M91, a magnitude 11.5 spiral galaxy in the constellation Coma Berenices
The New General Catalogue object NGC 91, a single star in the constellation Andromeda
In other fields
Ninety-one is also:
The code for international direct dial phone calls to India
In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar)
The ISBN Group Identifier for books published in Sweden.
Psalm 91 is known as the Psalm of Protection.
STS-91 Space Shuttle Discovery mission to the International Space Station, June 2, 1998
Swedish comic strip 91:an
The 91st Missile Wing (91 SW) is a Minuteman (missile) III unit of the United States Air Force, based at Minot Air Force Base, North Dakota
The number of the French department Essonne
References
Integers
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https://en.wikipedia.org/wiki/92%20%28number%29
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92 (ninety-two) is the natural number following 91 and preceding 93.
In mathematics
92 is a composite number; a square-prime, of the general form (p2, q) where q is a higher prime. It is the tenth of this form and the eighth of the form (22.q).
92 is the eighth pentagonal number, and an Erdős–Woods number, since it is possible to find sequences of 92 consecutive integers such that each inner member shares a factor with either the first or the last member.
With an aliquot sum of 76; itself a square-prime, within an aliquot sequence of five composite numbers (92,76,64,63,1,0) to the prime in the 63-aliquot tree.
For , there are 92 solutions in the n-Queens Problem.
There are 92 "atomic elements" in John Conway's look-and-say sequence, corresponding to the 92 non-transuranic elements in the chemist's periodic table.
92 is palindromic in bases 6 (2326), 7 (1617), 22 (4422), and 45 (2245).
The most faces or vertices an Archimedean or Catalan solid can have is 92: the snub dodecahedron has 92 faces while its dual polyhedron, the pentagonal hexecontahedron, has 92 vertices.
As a simple polyhedron, the final stellation of the icosahedron has 92 vertices.
There are 92 Johnson solids.
In science
The atomic number of uranium, an actinide.
Messier object M92, a magnitude 7.5 globular cluster in the constellation Hercules
The New General Catalogue object NGC 92, a magnitude 13.1 peculiar spiral galaxy in the constellation Phoenix, and a member of Robert's Quartet
In other fields
Ninety-two is also:
The code for international direct dial phone calls to Pakistan.
The numeric code for the Hauts-de-Seine department of France. The number is reflected in the department's postal code, plus the names of at least three local sports clubs, specifically Racing 92 in rugby union and Metropolitans 92 and Nanterre 92 in basketball.
In the title of the book Ninety-two in the Shade, by Thomas McGuane.
The 92nd Tiger book by Michael Gilbert.
The House on 92nd Street, a 1945 film.
The model number of the gray Texas Instruments TI-92 graphing calculator.
The Beretta 92 series of semi-automatic pistols.
The "Illustrious 92" or "Glorious 92": Massachusetts legislators who refused to rescind the Massachusetts Circular Letter soliciting other British colonies' support in resistance to the Townshend Acts prior to the American Revolution. Analogous to the number 45 in reference to the protests of John Wilkes against British corruption.
The ISBN Group Identifier for books published by international publishers such as UNESCO.
The number which runs through almost every single of British film-maker Peter Greenaway's films. This number has special association with the fictional character of Greenaway's creation, Tulse Luper. It is said the number itself is based on a mathematical error in calculations concerning John Cage's work Indeterminacy. See The Falls for extensive use of this number.
"92", a song by Avail from their 1996 album 4am Friday.
STS-92 Space
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https://en.wikipedia.org/wiki/93%20%28number%29
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93 (ninety-three) is the natural number following 92 and preceding 94.
In mathematics
93 is:
the 28th distinct semiprime and the 9th of the form (3.q) where q is a higher prime.
the first number in the 3rd triplet of consecutive semiprimes, 93, 94, 95.
with an aliquot sum of 35; itself a semiprime, within an aliquot sequence (93,35,13,1,0) of three numbers to the Prime 13 in the 13-Aliquot tree.
a Blum integer, since its two prime factors, 3 and 31 are both Gaussian primes.
a repdigit in base 5 (3335), and 30 (3330).
palindromic in bases 2, 5, and 30.
a lucky number.
a cake number.
an idoneal number.
There are 93 different cyclic Gilbreath permutations on 11 elements, and therefore there are 93 different real periodic points of order 11 on the Mandelbrot set.
In other fields
Ninety-three is:
The atomic number of neptunium, an actinide.
The code for international direct dial phone calls to Afghanistan.
One of two ISBN Group Identifiers for books published in India.
The number of the French department Seine-Saint-Denis, a Paris suburb with high proportions of immigrants and low-income people, and as such used by many French rappers and those emulating their speech.
In classical Persian finger counting, the number 93 is represented by a closed fist. Because of this, classical Arab and Persian poets around 1 CE referred to someone's lack of generosity by saying that the person's hand made "ninety-three".
See also
AD 93, a year in the Julian calendar
List of highways numbered 93
Ninety-Three (Quatrevingt-treize), a novel concerning the French Revolution by Victor Hugo
93 (Thelema), a greeting among Thelemites based on the numerological (gematric) value of Thelema (Will) and Agape (Love) in Greek letters.
Babia 93, an album from a Pakistani pop singer Sajjad Ali
London's 93 Feet East music venue
Current 93, a musical project of David Tibet
Los Angeles 93 KHJ radio
United Airlines Flight 93, one of the airplanes hijacked on September 11, 2001.
93 'til Infinity, the debut album by Oakland hip hop group Souls of Mischief.
References
External links
On the Number 93
Integers
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https://en.wikipedia.org/wiki/94%20%28number%29
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94 (ninety-four) is the natural number following 93 and preceding 95.
In mathematics
94 is:
the twenty-ninth distinct semiprime and the fourteenth of the form (2.q).
the ninth composite number in the 43-aliquot tree. The aliquot sum of 94 is 50 within the aliquot sequence; (94,50,43,1,0).
the second number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95
a 17-gonal number and a nontotient.
an Erdős–Woods number, since it is possible to find sequences of 94 consecutive integers such that each inner member shares a factor with either the first or the last member.
a Smith number in decimal.
In computing
The ASCII character set (and, more generally, ISO 646) contains exactly 94 graphic non-whitespace characters, which form a contiguous range of code points. These codes (0x21–0x7E, as corresponding high bit set bytes 0xA1–0xFE) also used in various multi-byte encoding schemes for languages of East Asia, such as ISO 2022, EUC and GB 2312. For this reason, code pages of 942 and even 943 code points were common in East Asia in 1980s–1990s.
In astronomy
Messier 94, a spiral galaxy in the constellation Canes Venatici
The New General Catalogue object NGC 94, a galaxy in the constellation Andromeda
In other fields
Ninety-four is:
The atomic number of plutonium, an actinide.
The designation of STS-94 Space Shuttle Columbia launched July 1, 1997
The code for international direct dial phone calls to Sri Lanka.
Part of the model number of
AN-94, a Russian assault rifle.
M-94, a piece of cryptographic equipment used by the United States army in use from 1922 to 1943.
The number of Haydn's Surprise Symphony (Symphony No. 94).
Used as a nonsense number by the British satire magazine Private Eye. Most commonly used in spoof articles end halfway through a sentence with "(continued p. 94)". The magazine never extends to 94 pages: this was originally a reference to the enormous size of some Sunday newspapers.
Each February, Respiratory Health Association of Metropolitan Chicago hosts Hustle Up the Hancock, a race up 94 floors of the John Hancock Center in Chicago to raise more than $1 million for lung disease research and programs."
The 94th Fighter Squadron is a squadron of the United States Air Force, currently part of the 1st Operations Group of the 1st Fighter Wing, and stationed at Langley Air Force Base in Virginia
The 94th Infantry Division was a unit of the United States Army in World War II, activated September 15, 1942.
Saab 94 was the model number Saab unofficially used for the first generation Saab Sonett
Form I-94 is the form used to declare to US Customs Officers by international travelers the items in their possession, purpose of visit, etc.
The number of the French department Val-de-Marne
In sports
The length of an NBA court is and width is .
Pascal Wehrlein, who drove for Sauber in Formula One in 2017, chose the number 94.
See also
List of highways numbered 94
References
Integers
Private Eye
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https://en.wikipedia.org/wiki/95%20%28number%29
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95 (ninety-five) is the natural number following 94 and preceding 96.
In mathematics
95 is:
the 30th distinct semiprime and the fifth of the form (5.q).
the third composite number in the 6-aliquot tree. The aliquot sum of 95 is 25, within the aliquot sequence (95,25,6).
the last member in the third triplet of distinct semiprimes 93, 94, and 95.
an 11-gonal number.
a Thabit number.
the lowest integer for which the Mertens function is greater than 1. (The lowest integer producing a Merten's value greater than that of 95 is 218).
In astronomy
The Messier object M95, a magnitude 11.0 spiral galaxy in the constellation Leo
The New General Catalogue object NGC 95, a magnitude 12.6 peculiar spiral galaxy in the constellation Pisces
In sports
NBA record for Most Assists in a 7-game playoff series, 95, Los Angeles Lakers Magic Johnson, 1984
NBA record for Most Free Throw Attempts in a 6-game playoff series, 95, Los Angeles Lakers Jerry West, 1965
In other uses
Ninety-five is also:
The atomic number of americium, an actinide.
The number of theses in Martin Luther's 95 Theses.
"95 Poems" by E.E. Cummings (1958)
The book The Prince, Utopia, Ninety-Five Thesis by Sir Thomas More
The designation of American
Interstate 95, a freeway that runs from Florida to Maine.
U.S. Highway 95, a freeway that runs through the western part of the United States.
Bay Ridge–95th Street subway station, Brooklyn, on the R Train
York Mills 95E, a bus route number from the Toronto Transit Commission
95th Street is a major east–west thoroughfare on Chicago's South Side, designated as 9500 South in the address system
Part of the name of:
Windows 95, a version of the Microsoft Windows graphical interface.
CommSuite 95 was a communications software suite of products launched by Delrina in 1995, created for use with Windows 95
Dogme 95, a movement in filmmaking developed in 1995
The model number of the automobile Saab 95 introduced in 1959, and Saab 9-5 introduced 1997
The racing number for Lightning McQueen (voiced by Owen Wilson), the main character in Disney-Pixar's film Cars (2006), is 95. This was used as his racing number because 1995 was the year that Toy Story was released however, his number was actually going to be 57, a reference to the year Pixar's CEO John Lasseter was born.
In Toy Story 3 (2010), Woody is seen driving a steam locomotive at the beginning of the film. The steam locomotive's number is 95 in reference to Lightning McQueen's racing number and (again) the year the first Toy Story was released.
Part of the designation of:
Z-95 Headhunter, a fictitious starfighter from the Star Wars Expanded Universe.
Tupolev Tu-95 (NATO reporting name Bear), a strategic bomber and missile carrier from the times of the Soviet Union
The number of the French department Val-d'Oise
In statistics, a 95% confidence interval is considered satisfactory for most purposes.
The movie 95 Miles to Go (2004) starring Ray Romano
95 Worlds and Counting, a 2000 documentary
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https://en.wikipedia.org/wiki/96%20%28number%29
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96 (ninety-six) is the natural number following 95 and preceding 97. It is a number that appears the same when turned upside down.
In mathematics
96 is:
an octagonal number.
a refactorable number.
an untouchable number.
a semiperfect number since it is a multiple of 6.
an abundant number since the sum of its proper divisors is greater than 96.
the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being 24.
the sum of Euler's totient function φ(x) over the first seventeen integers.
strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195).
palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195).
an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member.
divisible by the number of prime numbers (24) below 96.
the smallest natural number that can be expressed as the difference of two nonzero squares in more than three ways: , , or .
Skilling's figure, a degenerate uniform polyhedron, has a Euler characteristic
Every integer greater than 96 may be represented as a sum of distinct super-prime numbers.
In geography
Ninety Six, South Carolina
Ninety Six District, a historical judicial and military district of colonial America which extended through North and South Carolina
Ninety Six National Historic Site, in Ninety Six, South Carolina, derives its name from the original settlement's distance in miles from a Cherokee village in the Blue Ridge Mountains
In music
96Neko is a female Japanese singer
The song "96 Tears" by garage rock band Question Mark and the Mysterians
"96", a song by Uverworld, a Japanese band.
"96 Quite Bitter Beings", a song recorded by rock band CKY
"96 Degrees In The Shade", a song on an album with the same title (official song title "1865", recorded by Jamaican reggae band Third World.
In science
The atomic number of curium, an actinide.
Messier 96, a magnitude 10.5 spiral galaxy in the constellation Leo
The New General Catalogue object NGC 96, a spiral galaxy in the constellation Andromeda
In other fields
An Australian TV soap opera, Number 96 (broadcast 1972–1977)
A 1974 film based on the TV series, Number 96
Class of '96 was a short-lived Fox drama series which aired in 1993
96 dpi, the standard resolution of the monitor of an IBM-compatible computer running Microsoft Windows
The number of surat Al-Alaq in the Qur'an
According to Gurdjieff's Fourth Way symbolism, the number of the Moon level
The 96th United States Congress met January 1979 to January 1981 during the last two years of President Jimmy Carter's administration
The 96th Infantry Division (United States) was a unit of the United States Army in World War II
, a German U-boat during World War II and subject of the film Das Boot
The Saab 96 car produced from 1960 to 1966
STS-96, Space Shuttle Discovery mission launched
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https://en.wikipedia.org/wiki/97%20%28number%29
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97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number and the only prime in the nineties.
In mathematics
97 is:
the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101.
a Proth prime and a Pierpont prime as it is 3 × 25 + 1.
the eleventh member of the Mian–Chowla sequence.
a self number in base 10, since there is no integer that added to its own digits, adds up to 97.
the smallest odd prime that is not a cluster prime.
the highest two-digit number where the sum of its digits is a square.
the number of primes <= 29.
The numbers 97, 907, 9007, 90007 and 900007 are all primes, and they are all happy primes. However, 9000007 (read as nine million seven) is composite and has the factorisation 277 × 32491.
an emirp
an isolated prime, since 95 and 99 aren't prime.
In science
Ninety-seven is:
The atomic number of berkelium, an actinide.
In astronomy
Messier object M97, a magnitude 12.0 planetary nebula in the constellation Ursa Major, also known as the Owl Nebula
The New General Catalogue object NGC 97, an elliptical galaxy in the constellation Andromeda
In other fields
Ninety-seven is:
The 97th United States Congress met during the Ronald Reagan administration, from January 1981 to January 1983
The 10-97 police code means "arrived on the scene"
STS-97 Space Shuttle Endeavour mission launched November 30, 2000
The 97th Infantry Division was a unit of the United States Army in World War I and World War II
Madden NFL 97 was the first John Madden NFL American football game to be created in the 32-bit gaming era
Radio stations broadcasting on frequencies near 97, such as Hot 97, New York City and 97X, Tampa, Florida
The decimal unicode number representing the Latin lowercase "a"
In music
A song "Baby Boy / Saturday Night '97" by Whigfield
The number of the Southern Railway train in the Wreck of the Old 97, a ballad recorded by numerous artists, including Flatt and Scruggs, Woody Guthrie, Johnny Cash, Nine Pound Hammer, and Hank Snow.
The Old 97's are an alt-country band, which took their name from the song "The Wreck of the Old '97".
A song by Alkaline Trio off their self-titled album
The Marching 97, marching band of Lehigh University.
District 97, a progressive rock band from the Chicago area (named after the elementary school District 97 in Oak Park, Illinois).
See also
List of highways numbered 97
References
Integers
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https://en.wikipedia.org/wiki/98%20%28number%29
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98 (ninety-eight) is the natural number following 97 and preceding 99.
In mathematics
98 is:
Wedderburn–Etherington number
nontotient
number of non-isomorphic set-systems of weight 7
In astronomy
98 Ianthe, a main-belt asteroid
Messier 98, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices.
The New General Catalogue object NGC 98, a magnitude 12.7 spiral galaxy in the constellation Phoenix.
In computing
Windows 98, a Microsoft operating system for personal computers
Microsoft Flight Simulator 98, a flight simulator program
In space travel
MPTA-098, the Main Propulsion Test Article built as a systems testbed for the Space Shuttle program
Pathfinder (OV-098), a Space Shuttle simulator built by NASA in 1977
STS-98, Space Shuttle Atlantis mission launched February 7, 2001
In other fields
Ninety-eight is:
The atomic number of californium, an actinide
+98, the code for international direct dial phone calls to Iran
98 Degrees, an American adult contemporary boy band
98.6 degrees Fahrenheit is normal body temperature
"98.6", a 1967 hit song by Keith
10-98 code in police code means "Assignment Completed"
The number of sons of Ater in the census of men of Israel upon return from exile (Bible, Ezra 2:16)
Beach 98th Street, often referred as Beach 98th Street–Playland, a station on the New York City Subway's IND Rockaway Line
Expo '98, a World's Fair held in Lisbon, Portugal, from May to September 1998
Oldsmobile 98, a full-size automobile and the highest-end of the Oldsmobile division of General Motors
Power 98 (radio station), an English radio station of So Drama! Entertainment in Singapore
Power 98 (film), a 1996 film starring Eric Roberts about a Los Angeles talk radio station.
"Power 98", official nickname of radio station WPEG, in Charlotte, North Carolina
Saab 98, a project by automaker Saab for a combi coupé based on the Saab 95
Spirit of '98, a ship owned by Cruise West built to accommodate 98 passengers
The Trail of '98, a 1928 western film
In sports
The highest jersey number allowed in the National Hockey League, as 99 was retired by the entire league to honor Wayne Gretzky and major-league sports only allow one- or two-digit uniform numbers.
See also
List of highways numbered 98
References
Integers
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https://en.wikipedia.org/wiki/102%20%28number%29
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102 (one hundred [and] two) is the natural number following 101 and preceding 103.
In mathematics
102 is an abundant number and a semiperfect number. It is a sphenic number.
The sum of Euler's totient function φ(x) over the first eighteen integers is 102.
102 is the first three-digit base 10 polydivisible number, since 1 is divisible by 1, 10 is divisible by 2 and 102 is divisible by 3. This also shows that 102 is a Harshad number. 102 is the first 3-digit number divisible by the numbers 3, 6, 17, 34 and 51.
10264 + 1 is a prime number
There are 102 vertices in the Biggs-Smith graph.
In science
The atomic number of nobelium, an actinide.
In other fields
102 is also:
The emergency telephone number for police in Azerbaijan, Ukraine and Belarus
The emergency telephone number for fire in Israel
The emergency telephone number for ambulance in parts of India
The emergency telephone number for ambulance in Maldives
See also
List of highways numbered 102
One Hundred and Two, a song by The Judds
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 133
Integers
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https://en.wikipedia.org/wiki/103%20%28number%29
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103 (one hundred [and] three) is the natural number following 102 and preceding 104.
In mathematics
103 is a prime number, the largest prime factor of . The previous prime is 101, making them both twin primes. It is the fifth irregular prime, because it divides the numerator of the Bernoulli number
The equation makes 103 part of a "Fermat near miss".
There are 103 different connected series-parallel partial orders on exactly six unlabeled elements.
103 is conjectured to be the smallest number for which repeatedly reversing the digits of its ternary representation, and adding the number to its reversal, does not eventually reach a ternary palindrome.
See also
103 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/104%20%28number%29
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104 (one hundred [and] four) is the natural number following 103 and preceding 105.
In mathematics
104 forms the fifth Ruth-Aaron pair with 105, since the distinct prime factors of 104 (2 and 13) and 105 (3, 5, and 7) both add up to 15. Also, the sum of the divisors of 104 aside from unitary divisors, is 105. With eight total divisors where 8 is the fourth largest, 104 is the seventeenth refactorable number. 104 is also the twenty-fifth primitive semiperfect number.
The sum of all its divisors is σ(104) = 210, which is the sum of the first twenty nonzero integers, as well as the product of the first four prime numbers (2 × 3 × 5 × 7).
Its Euler totient, or the number of integers relatively prime with 104, is 48. This value is also equal to the totient of its sum of divisors, φ(104) = φ(σ(104)).
The smallest known 4-regular matchstick graph has 104 edges and 52 vertices, where four unit line segments intersect at every vertex.
A row of four adjacent congruent rectangles can be divided into a maximum of 104 regions, when extending diagonals of all possible rectangles.
Regarding the second largest sporadic group , its McKay–Thompson series representative of a principal modular function is , with constant term :
The Tits group , which is the only finite simple group to classify as either a non-strict group of Lie type or sporadic group, holds a minimal faithful complex representation in 104 dimensions. This is twice the dimensional representation of exceptional Lie algebra in 52 dimensions, whose associated lattice structure forms the ring of Hurwitz quaternions that is represented by the vertices of the 24-cell — with this regular 4-polytope one of 104 total four-dimensional uniform polychora, without taking into account the infinite families of uniform antiprismatic prisms and duoprisms.
In other fields
104 is also:
The atomic number of rutherfordium.
The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built in Greece.
The number of Symphonies written by Joseph Haydn upon which numbers are agreed (though in fact, he wrote two more: see list of symphonies by Joseph Haydn).
See also
List of highways numbered 104
The years 104 BC and AD 104.
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 133
Integers
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https://en.wikipedia.org/wiki/105%20%28number%29
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105 (one hundred [and] five) is the natural number following 104 and preceding 106.
In mathematics
105 is a triangular number, a dodecagonal number, and the first Zeisel number. It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers.
105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such numbers less than a thousand are 9, 15, 195, and 825.
105 is also the middle of the only prime sextuplet (97, 101, 103, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes () and less than the square of the next prime (11) by > 8, for , n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).
105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83, and 97. The distinct prime factors of 105 add up to 15, and so do those of 104; hence, the two numbers form a Ruth-Aaron pair under the first definition.
105 is also a number n for which is prime, for . (This even works up to , ignoring the negative sign.)
105 is the smallest integer such that the factorization of over Q includes non-zero coefficients other than . In other words, the 105th cyclotomic polynomial, Φ105, is the first with coefficients other than .
105 is the number of parallelogram polyominoes with 7 cells.
In science
The atomic number of dubnium.
In other fields
105 is also:
A Shimano Road groupset since 1984
See also
List of highways numbered 105
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134
Integers
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https://en.wikipedia.org/wiki/106%20%28number%29
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106 (one hundred [and] six) is the natural number following 105 and preceding 107.
In mathematics
106 is a centered pentagonal number, a centered heptagonal number, and a regular 19-gonal number.
There are 106 mathematical trees with ten vertices.
See also
106 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/107%20%28number%29
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107 (one hundred [and] seven) is the natural number following 106 and preceding 108.
In mathematics
107 is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime.
Plugged into the expression , 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime.
It is the fourth Busy beaver number, the maximum number of steps that any Turing machine with 2 symbols and 4 states can make before eventually halting.
It is the number of triangle-free graphs on 7 vertices.
It is the ninth emirp, because reversing its digits gives another prime number (701)
In other fields
As "one hundred and seven", it is the smallest positive integer requiring six syllables in English (without the "and" it only has five syllables and seventy-seven is a smaller 5-syllable number).
107 is also:
The atomic number of bohrium.
The emergency telephone number in Argentina and Cape Town.
The telephone of the police in Hungary.
A common designation for the fair use exception in copyright law (from 17 U.S.C. 107)
Peugeot 107 model of car
In sports
The 107% rule, a Formula One Sporting Regulation in operation from 1996 to 2002 and 2011 onward.
The number 107 is also associated with the Timbers Army supporters group of the Portland Timbers soccer team, in reference to the stadium seating section where the group originally congregated.
See also
List of highways numbered 107
References
Integers
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https://en.wikipedia.org/wiki/108%20%28number%29
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108 (one hundred [and] eight) is the natural number following 107 and preceding 109.
In mathematics
108 is:
an abundant number.
a semiperfect number.
a tetranacci number.
the hyperfactorial of 3 since it is of the form .
divisible by the value of its φ function, which is 36.
divisible by the total number of its divisors (12), hence it is a refactorable number.
the angle in degrees of the interior angles of a regular pentagon in Euclidean space.
palindromic in bases 11 (9911), 17 (6617), 26 (4426), 35 (3335) and 53 (2253)
a Harshad number in bases 2, 3, 4, 6, 7, 9, 10, 11, 12, 13 and 16
a self number.
an Achilles number because it is a powerful number but not a perfect power.
nine dozen
There are 108 free polyominoes of order 7.
The equation results in the golden ratio.
This could be restated as saying that the "chord" of 108 degrees is , the golden ratio.
Religion and the arts
The number 108 is considered sacred by the Dharmic Religions, such as Hinduism, Buddhism, and Jainism.
Hinduism
In Hindu tradition, the Mukhya Shivaganas (attendants of Shiva) are 108 in number and hence Shaiva religions, particularly Lingayats, use malas of 108 beads for prayer and meditation.
Similarly, in Gaudiya Vaishnavism, Lord Krishna in Brindavan had 108 followers known as gopis. Recital of their names, often accompanied by the counting of a 108-beaded mala, is often done during religious ceremonies.
The Sri Vaishnavite Tradition has 108 Divya Desams (temples of Vishnu) that are revered by the 12 Alvars in the Divya Prabandha, a collection of 4,000 Tamil verses. There are also 18 pithas (sacred places).
The Sudarshana Chakra is a spinning, discus weapon with 108 serrated edges, generally portrayed on the right rear hand of the four hands of Vishnu.
The total number of Upanishads is 108 as per Muktikā canon.
Jainism
In Jainism, the total number of ways of Karma influx (Aasrav). 4 Kashays (anger, pride, conceit, greed) x 3 karanas (mind, speech, bodily action) x 3 stages of planning (planning, procurement, commencement) x 3 ways of execution (own action, getting it done, supporting or approval of action).
Buddhism
In Buddhism, according to Bhante Gunaratana this number is reached by multiplying the senses smell, touch, taste, hearing, sight, and consciousness by whether they are painful, pleasant or neutral, and then again by whether these are internally generated or externally occurring, and yet again by past, present and future, finally we get 108 feelings. 6 × 3 × 2 × 3 = 108.
Tibetan Buddhist malas or rosaries (Tib. ཕྲེང་བ Wyl. phreng ba, "Trengwa") are usually 108 beads; sometimes 111 including the guru bead(s), reflecting the words of the Buddha called in Tibetan the Kangyur (Wylie: Bka'-'gyur) in 108 volumes.
Zen priests wear juzu (a ring of prayer beads) around their wrists, which consists of 108 beads.
The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions and another section where Buddha lists 108
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https://en.wikipedia.org/wiki/109%20%28number%29
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109 (one hundred [and] nine) is the natural number following 108 and preceding 110.
In mathematics
109 is the 29th prime number. As 29 is itself prime, 109 is the tenth super-prime. The previous prime is 107, making them both twin primes.
109 is a centered triangular number.
There are exactly:
109 different families of subsets of a three-element set whose union includes all three elements.
109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements.
109 squares on an infinite chessboard that can be reached by a knight within three moves.
There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings.
The decimal expansion of 1/109 can be computed using the alternating series, with the Fibonacci number:
The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime in decimal. The last six digits of the 108-digit cycle are 853211, the first six Fibonacci numbers in descending order.
See also
109 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/110%20%28number%29
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110 (one hundred [and] ten) is the natural number following 109 and preceding 111.
In mathematics
110 is a sphenic number and a pronic number. Following the prime quadruplet (101, 103, 107, 109), at 110, the Mertens function reaches a low of −5.
110 is the sum of three consecutive squares, .
RSA-110 is one of the RSA numbers, large semiprimes that are part of the RSA Factoring Challenge.
In base 10, the number 110 is a Harshad number and a self number.
In science
The atomic number of darmstadtium.
In religion
According to the Bible, the figures Joseph and Joshua both died aged 110.
In sports
Olympic male track and field athletics run 110 metre hurdles. (Female athletes run the 100 metre hurdles instead.)
The International 110, or the 110, is a one-design racing sailboat designed in 1939 by C. Raymond Hunt.
In other fields
110 is also:
The year AD 110 or 110 BC
A common name for mains electricity in North America, despite the nominal voltage actually being 120 V (range 110–120 V). Normally spoken as "one-ten".
1-1-0, the emergency telephone number used to reach police services in Iran, Germany, Estonia, China, Indonesia, and Japan. Also used to reach the fire and rescue services in Norway and Turkey.
The age a person must attain in order to be considered a supercentenarian.
A card game related to Forty-five (card game).
A percentage in the expression "To give 110%", meaning to give a little more effort than one's maximum effort
The number of stories of each of the towers of the former World Trade Center in New York.
The number of stories (by common reckoning) of the Sears Tower in Chicago.
The TCP port used for POP3 email protocol
A 110 block is a type of punch block used to connect sets of wires in a structured cabling system.
The abjad (ابجد) translation of word "علی" (Ali) in Arabic and Persian.
It is also known as "eleventy", a term made famous by linguist and author J. R. R. Tolkien (Bilbo Baggins celebrates his eleventy-first birthday at the beginning of The Lord of the Rings) and derived from the Old English . When the word eleventy is used, it may indicate the exact number (110), or more commonly an indefinite large number such as gazillion.
Eleventy is used in the comic reading of a phone number in the Irish TV series "The Savage Eye" by Dave McSavage playing an opiate user advertising life insurance.
Lowest number to not be considered a favorite by anyone among 44,000 people surveyed in a 2014 online poll and subsequently adopted by British television show QI as the show's favourite number in 2017.
See also
110s decade
List of highways numbered 110
List of 110th Street, New York City Subway stations
110 film
Integers between 111 and 112
111
112
113
114
115
116
117
118
119
References
Integers
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https://en.wikipedia.org/wiki/112%20%28number%29
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112 (one hundred [and] twelve) is the natural number following 111 and preceding 113.
Mathematics
112 is an abundant number, a heptagonal number, and a Harshad number.
112 is the number of connected graphs on 6 unlabeled nodes.
If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices. This is the smallest possible side length of an equilateral triangle that contains a point at integer distances from the vertices.
See also
112 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/113%20%28number%29
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113 (one hundred [and] thirteen) is the natural number following 112 and preceding 114.
Mathematics
113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 113 is also an Eisenstein prime with no imaginary part and real part of the form . In decimal, this prime is a primeval number and a permutable prime with 131 and 311.
113 is a highly cototient number and a centered square number.
113 is the denominator of 355/113, an accurate approximation to .
See also
113 (disambiguation)
A113 is a Pixar recurring inside joke or Easter Egg, e.g.: (WALL-E) = (W-A113).
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134
Integers
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https://en.wikipedia.org/wiki/114%20%28number%29
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114 (one hundred [and] fourteen) is the natural number following 113 and preceding 115.
In mathematics
114 is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H(0). At 114, the Mertens function sets a new low of -6, a record that stands until 197.
114 is the smallest positive integer* which has yet to be represented as a3 + b3 + c3, where a, b, and c are integers. It is conjectured that 114 can be represented this way. (*Excluding integers of the form 9k ± 4, for which solutions are known not to exist.)
There is no answer to the equation φ(x) = 114, making 114 a nontotient.
114 appears in the Padovan sequence, preceded by the terms 49, 65, 86 (it is the sum of the first two of these).
114 is a repdigit in base 7 (222).
In religion
There are 114 chapters, or surahs, in the Quran.
There are 114 sayings in The Gospel of Thomas.
See also
114 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/100
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100 or one hundred (Roman numeral: C) is the natural number following 99 and preceding 101.
In mathematics
100 is the square of 10 (in scientific notation it is written as 102). The standard SI prefix for a hundred is "hecto-".
100 is the basis of percentages (per cent meaning "per hundred" in Latin), with 100% being a full amount.
100 is a Harshad number in decimal, and also in base-four, a base in-which it is also a self-descriptive number.
100 is the sum of the first nine prime numbers, from 2 through 23. It is also divisible by the number of primes below it, 25.
100 cannot be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient.
100 has a reduced totient of 20, and an Euler totient of 40. A totient value of 100 is obtained from four numbers: 101, 125, 202, and 250.
100 can be expressed as a sum of some of its divisors, making it a semiperfect number. The geometric mean of its nine divisors is 10.
100 is the sum of the cubes of the first four positive integers (100 = 13 + 23 + 33 + 43). This is related by Nicomachus's theorem to the fact that 100 also equals the square of the sum of the first four positive integers: .
100 = 26 + 62, thus 100 is the seventh Leyland number. 100 is also the seventeenth Erdős–Woods number, and the fourth 18-gonal number.
The 100th prime number is 541, which returns for the Mertens function. It is the 10th star number (whose digit sum also adds to 10 in decimal).
There are exactly 100 prime numbers in base-ten whose digits are in strictly ascending order (e.g. 239, 2357 etc.). The last such prime number is 23456789, which contains eight consecutive integers as digits.
In science
One hundred is the atomic number of fermium, an actinide and the last of the heavy metals that can be created through neutron bombardment.
On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level.
The Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is commonly used to define the boundary between Earth's atmosphere and outer space.
In history
In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120.
In religion
There are 100 blasts of the Shofar heard in the service of Rosh Hashana, the Jewish New Year.
A religious Jew is expected to utter at least 100 blessings daily.
In the Hindu epic of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas.
In politics
The Hundred (county division) is a largely historical division of a county or similar larger administrative unit.
The United States Senate has 100 Senators.
In money
Most of the world's currencies are divided into 100 subunits; for example, one euro is one hundred cents and one pound sterling is one hundred pence.
By specification, 100 euro notes feature a picture of a Rococo gateway on the ob
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https://en.wikipedia.org/wiki/1001%20%28number%29
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1001 is the natural number following 1000 and followed by 1002.
In mathematics
One thousand and one is a sphenic number, a pentagonal number, a pentatope number and the first four-digit palindromic number. Scheherazade numbers always have 1001 as a factor.
Divisibility by 7, 11 and 13
Two properties of 1001 are the basis of a divisibility test for 7, 11 and 13. The method is along the same lines as the divisibility rule for 11 using the property 10 ≡ -1 (mod 11). The two properties of 1001 are
1001 = 7 × 11 × 13 in prime factors
103 ≡ -1 (mod 1001)
The method simultaneously tests for divisibility by any of the factors of 1001. First, the digits of the number being tested are grouped in blocks of three. The odd numbered groups are summed. The sum of the even numbered groups is then subtracted from the sum of the odd numbered groups. The test number is divisible by 7, 11 or 13 iff the result of the summation is divisible by 7, 11 or 13 respectively.
Example:
Number under test, N = 22 872 563 219
Sum of odd groups, So = 219 + 872 = 1091
Sum of even groups, Se = 563 + 22 = 585
Total sum, S = So - Se = 1091 - 585 = 506
506 = 46 × 11
Since 506 is divisible by 11 then N is also divisible by 11. If the total sum is still too large to conveniently test for divisibility, and is longer than three digits, then the algorithm can be repeated to obtain a smaller number.
In other fields
In The Book of One Thousand and One Nights, Scheherazade tells her husband the king a new story every night for 1,001 nights, staving off her execution. From this, 1001 is sometimes used as a generic term for "a very large number", starting with a large number (1000) and going beyond it:
1001 uses for...
1001 ways to...
In Arabic, this is usually phrased as "one thousand things and one thing", e.g.:
The Book of One Thousand and One Nights, in Arabic Alf layla wa layla (), literally "One thousand nights and a night".
1001 was the name of a popular British detergent in the 1960s, supposedly with "1001 uses".
In the Mawlawiyyah order of Sufi Islam, a novice must complete 1001 days of prayer before becoming a dada, or junior teacher of the faith.
In many cases, including the title "Thousand and One Nights", 1001 is meant to indicate a "big number", and need not be taken literally. A book published in 2007 titled 40 Days & 1001 Nights describes a journey through the Islamic world.
Among them are recent books aiming to introduce significant works in various fields:
1001 Books You Must Read Before You Die
1001 Movies You Must See Before You Die
1001 Albums You Must Hear Before You Die
There are also many film titles starting with 1001. For example:
Bugs Bunny's 3rd Movie: 1001 Rabbit Tales
The NBA draft lottery uses a lottery with 1,001 combinations by selecting four balls out of 14, then disregards the combination 11, 12, 13 and 14 to produce 1,000 outcomes.
1001 was a hidden track on the Australian release of Two Shoes, the second album by The Cat Emp
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https://en.wikipedia.org/wiki/Witch%20of%20Agnesi
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In mathematics, the witch of Agnesi () is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sailing sheet. Before Agnesi, the same curve was studied by Fermat, Grandi, and Newton.
The graph of the derivative of the arctangent function forms an example of the witch of Agnesi. As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. It also gives rise to Runge's phenomenon in the approximation of functions by polynomials, has been used to approximate the energy distribution of spectral lines, and models the shape of hills.
The witch is tangent to its defining circle at one of the two defining points, and asymptotic to the tangent line to the circle at the other point. It has a unique vertex (a point of extreme curvature) at the point of tangency with its defining circle, which is also its osculating circle at that point. It also has two finite inflection points and one infinite inflection point. The area between the witch and its asymptotic line is four times the area of the defining circle, and the volume of revolution of the curve around its defining line is twice the volume of the torus of revolution of its defining circle.
Construction
To construct this curve, start with any two points O and M, and draw a circle with OM as diameter. For any other point A on the circle, let N be the point of intersection of the secant line OA and the tangent line at M.
Let P be the point of intersection of a line perpendicular to OM through A, and a line parallel to OM through N. Then P lies on the witch of Agnesi. The witch consists of all the points P that can be constructed in this way from the same choice of O and M. It includes, as a limiting case, the point M itself.
Equations
Suppose that point O is at the origin and point M lies on the positive -axis, and that the circle with diameter OM has
Then the witch constructed from O has the Cartesian equation
This equation can be simplified, by choosing to the form
or equivalently, by clearing denominators, as the cubic algebraic equation
In its simplified form, this curve is the graph of the derivative of the arctangent function.
The witch of Agnesi can also be described by parametric equations whose parameter is the angle between OM and OA, measured clockwise:
Properties
The main properties of this curve can be derived from integral calculus.
The area between the witch and its asymptotic line is four times the area of the fixed circle,
The volume of revolution of the witch of Agnesi about its asymptote This is two times the volume of the torus formed by revolving the defining circle of the witch around the same line.
The curve has a unique vertex at the point of tangency with its defining circle. That is, this point is the only point where the curvature reaches a local minimum or l
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https://en.wikipedia.org/wiki/List%20of%20probability%20topics
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This is a list of probability topics.
It overlaps with the (alphabetical) list of statistical topics. There are also the outline of probability and catalog of articles in probability theory. For distributions, see List of probability distributions. For journals, see list of probability journals. For contributors to the field, see list of mathematical probabilists and list of statisticians.
General aspects
Probability
Randomness, Pseudorandomness, Quasirandomness
Randomization, hardware random number generator
Random number generation
Random sequence
Uncertainty
Statistical dispersion
Observational error
Equiprobable
Equipossible
Average
Probability interpretations
Markovian
Statistical regularity
Central tendency
Bean machine
Relative frequency
Frequency probability
Maximum likelihood
Bayesian probability
Principle of indifference
Credal set
Cox's theorem
Principle of maximum entropy
Information entropy
Urn problems
Extractor
Free probability
Exotic probability
Schrödinger method
Empirical measure
Glivenko–Cantelli theorem
Zero–one law
Kolmogorov's zero–one law
Hewitt–Savage zero–one law
Law of truly large numbers
Littlewood's law
Infinite monkey theorem
Littlewood–Offord problem
Inclusion–exclusion principle
Impossible event
Information geometry
Talagrand's concentration inequality
Foundations of probability theory
Probability theory
Probability space
Sample space
Standard probability space
Random element
Random compact set
Dynkin system
Probability axioms
Normalizing constant
Event (probability theory)
Complementary event
Elementary event
Mutually exclusive
Boole's inequality
Probability density function
Cumulative distribution function
Law of total cumulance
Law of total expectation
Law of total probability
Law of total variance
Almost surely
Cox's theorem
Bayesianism
Prior probability
Posterior probability
Borel's paradox
Bertrand's paradox
Coherence (philosophical gambling strategy)
Dutch book
Algebra of random variables
Belief propagation
Transferable belief model
Dempster–Shafer theory
Possibility theory
Random variables
Discrete random variable
Probability mass function
Constant random variable
Expected value
Jensen's inequality
Variance
Standard deviation
Geometric standard deviation
Multivariate random variable
Joint probability distribution
Marginal distribution
Kirkwood approximation
Independent identically-distributed random variables
Independent and identically-distributed random variables
Statistical independence
Conditional independence
Pairwise independence
Covariance
Covariance matrix
De Finetti's theorem
Correlation
Uncorrelated
Correlation function
Canonical correlation
Convergence of random variables
Weak convergence of measures
Helly–Bray theorem
Slutsky's theorem
Skorokhod's representation theorem
Lévy's continuity theorem
Uniform integrability
Markov's inequality
Chebyshev's inequality = Chernoff bound
Chernoff
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https://en.wikipedia.org/wiki/Mahler%27s%20theorem
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In mathematics, Mahler's theorem, introduced by , expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval.
Statement
Let be the forward difference operator. Then for any p-adic function , Mahler's theorem states that is continuous if and only if its Newton series converges everywhere to , so that for all we have
where
is the th binomial coefficient polynomial. Here, the th forward difference is computed by the binomial transform, so thatMoreover, we have that is continuous if and only if the coefficients in as .
It is remarkable that as weak an assumption as continuity is enough in the p-adic setting to establish convergence of Newton series. By contrast, Newton series on the field of complex numbers are far more tightly constrained, and require Carlson's theorem to hold.
References
Factorial and binomial topics
Theorems in analysis
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https://en.wikipedia.org/wiki/Cauchy%20principal%20value
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In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is not covered by the integral).
Formulation
Depending on the type of singularity in the integrand , the Cauchy principal value is defined according to the following rules:
In some cases it is necessary to deal simultaneously with singularities both at a finite number and at infinity. This is usually done by a limit of the form
In those cases where the integral may be split into two independent, finite limits,
and
then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value".
The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function with with a pole on a contour . Define to be that same contour, where the portion inside the disk of radius around the pole has been removed. Provided the function is integrable over no matter how small becomes, then the Cauchy principal value is the limit:
In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.
If the function is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.
Principal value integrals play a central role in the discussion of Hilbert transforms.
Distribution theory
Let be the set of bump functions, i.e., the space of smooth functions with compact support on the real line . Then the map
defined via the Cauchy principal value as
is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the sign function and the Heaviside step function.
Well-definedness as a Distribution
To prove the existence of the limit
for a Schwartz function , first observe that is continuous on as
and hence
since is continuous and L'Hopital's rule applies.
Therefore, exists and by applying the mean value theorem to we get:
And furthermore:
we note that the map
is bounded by the usual seminorms for Schwartz functions . Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.
Note that the proof needs merely to be continuously differentiable in a neighbourhood of 0 and to be bounded towards infinity. The principal value therefore is defined on even weaker assumpti
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https://en.wikipedia.org/wiki/Principal%20value
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In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as
Motivation
Consider the complex logarithm function . It is defined as the complex number such that
Now, for example, say we wish to find . This means we want to solve
for . The value is a solution.
However, there are other solutions, which is evidenced by considering the position of in the complex plane and in particular its argument . We can rotate counterclockwise radians from 1 to reach initially, but if we rotate further another we reach again. So, we can conclude that is also a solution for . It becomes clear that we can add any multiple of to our initial solution to obtain all values for .
But this has a consequence that may be surprising in comparison of real valued functions: does not have one definite value. For , we have
for an integer , where is the (principal) argument of defined to lie in the interval . As the principal argument is unique for a given complex number , is not included in the interval. Each value of determines what is known as a branch (or sheet), a single-valued component of the multiple-valued log function.
The branch corresponding to is known as the principal branch, and along this branch, the values the function takes are known as the principal values.
General case
In general, if is multiple-valued, the principal branch of is denoted
such that for in the domain of , is single-valued.
Principal values of standard functions
Complex valued elementary functions can be multiple-valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.
Logarithm function
We have examined the logarithm function above, i.e.,
Now, is intrinsically multivalued. One often defines the argument of some complex number to be between (exclusive) and (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch (with the leading capital A). Using instead of , we obtain the principal value of the logarithm, and we write
Square root
For a complex number the principal value of the square root is:
with argument
Complex argument
The principal value of complex number argument measured in radians can be defined as:
values in the range
values in the range
To compute these values one can use functions :
atan2 with principal value in the range
atan with principal value in the range
See also
Principal branch
Branch point
References
Complex analysis
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https://en.wikipedia.org/wiki/Taxicab%20geometry
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A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 distance, L1 distance or norm (see Lp space), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets.
The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski.
In , the taxicab distance between two points and is . That is, it is the sum of the absolute values of the differences in both coordinates.
Formal definition
The taxicab distance, , between two vectors in an n-dimensional real vector space with fixed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. More formally,For example, in , the taxicab distance between and is
History
The L1 metric was used in regression analysis in 1757 by Roger Joseph Boscovich. The geometric interpretation dates to the late 19th century and the development of non-Euclidean geometries, notably by Hermann Minkowski and his Minkowski inequality, of which this geometry is a special case, particularly used in the geometry of numbers, . The formalization of Lp spaces is credited to .
Properties
Taxicab distance depends on the rotation of the coordinate system, but does not depend on its reflection about a coordinate axis or its translation. Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry) except for the side-angle-side axiom, as two triangles with equally "long" two sides and an identical angle between them are typically not congruent unless the mentioned sides are parallel.
Balls
A topological ball is a set of points with a fixed distance, called the radius, from a point called the center. In n-dimensional Euclidean geometry, the balls are spheres. In taxicab geometry, distance is determined by a different metric than in Euclidean geometry, and the shape of the ball changes as well. In n dimensions, a taxicab ball is in the shape of an n-dimensional orthoplex. In two dimensions, these are squares with sides oriented at a 45° angle to the coordinate axes. The image to the right shows why this is true, by showing in red the set of all points with a fixed distance from a center, shown in blue. As the size of the city blocks diminishes, the points become more numerous and become a rotated square in a continuous taxicab geometry. While each side would ha
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https://en.wikipedia.org/wiki/Trig%20%28disambiguation%29
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Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
Trig also may refer to:
Trig functions
TriG (syntax), a format for storing and transmitting Resource Description Framework (RDF) data
Trig points, also known as triangulation stations
Trig Paxson Van Palin, son of former Alaska Governor Sarah Palin
Trigedasleng, or Trig, a language used on the TV series The 100
Phyllopalpus pulchellus, species of cricket commonly known as the handsome trig
See also
Spherical trigonometry
Non-euclidean geometry
Celestial navigation
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https://en.wikipedia.org/wiki/Kissing%20number
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In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number.
In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.
Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century. Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others investigations have determined upper and lower bounds, but not exact solutions.
Known greatest kissing numbers
One dimension
In one dimension, the kissing number is 2:
Two dimensions
In two dimensions, the kissing number is 6:
Proof: Consider a circle with center C that is touched by circles with centers C1, C2, .... Consider the rays C Ci. These rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°.
Assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, say C C1 and C C2, are separated by an angle of less than 60°. The segments C Ci have the same length – 2r – for all i. Therefore, the triangle C C1 C2 is isosceles, and its third side – C1 C2 – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction.
Three dimensions
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, mo
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https://en.wikipedia.org/wiki/Primorial%20prime
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In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).
Primality tests show that
pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ...
pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ...
The first term of the second sequence is 0 because p0# = 1 is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because p1# = 2, and 2 − 1 = 1 is not prime.
The first few primorial primes are
2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309
, the largest known primorial prime (of the form pn# − 1) is 3267113# − 1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.
, the largest known prime of the form pn# + 1 is 392113# + 1 (n = 33,237) with 169,966 digits, found in 2001 by Daniel Heuer.
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:
Assume that the first n consecutive primes including 2 are the only primes that exist. If either pn# + 1 or pn# − 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence all its prime factors are larger than pn).
See also
Compositorial
Euclid number
Factorial prime
References
See also
A. Borning, "Some Results for and " Math. Comput. 26 (1972): 567–570.
Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
Harvey Dubner, "Factorial and Primorial Primes." J. Rec. Math. 19 (1987): 197–203.
Paulo Ribenboim, The New Book of Prime Number Records. New York: Springer-Verlag (1989): 4.
Integer sequences
Classes of prime numbers
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https://en.wikipedia.org/wiki/Statistics%20Netherlands
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Statistics Netherlands, founded in 1899, is a Dutch governmental institution that gathers statistical information about the Netherlands. In Dutch it is known as the Centraal Bureau voor de Statistiek (Central Agency for Statistics), often abbreviated to CBS. It is located in The Hague and Heerlen. Since 3 January 2004, Statistics Netherlands has been a self-standing organisation, or quango. Its independent status in law guarantees the reliable collection and dissemination of information supporting public debate, policy development and decision-making.
The CBS collects statistical information about, amongst others:
Count of the population
Consumer pricing
Economic growth
Income of persons and households
Unemployment
Religion
The CBS carries out a program that needs to be ratified by the Central Commission for Statistics. This commission was replaced in 2016 by an Advisory Board. This independent board must guard the impartiality, independence, quality, relevance, and continuity of the CBS, according to the Law on the CBS of 1996 (Wet op het Centraal bureau en de Centrale commissie voor de statistiek) and 2003.
History
CBS was established in 1899 in response to the need for independent and reliable information that advances the understanding of social issues and supports public decision making. This is still the main role of CBS. Philip Idenburg, who worked at the CBS from 1929–1966, played a key role in salvaging the work of the Mundaneum offices in The Hague, arranging for Gerd Arntz to be involved in setting up the Dutch Foundation for Statistics, which used the Isotypes previously developed by Arntz and Otto Neurath.
Offices
The CBS has offices in The Hague and Heerlen. The office in Heerlen was located there by the government in 1973 to compensate the area for the loss of ten of thousands of jobs because of closing the coal mines. The office in The Hague with the name 'Double U' was designed by Branimir Medić and Pero Puljiz. It has a surface of and the total cost was €41,000,000. The office in Heerlen was designed by Meyer en Van Schooten Architects in 2009. The office has a surface of and parking spaces for 296 cars. Glass was used everywhere in the building. The main hall has a glass roof and the outside walls are fully glass. The several straight staircases in the main hall have glass balustrades with a RVS handrail and were manufactured by EeStairs. Queen Beatrix of the Netherlands officially opened the building on 30 September 2009.
See also
Eurostat
References
External links
Statistics Netherlands
StatLine database
Netherlands
Economy of the Netherlands
Independent government agencies of the Netherlands
Organisations based in The Hague
Buildings and structures in Heerlen
Buildings and structures in The Hague
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https://en.wikipedia.org/wiki/Rotation%20%28mathematics%29
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire -dimensional flat of fixed points in a -dimensional space.
Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations.
Related definitions and terminology
The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions.
For a particular rotation:
The axis of rotation is a line of its fixed points. They exist only in .
The plane of rotation is a plane that is invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are orthogonal.
A representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory.
Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations. See the article below for details.
Definitions and representations
In Euclidean geometry
A motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the orientation structure. The "improper rotation" term refers to isometries that reverse (flip) the orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be repres
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https://en.wikipedia.org/wiki/119%20%28number%29
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119 (one hundred [and] nineteen) is the natural number following 118 and preceding 120.
Mathematics
119 is a Perrin number, preceded in the sequence by 51, 68, 90 (it is the sum of the first two mentioned).
119 is the sum of five consecutive primes (17 + 19 + 23 + 29 + 31).
119 is the sum of seven consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29).
119 is a highly cototient number.
119 is the order of the largest cyclic subgroups of the monster group.
119 is the smallest composite number that is 1 less than a factorial (120 is 5!).
119 is a semiprime, and the third in the {7×q} family.
Telephony
119 is an emergency telephone number in some countries
A number to report youth at risk in France
119 is the emergency number in Afghanistan that belongs to police and interior ministry.
The South Korean emergency call number
The Chinese fire station call number
119 is the number for the UK's NHS Test and Trace service (created in response to the COVID-19 pandemic)
In other fields
119 is the default port for unencrypted NNTP connections.
Project 119 is a governmental program of the People's Republic of China targeting sports that China has not traditionally excelled in at the Summer Olympics, to maximize the number of medals won during the games.
119 is also the atomic number of the theoretical element ununennium.
Union Pacific No. 119, a 4-4-0 American Type standard gauge steam locomotive of the Union Pacific Railroad that was memorialized in railroading history on the right-hand side of Andrew J. Russell's famous "Joining of the Lines" photograph taken on 10 May 1869, at Promontory, Utah, during the celebration of the completion of the First transcontinental railroad, where it was cowcatcher to cowcatcher with Central Pacific Railroad's Jupiter (locomotive).
See also
List of highways numbered 119
References
Integers
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https://en.wikipedia.org/wiki/Leibniz%27s%20notation
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In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just as and represent finite increments of and , respectively.
Consider as a function of a variable , or = . If this is the case, then the derivative of with respect to , which later came to be viewed as the limit
was, according to Leibniz, the quotient of an infinitesimal increment of by an infinitesimal increment of , or
where the right hand side is Joseph-Louis Lagrange's notation for the derivative of at . The infinitesimal increments are called . Related to this is the integral in which the infinitesimal increments are summed (e.g. to compute lengths, areas and volumes as sums of tiny pieces), for which Leibniz also supplied a closely related notation involving the same differentials, a notation whose efficiency proved decisive in the development of continental European mathematics.
Leibniz's concept of infinitesimals, long considered to be too imprecise to be used as a foundation of calculus, was eventually replaced by rigorous concepts developed by Weierstrass and others in the 19th century. Consequently, Leibniz's quotient notation was re-interpreted to stand for the limit of the modern definition. However, in many instances, the symbol did seem to act as an actual quotient would and its usefulness kept it popular even in the face of several competing notations. Several different formalisms were developed in the 20th century that can give rigorous meaning to notions of infinitesimals and infinitesimal displacements, including nonstandard analysis, tangent space, O notation and others.
The derivatives and integrals of calculus can be packaged into the modern theory of differential forms, in which the derivative is genuinely a ratio of two differentials, and the integral likewise behaves in exact accordance with Leibniz notation. However, this requires that derivative and integral first be defined by other means, and as such expresses the self-consistency and computational efficacy of the Leibniz notation rather than giving it a new foundation.
History
The Newton–Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton worked with fluxions and fluents, Leibniz based his approach on generalizations of sums and differences. Leibniz was the first to use the character. He based the character on the Latin word summa ("sum"), which he wrote with the elongated s commonly used in Germany at the time. Viewing differences as the inverse operation of summation, he used the symbol , the first letter of the Latin differentia, to indicate this inverse operation. Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them. Notations he used for the differential of ranged succe
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https://en.wikipedia.org/wiki/Geometric%20standard%20deviation
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In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor. When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from (the geometric mean divided by the geometric SD factor) to (the geometric mean multiplied by the geometric SD factor), and one cannot add/subtract "geometric SD factor" to/from geometric mean.
Definition
If the geometric mean of a set of numbers is denoted as then the geometric standard deviation is
Derivation
If the geometric mean is
then taking the natural logarithm of both sides results in
The logarithm of a product is a sum of logarithms (assuming is positive for all so
It can now be seen that is the arithmetic mean of the set therefore the arithmetic standard deviation of this same set should be
This simplifies to
Geometric standard score
The geometric version of the standard score is
If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score can be reconstructed by
Relationship to log-normal distribution
The geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean. As the log-transform of a log-normal distribution results in a normal distribution, we see that the
geometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e.
As such, the geometric mean and the geometric standard deviation of a sample of
data from a log-normally distributed population may be used to find the bounds of confidence intervals analogously to the way the arithmetic mean and standard deviation are used to bound confidence intervals for a normal distribution. See discussion in log-normal distribution for details.
References
External links
Non-Newtonian calculus website
Scale statistics
Non-Newtonian calculus
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https://en.wikipedia.org/wiki/Charts%20on%20SO%283%29
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In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another.
The space of rotations
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space. A length-preserving transformation which reverses orientation is called an improper rotation. Every improper rotation of three-dimensional Euclidean space is a rotation followed by a reflection in a plane through the origin.
Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) for reasons explained below.
The space of rotations is isomorphic with the set of rotation operators and the set of orthonormal matrices with determinant +1. It is also closely related (double covered) with the set of quaternions with their internal product, as well as to the set of rotation vectors (though here the relation is harder to describe, see below for details), with a different internal composition operation given by the product of their equivalent matrices.
Rotation vectors notation arise from the Euler's rotation theorem which states that any rotation in three dimensions can be described by a rotation by some angle about some axis. Considering this, we can then specify the axis of one of these rotations by two angles, and we can use the radius of the vector to specify the angle of rotation. These vectors represent a ball in 3D with an unusual topology.
This 3D solid sphere is equivalent to the surface of a 4D disc, which is also a 3D variety. For doing this equivalence, we will have to define how will we represent a rotation with this 4D-embedded surface.
The hypersphere of rotations
Visualizing the hypersphere
It is interesting to consider the space as the three-dimensional sphere S3, the boundary of a disk in 4-dimensional Euclidean space. For doing this, we will have to define how we represent a rotation with this 4D-embedded surface.
The way in which the radius can be used to specify the angle of rotation is not straightfor
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https://en.wikipedia.org/wiki/Spin%20group
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In mathematics the spin group Spin(n) is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
The group multiplication law on the double cover is given by lifting the multiplication on .
As a Lie group, Spin(n) therefore shares its dimension, , and its Lie algebra with the special orthogonal group.
For , Spin(n) is simply connected and so coincides with the universal cover of SO(n).
The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −.
Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations.
Motivation and physical interpretation
The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; however, space is not zero-dimensional, and so the spin group is used to define spin structures on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection can simplify calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not).
Construction
Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space V with a definite quadratic form q. The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal. The tensor algebra (over the reals) may be written as
The Clifford algebra Cl(V) is then the quotient algebra
where is the quadratic form applied to a vector . The resulting space is finite dimensional, naturally graded (as a vector space), and can be written as
where is the dimension of , and . The spin algebra is defined as
where the last is a short-hand for V being a real vector space of real dimension n. It is a Lie algebra; it has a natural action on V, and in this way can be shown to be isomorphic to the Lie algebra of the special orthogonal group.
The pin group is a subgroup of 's Clifford group of all elements of the form
where each is of unit length:
The spin group is then defined as
where
is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(V) consists of all elements of Pin(V), given above, with the restriction to k being an even nu
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https://en.wikipedia.org/wiki/Real%20projective%20plane
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In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin.
The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.
Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together with a half-twist, the real projective plane can thus be represented as a unit square (that is, ) with its sides identified by the following equivalence relations:
for
and
for
as in the leftmost diagram shown here.
Examples
Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. Some of the more important examples are described below.
The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be the projective plane, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.
The projective sphere
Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane:
any pair of distinct great circles meet at a pair of antipodal points; and
any two distinct pairs of antipodal points lie on a single great circle.
If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = x or y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R3.
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https://en.wikipedia.org/wiki/Random%20Fibonacci%20sequence
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In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation , where the signs + or − are chosen at random with equal probability , independently for different . By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943... , a mathematical constant that was later named Viswanath's constant.
Description
A random Fibonacci sequence is an integer random sequence given by the numbers for natural numbers , where and the subsequent terms are chosen randomly according to the random recurrence relation
An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence, the next element is either their sum or their difference with probability 1/2, independently of all the choices made previously. If in the random Fibonacci sequence the plus sign is chosen at each step, the corresponding instance is the Fibonacci sequence (Fn),
If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence
However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:
Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:
where the signs are chosen independently for different n with equal probabilities for + or −. Thus
where (Mk) is a sequence of independent identically distributed random matrices taking values A or B with probability 1/2:
Growth rate
Johannes Kepler discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence (Fn) approaches the golden ratio which is approximately 1.61803. In 1765, Leonhard Euler published an explicit formula, known today as the Binet formula,
It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio φ.
In 1960, Hillel Furstenberg and Harry Kesten showed that for a general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the nth root of |fn| converges to a constant value almost surely, or with probability one:
An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating point arithmetic validated by an analysis of the rounding error.
Generalization
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https://en.wikipedia.org/wiki/Abelian%20and%20Tauberian%20theorems
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In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.
There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense.
In the theory of integral transforms, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.
Abelian theorems
For any summation method L, its Abelian theorem is the result that if c = (cn) is a convergent sequence, with limit C, then L(c) = C.
An example is given by the Cesàro method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity. One can prove that if c does converge to C, then so does the sequence (dN) where
To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take N large enough to make the initial segment of terms up to cN average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded.
The name derives from Abel's theorem on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term
anzn
and set z = r ·eiθ. That theorem has its main interest in the case that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum of the an. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless, if the sum of the an exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
Tauberian theorems
Partial converses to Abelian theorems are called Tauberian theorems. The origina
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https://en.wikipedia.org/wiki/Symmetry%20of%20second%20derivatives
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In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
of variables without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity
so that they form an symmetric matrix, known as the function's Hessian matrix. Sufficient conditions for the above symmetry to hold are established by a result known as Schwarz's theorem, Clairaut's theorem, or Young's theorem.
In the context of partial differential equations it is called the Schwarz integrability condition.
Formal expressions of symmetry
In symbols, the symmetry may be expressed as:
Another notation is:
In terms of composition of the differential operator which takes the partial derivative with respect to :
.
From this relation it follows that the ring of differential operators with constant coefficients, generated by the , is commutative; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to monomials, so that one can take polynomials in the as a domain. In fact smooth functions are another valid domain.
History
The result on the equality of mixed partial derivatives under certain conditions has a long history. The list of unsuccessful proposed proofs started with Euler's, published in 1740, although already in 1721 Bernoulli had implicitly assumed the result with no formal justification. Clairaut also published a proposed proof in 1740, with no other attempts until the end of the 18th century. Starting then, for a period of 70 years, a number of incomplete proofs were proposed. The proof of Lagrange (1797) was improved by Cauchy (1823), but assumed the existence and continuity of the partial derivatives and . Other attempts were made by P. Blanchet (1841), Duhamel (1856), Sturm (1857), Schlömilch (1862), and Bertrand (1864). Finally in 1867 Lindelöf systematically analyzed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal.
Six years after that, Schwarz succeeded in giving the first rigorous proof. Dini later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by Jordan in 1883 that is still the proof found in most textbooks. Minor variants of earlier proofs were published by Laurent (1885), Peano (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), Vivanti (1899) and Pierpont (1905). Further progress was made in 1907-1909 when E. W. Hobson and W. H. Young found proofs with weaker conditions than those of Schwarz and Dini. In 1918, Carathéodory gave a different proof based on the Lebesgue integral.
Schwarz's theorem
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed
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https://en.wikipedia.org/wiki/Hessian%20matrix
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In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or, ambiguously, by ∇2.
Definitions and properties
Suppose is a function taking as input a vector and outputting a scalar If all second-order partial derivatives of exist, then the Hessian matrix of is a square matrix, usually defined and arranged as
That is, the entry of the th row and the th column is
If furthermore the second partial derivatives are all continuous, the Hessian matrix is a symmetric matrix by the symmetry of second derivatives.
The determinant of the Hessian matrix is called the .
The Hessian matrix of a function is the transpose of the Jacobian matrix of the gradient of the function ; that is:
Applications
Inflection points
If is a homogeneous polynomial in three variables, the equation is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most inflection points, since the Hessian determinant is a polynomial of degree
Second-derivative test
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point is a local maximum, local minimum, or a saddle point, as follows:
If the Hessian is positive-definite at then attains an isolated local minimum at If the Hessian is negative-definite at then attains an isolated local maximum at If the Hessian has both positive and negative eigenvalues, then is a saddle point for Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite.
For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of Morse theory.
The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then is a local minimum, and if it is negative, then is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is
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https://en.wikipedia.org/wiki/Cartan%20connection
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In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.
The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile). The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
Cartan reformulated the differential geometry of (pseudo) Riemannian geometry, as well as the differential geometry of manifolds equipped with some non-metric structure, including Lie groups and homogeneous spaces. The term 'Cartan connection' most often refers to Cartan's formulation of a (pseudo-)Riemannian, affine, projective, or conformal connection. Although these are the most commonly used Cartan connections, they are special cases of a more general concept.
Cartan's approach seems at first to be coordinate dependent because of the choice of frames it involves. However, it is not, and the notion can be described precisely using the language of principal bundles. Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport. They have many applications in geometry and physics: see the method of moving frames, Cartan formalism and Einstein–Cartan theory for some examples.
Introduction
At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries—those with zero curvature—are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
A Klein geometry consists of a Lie group G together with a Lie subgroup H of G. Together G and H determine a homogeneous space G/H, on which the group G acts by left-translation. Klein's aim was then to study objects living on the homogeneous space which were congruent by the action of G. A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a manifold a copy of a Klein geometry, and to regard this copy as tangent to the manifold. Thus the geometry of the manifold is infinit
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https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil%20integral
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In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like
This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval and then let .
Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations on the notions of absolute continuity), and by Oskar Perron, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical.
Later, in 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral. Ralph Henstock independently introduced a similar integral that extended the theory in 1961, citing his investigations of Ward's extensions to the Perron integral. Due to these two important contributions, it is now commonly known as the Henstock–Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.
Definition
Given a tagged partition of , that is,
together with each subinterval's tag defined as a point
we define the Riemann sum for a function to be
where This is the summation of each subinterval's length () multiplied by the function evaluated at that subinterval's tag ().
Given a positive function
which we call a gauge, we say a tagged partition P is -fine if
We now define a number to be the Henstock–Kurzweil integral of if for every there exists a gauge such that whenever is -fine, we have
If such an exists, we say that is Henstock–Kurzweil integrable on .
Cousin's theorem states that for every gauge , such a -fine partition P does exist, so this condition cannot be satisfied vacuously. The Riemann integral can be regarded as the special case where we only allow constant gauges.
Properties
Let be any function.
Given , is Henstock–Kurzweil integrable on if and only if it is Henstock–Kurzweil integrable on both and ; in which case,
Henstock–Kurzweil integrals are linear. Given integrable functions
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https://en.wikipedia.org/wiki/Glide%20reflection
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In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection and translation can look different from the starting configuration, so objects with glide symmetry are in general, not symmetrical under reflection alone. In group theory, the glide plane is classified as a type of opposite isometry of the Euclidean plane.
A single glide is represented as frieze group p11g. A glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. It can also be given a Schoenflies notation as S2∞, Coxeter notation as [∞+,2+], and orbifold notation as ∞×.
Description
The combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. However, a glide reflection cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide reflection, with as special case just a reflection. These are the two kinds of indirect isometries in 2D.
For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In coordinates, it takes
This isometry maps the x-axis to itself; any other line which is parallel to the x-axis gets reflected in the x-axis, so this system of parallel lines is left invariant.
The isometry group generated by just a glide reflection is an infinite cyclic group.
Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group.
In the case of glide reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type is frieze group p11g.
Example pattern with this symmetry group:
Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C2.
Example pattern with this symmetry group:
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach.
For any symmetry group containing some glide reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry.
Glide reflection symmetry with respect to two parallel lines with the same translation implies that there is also translational symmetry in the direction perpendicular to these lines, with a translation distance whic
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https://en.wikipedia.org/wiki/Pincherle%20derivative
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In mathematics, the Pincherle derivative of a linear operator on the vector space of polynomials in the variable x over a field is the commutator of with the multiplication by x in the algebra of endomorphisms . That is, is another linear operator
(for the origin of the notation, see the article on the adjoint representation) so that
This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
Properties
The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators and belonging to
;
where is the composition of operators.
One also has where is the usual Lie bracket, which follows from the Jacobi identity.
The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is
This formula generalizes to
by induction. This proves that the Pincherle derivative of a differential operator
is also a differential operator, so that the Pincherle derivative is a derivation of .
When has characteristic zero, the shift operator
can be written as
by the Taylor formula. Its Pincherle derivative is then
In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars .
If T is shift-equivariant, that is, if T commutes with Sh or , then we also have , so that is also shift-equivariant and for the same shift .
The "discrete-time delta operator"
is the operator
whose Pincherle derivative is the shift operator .
See also
Commutator
Delta operator
Umbral calculus
References
External links
Weisstein, Eric W. "Pincherle Derivative". From MathWorld—A Wolfram Web Resource.
Biography of Salvatore Pincherle at the MacTutor History of Mathematics archive.
Differential algebra
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https://en.wikipedia.org/wiki/Convex
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Convex or convexity may refer to:
Science and technology
Convex lens, in optics
Mathematics
Convex set, containing the whole line segment that joins points
Convex polygon, a polygon which encloses a convex set of points
Convex polytope, a polytope with a convex set of points
Convex metric space, a generalization of the convexity notion in abstract metric spaces
Convex function, when the line segment between any two points on the graph of the function lies above or on the graph
Convex conjugate, of a function
Convexity (algebraic geometry), a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces
Economics and finance
Convexity (finance), second derivatives in financial modeling generally
Convexity in economics
Bond convexity, a measure of the sensitivity of the duration of a bond to changes in interest rates
Convex preferences, an individual's ordering of various outcomes
Other uses
Convex Computer, a former company that produced supercomputers
See also
List of convexity topics
Non-convexity (economics), violations of the convexity assumptions of elementary economics
Obtuse angle
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https://en.wikipedia.org/wiki/Statistical%20parameter
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In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarizes or describes an aspect of the population, such as a mean or a standard deviation. If a population exactly follows a known and defined distribution, for example the normal distribution, then a small set of parameters can be measured which completely describes the population, and can be considered to define a probability distribution for the purposes of extracting samples from this population.
A parameter is to a population as a statistic is to a sample; that is to say, a parameter describes the true value calculated from the full population, whereas a statistic is an estimated measurement of the parameter based on a sample. Thus a "statistical parameter" can be more specifically referred to as a population parameter.
Discussion
Parameterised distributions
Suppose that we have an indexed family of distributions. If the index is also a parameter of the members of the family, then the family is a parameterized family. Among parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential family of distributions. For example, the family of normal distributions has two parameters, the mean and the variance: if those are specified, the distribution is known exactly. The family of chi-squared distributions can be indexed by the number of degrees of freedom: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.
Measurement of parameters
In statistical inference, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to estimate or infer what they can about the parameter based on a random sample of observations taken from the full population. Estimators of a set of parameters of a specific distribution are often measured for a population, under the assumption that the population is (at least approximately) distributed according to that specific probability distribution. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a Pearson's chi-squared test). Even if a family of distributions is not specified, quantities such as the mean and variance can generally still be regarded as statistical parameters of the population, and statistical procedures can still attempt to make inferences about such population parameters.
Types of parameters
Parameters are given names appropriate to their roles, including the following:
location parameter
dispersion parameter or scale parameter
shape parameter
Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term concentration parameter is used for quantities that index how variable the outcomes wou
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https://en.wikipedia.org/wiki/Moment
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Moment or Moments may refer to:
Science
Moment (mathematics), a concept in probability theory and statistics
Moment (physics), a combination of a physical quantity and a distance
Moment of force, torque
Time
Present time
An instant
Moment (unit), a medieval unit of time
Technology
Moment space surveillance complex, a Russian military apparatus
Samsung M900 Moment, an Android phone
Arts
Film and television
Moment (film), a 1978 Yugoslav film
Moments (1974 film), a British drama starring Angharad Rees
Moments (1979 film), a French-Israeli film
Moments (talk show), a Philippine TV celebrity talk show
Music and dance
Moment form, a musical concept developed by Karlheinz Stockhausen
Moment Rustica (ballet), a Martha Graham ballet
Momente or Moments, a musical composition by Stockhausen
The Moments, American R&B vocal group
Albums
Moment (Dark Tranquillity album), 2020
Moment (Speed album), 1998
Moment (EP), a 2019 EP by Peggy Gou
Moments (Darude album), 2015
Moments (Christine Guldbrandsen album), 2004
Moments (Mark Holden album), 1995
Moments (Leo Ku album), 2007
Moments (Barbara Mandrell album), 1986
Moments (Andrew Rayel album), 2017
Moments (Boz Scaggs album), 1971
Moments, a 2015 album by Maudy Ayunda
Songs
"Moment" (Blanche song), 2018
"Moment" (SMAP song), 2012
"Moments", a song by Westlife from Westlife, 1999
"Moment", a song by Aiden from Conviction, 2007
"Moment", a song by Young Money from Young Money: Rise of an Empire, featuring Lil Wayne, 2014
"Moment", a song by Victoria Monét from Jaguar, 2020
"Moments" (Ayumi Hamasaki song), 2004
"Moments" (Emerson Drive song), 2007
"Moments" (Hans Bollandsås song), 2010
"Moments" (One Direction song), 2011
"Moments" (Tove Lo song), 2014
"Moments" (Freddy Verano song), 2015
"Lost in the Moment", a song by NF from Therapy Session, featuring Jonathan Thulin, 2016
Publications and literature
Moment (magazine), an American Jewish publication
"Moments" (poem), a poem wrongly attributed to Jorge Luis Borges
See also
Momentum (disambiguation)
The Moment (disambiguation)
Theory of moments
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https://en.wikipedia.org/wiki/Syzygy
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Syzygy (from Greek Συζυγία "conjunction, yoked together") may refer to:
Science
Syzygy (astronomy), a collinear configuration of three celestial bodies
Syzygy (mathematics), linear relation between generators of a module
Syzygy, in biology, the pairing of chromosomes during meiosis
Syzygy endgame tablebases, used by chess engines
Philosophy
Syzygy, a concept in the philosophy of Vladimir Solovyov denoting "close union"
Syzygy, a term used by Carl Jung to mean a union of opposites, e.g. anima and animus
Syzygy, female–male pairings of the emanations known as Aeon (Gnosticism)
Literature
Epirrhematic syzygy: a system of symmetrically corresponding verse forms in Greek Old Comedy
"It Wasn't Syzygy", a short story by Theodore Sturgeon
Syzygy, a novel by Michael G. Coney
Syzygy (novel), a novel by Frederik Pohl
Syzygy (poetry), the combination of two metrical feet into a single unit
Syzygy Darklock, a fictional character in the comic book series Dreadstar
Syzygy Publishing, an American comics publisher founded by Chris Ryall and Ashley Wood.
Film, television, and games
Atari, Inc., the successor to the Syzygy Co.
"Syzygy" (The X-Files), a 1996 episode of the science fiction series
Syzygy, a Robot character from the video game Unreal Tournament 2003
Syzygy, a game for the Dragon 32 home computer, published by Microdeal
Syzygy, a linking word game by Lewis Carroll, published in The Lady magazine
Syzygy, a Great Old One in the game Eldritch Horror (board game), introduced in the expansion Strange Remnants
Syzygy Co., an arcade game engineering company co-founded by Nolan Bushnell
SYZYGY, the title of Chapter 4 of Part 2 of the Netflix series The OA
Music
Syzygy, an alternative electronica music duo featuring Dominic Glynn
Syzygy, composition by Del Tredici
Syzygys (band), a Japanese band
Albums
Syzygy (EP), 1998, by Lynch Mob
Syzygy (LP), 2013, by Lucrecia Dalt
Songs
"Syzygy", a track from the 1987 album Michael Brecker
"Syzygy", a track by Gene Loves Jezebel on the 1990 album Kiss of Life
"Syzygy, Part I", "Syzygy, Part II", and "Syzygy, Part III", tracks from the 2004 album Suns of the Tundra
"Syzygy", a track by Los Angeles Guitar Quartet on the 2012 LAGQ: Latin album
"Syzygy", a track by Mickey Factz on the 2015 mixtape "Y3"
"Syzygy", a track by Laurel Halo on the 2017 album Dust
See also
Caledonian Antisyzygy, a term referring to the Scottish psyche and literature
Szyzyg, the YouTube username of Scott Manley
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https://en.wikipedia.org/wiki/Set%20theory%20%28music%29
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Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, melodic inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well.
Mathematical set theory versus musical set theory
Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).
Moreover, musical set theory is more closely related to group theory and combinatorics than to mathematical set theory, which concerns itself with such matters as, for example, various sizes of infinitely large sets. In combinatorics, an unordered subset of objects, such as pitch classes, is called a combination, and an ordered subset a permutation. Musical set theory is better regarded as an application of combinatorics to music theory than as a branch of mathematical set theory. Its main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets.
Set and set types
The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes. More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates). The elements of a set may be manifested in music as simultaneous chords, successive tones (as in a melody), or both. Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {}, or square brackets: [].
Some theorists use angle brackets to denote ordered sequences, while others distinguish ordered sets by separating the numbers with spaces. Thus one might notate the unordered set of pitch classes 0, 1, and 2 (corresponding in this case to C, C, and D) as {0,1,2}. The ordered sequence C-C-D would be notated or (0,1,2). Although C is considered zero in this example, this
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https://en.wikipedia.org/wiki/Nonagon
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In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon.
The name nonagon is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name enneagon comes from Greek enneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common than "nonagon".
Regular nonagon
A regular nonagon is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length a is given by
where the radius r of the inscribed circle of the regular nonagon is
and where R is the radius of its circumscribed circle:
Construction
Although a regular nonagon is not constructible with compass and straightedge (as 9 = 32, which is not a product of distinct Fermat primes), there are very old methods of construction that produce very close approximations.
It can be also constructed using neusis, or by allowing the use of an angle trisector.
Symmetry
The regular enneagon has Dih9 symmetry, order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, and 3 cyclic group symmetries: Z9, Z3, and Z1.
These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g9 subgroup has no degrees of freedom but can seen as directed edges.
Tilings
The regular enneagon can tessellate the euclidean tiling with gaps. These gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H(*;3;*;[2]) with H representing *632 hexagonal symmetry in the plane.
Graphs
The K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex.
Pop culture references
They Might Be Giants have a song entitled "Nonagon" on their children's album Here Come the 123s. It refers to both an attendee at a party at which "everybody in the party is a many-sided polygon" and a dance they perform at this party.
Slipknot's logo is also a version of a nonagon, being a nine-pointed star made of three triangles, referring to the nine members.
King Gizzard & the Lizard Wizard have an album titled 'Nonagon Infinity', the album art featuring a nonagonal complete graph. The album consists of nine songs and repeats cyclically.
Architecture
Temples of the Baháʼí Faith, called Baháʼí Houses of Worship, are
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https://en.wikipedia.org/wiki/Rewriting
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In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.
Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting.
Example cases
Logic
In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a rewriting system. The rules of an example of such a system would be:
(double negation elimination)
(De Morgan's laws)
(distributivity)
where the symbol () indicates that an expression matching the left hand side of the rule can be rewritten to one formed by the right hand side, and the symbols each denote a subexpression. In such a system, each rule is chosen so that the left side is equivalent to the right side, and consequently when the left side matches a subexpression, performing a rewrite of that subexpression from left to right maintains logical consistency and value of the entire expression.
Arithmetic
Term rewriting systems can be employed to compute arithmetic operations on natural numbers.
To this end, each such number has to be encoded as a term.
The simplest encoding is the one used in the Peano axioms, based on the constant 0 (zero) and the successor function S. For example, the numbers 0, 1, 2, and 3 are represented by the terms 0, S(0), S(S(0)), and S(S(S(0))), respectively.
The following term rewriting system can then be used to compute sum and product of given natural numbers.
For example, the computation of 2+2 to result in 4 can be duplicated by term rewriting as follows:
where the rule numbers are given above the rewrites-to arrow.
As another example, the computation of 2⋅2 looks like:
where the last step comprises the previous example computation.
Linguistics
In linguistics, phrase structure rules, also called rewrite rules, are used in some systems of generative grammar, as a means of generating the grammatically correct sentences of a language. Such a rule typically takes the form , where A is a syntactic category label, such as noun phrase or sentence, and X is a sequence of such labels or morphemes, expressing the fact that A can be replaced by X in generating the constituent structure of a sentence. For example, the rule means that a sentence can consist of a noun phrase (NP) followed by a verb phrase (VP); further
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https://en.wikipedia.org/wiki/Frank%20J.%20Tipler
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Frank Jennings Tipler (born February 1, 1947) is an American mathematical physicist and cosmologist, holding a joint appointment in the Departments of Mathematics and Physics at Tulane University. Tipler has written books and papers on the Omega Point based on Pierre Teilhard de Chardin's religious ideas, which he claims is a mechanism for the resurrection of the dead. He is also known for his theories on the Tipler cylinder time machine. His work has attracted criticism, most notably from Quaker and systems theorist George Ellis who has argued that his theories are largely pseudoscience.
Biography
Tipler was born in Andalusia, Alabama, to Frank Jennings Tipler Jr., a lawyer, and Anne Tipler, a homemaker. Tipler attended the Massachusetts Institute of Technology from 1965 to 1969, where he completed a Bachelor of Science degree in physics. In 1976 he completed his PhD with the University of Maryland. Tipler was hired in a series of postdoctoral research positions at three universities, with the final one being at the University of Texas, working under John Archibald Wheeler, Abraham Taub, Rainer K. Sachs, and Dennis W. Sciama. Tipler became an associate professor in mathematical physics in 1981 and a full professor in 1987 at Tulane University, where he has been a faculty member ever since.
The Omega Point cosmology
The Omega Point is a term Tipler uses to describe a cosmological state in the distant proper-time future of the universe. He claims that this point is required to exist due to the laws of physics. According to him, it is required, for the known laws of physics to be consistent, that intelligent life take over all matter in the universe and eventually force its collapse. During that collapse, the computational capacity of the universe diverges to infinity, and environments emulated with that computational capacity last for an infinite duration as the universe attains a cosmological singularity. This singularity is Tipler's Omega Point. With computational resources diverging to infinity, Tipler states that a society in the far future would be able to resurrect the dead by emulating alternative universes. Tipler identifies the Omega Point with God, since, in his view, the Omega Point has all the properties of God claimed by most traditional religions.
Tipler's argument of the Omega Point being required by the laws of physics is a more recent development that arose after the publication of his 1994 book The Physics of Immortality. In that book (and in papers he had published up to that time), Tipler had offered the Omega Point cosmology as a hypothesis, while still claiming to confine the analysis to the known laws of physics.
Tipler, along with co-author physicist John D. Barrow, defined the "final anthropic principle" (FAP) in their 1986 book The Anthropic Cosmological Principle as a generalization of the anthropic principle:
One paraphrasing of Tipler's argument for FAP runs as follows: For the universe to physically exist, it
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https://en.wikipedia.org/wiki/Exploratory%20data%20analysis
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In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling and thereby contrasts traditional hypothesis testing. Exploratory data analysis has been promoted by John Tukey since 1970 to encourage statisticians to explore the data, and possibly formulate hypotheses that could lead to new data collection and experiments. EDA is different from initial data analysis (IDA), which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, and handling missing values and making transformations of variables as needed. EDA encompasses IDA.
Overview
Tukey defined data analysis in 1961 as: "Procedures for analyzing data, techniques for interpreting the results of such procedures, ways of planning the gathering of data to make its analysis easier, more precise or more accurate, and all the machinery and results of (mathematical) statistics which apply to analyzing data."
Exploratory data analysis is an analysis technique to analyze and investigate the data set and summaries the main characteristics of the dataset. Main advantage of EDA is providing the data visualization of data after conducting the analysis.
Tukey's championing of EDA encouraged the development of statistical computing packages, especially S at Bell Labs. The S programming language inspired the systems S-PLUS and R. This family of statistical-computing environments featured vastly improved dynamic visualization capabilities, which allowed statisticians to identify outliers, trends and patterns in data that merited further study.
Tukey's EDA was related to two other developments in statistical theory: robust statistics and nonparametric statistics, both of which tried to reduce the sensitivity of statistical inferences to errors in formulating statistical models. Tukey promoted the use of five number summary of numerical data—the two extremes (maximum and minimum), the median, and the quartiles—because these median and quartiles, being functions of the empirical distribution are defined for all distributions, unlike the mean and standard deviation; moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries (the mean and standard deviation). The packages S, S-PLUS, and R included routines using resampling statistics, such as Quenouille and Tukey's jackknife and Efron bootstrap, which are nonparametric and robust (for many problems).
Exploratory data analysis, robust statistics, nonparametric statistics, and the development of statistical programming languages facilitated statisticians' work on scientific and engineering problems. Such problems included the fabrication of semiconductors and the understanding of communications networks
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https://en.wikipedia.org/wiki/Cross-validation%20%28statistics%29
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Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set.
Cross-validation is a resampling method that uses different portions of the data to test and train a model on different iterations. It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (called the validation dataset or testing set). The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias and to give an insight on how the model will generalize to an independent dataset (i.e., an unknown dataset, for instance from a real problem).
One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance.
In summary, cross-validation combines (averages) measures of fitness in prediction to derive a more accurate estimate of model prediction performance.
Motivation
Assume a model with one or more unknown parameters, and a data set to which the model can be fit (the training data set). The fitting process optimizes the model parameters to make the model fit the training data as well as possible. If an independent sample of validation data is taken from the same population as the training data, it will generally turn out that the model does not fit the validation data as well as it fits the training data. The size of this difference is likely to be large especially when the size of the training data set is small, or when the number of parameters in the model is large. Cross-validation is a way to estimate the size of this effect.
Example: linear regression
In linear regression, there exist real response values , and n p-dimensional vector covariates x1, ..., xn. The components of the vector xi are denoted xi1, ..., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1 ≤ i ≤ n, then the fit can be assessed using the mean squared error (MSE). The MSE for given estimated parameter values a and β on the training set (xi, yi) 1 ≤ i ≤ n is defined as:
If the model is correctly specified, it can be shown under mild assumptions that
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https://en.wikipedia.org/wiki/Square%20triangular%20number
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In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
0, 1, 36, , , , , , ,
Explicit formulas
Write for the th square triangular number, and write and for the sides of the corresponding square and triangle, so that
Define the triangular root of a triangular number to be . From this definition and the quadratic formula,
Therefore, is triangular ( is an integer) if and only if is square. Consequently, a square number is also triangular if and only if is square, that is, there are numbers and such that . This is an instance of the Pell equation with . All Pell equations have the trivial solution for any ; this is called the zeroth solution, and indexed as . If denotes the th nontrivial solution to any Pell equation for a particular , it can be shown by the method of descent that
Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever is not a square. The first non-trivial solution when is easy to find: it is (3,1). A solution to the Pell equation for yields a square triangular number and its square and triangular roots as follows:
Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from , is 36.
The sequences , and are the OEIS sequences , , and respectively.
In 1778 Leonhard Euler determined the explicit formula
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for and are:
Pell's equation
The problem of finding square triangular numbers reduces to Pell's equation in the following way.
Every triangular number is of the form . Therefore we seek integers , such that
Rearranging, this becomes
and then letting and , we get the Diophantine equation
which is an instance of Pell's equation. This particular equation is solved by the Pell numbers as
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
Recurrence relations
There are recurrence relations for the square triangular numbers, as well as for the sides of the square and triangle involved. We have
We have
Other characterizations
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of .
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers: If the th triangular number is square, then so is the larger th triangular number, since:
As the product of three squares, the right hand side is square. The triangular roots are alternately simultaneously one less than a square and twice a square if is even, and simultaneously a square and one less than twice a square if is odd. Thus,
49 = 72 = 2 × 52 − 1,
288 = 172 − 1 = 2 ×
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https://en.wikipedia.org/wiki/Testing%20hypotheses%20suggested%20by%20the%20data
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In statistics, hypotheses suggested by a given dataset, when tested with the same dataset that suggested them, are likely to be accepted even when they are not true. This is because circular reasoning (double dipping) would be involved: something seems true in the limited data set; therefore we hypothesize that it is true in general; therefore we wrongly test it on the same, limited data set, which seems to confirm that it is true. Generating hypotheses based on data already observed, in the absence of testing them on new data, is referred to as post hoc theorizing (from Latin post hoc, "after this").
The correct procedure is to test any hypothesis on a data set that was not used to generate the hypothesis.
The general problem
Testing a hypothesis suggested by the data can very easily result in false positives (type I errors). If one looks long enough and in enough different places, eventually data can be found to support any hypothesis. Yet, these positive data do not by themselves constitute evidence that the hypothesis is correct. The negative test data that were thrown out are just as important, because they give one an idea of how common the positive results are compared to chance. Running an experiment, seeing a pattern in the data, proposing a hypothesis from that pattern, then using the same experimental data as evidence for the new hypothesis is extremely suspect, because data from all other experiments, completed or potential, has essentially been "thrown out" by choosing to look only at the experiments that suggested the new hypothesis in the first place.
A large set of tests as described above greatly inflates the probability of type I error as all but the data most favorable to the hypothesis is discarded. This is a risk, not only in hypothesis testing but in all statistical inference as it is often problematic to accurately describe the process that has been followed in searching and discarding data. In other words, one wants to keep all data (regardless of whether they tend to support or refute the hypothesis) from "good tests", but it is sometimes difficult to figure out what a "good test" is. It is a particular problem in statistical modelling, where many different models are rejected by trial and error before publishing a result (see also overfitting, publication bias).
The error is particularly prevalent in data mining and machine learning. It also commonly occurs in academic publishing where only reports of positive, rather than negative, results tend to be accepted, resulting in the effect known as publication bias.
Correct procedures
All strategies for sound testing of hypotheses suggested by the data involve including a wider range of tests in an attempt to validate or refute the new hypothesis. These include:
Collecting confirmation samples
Cross-validation
Methods of compensation for multiple comparisons
Simulation studies including adequate representation of the multiple-testing actually involved
Henry Scheffé's
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https://en.wikipedia.org/wiki/Cellular
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Cellular may refer to:
Cellular automaton, a model in discrete mathematics
Cell biology, the evaluation of cells work and more
Cellular (film), a 2004 movie
Cellular frequencies, assigned to networks operating in cellular RF bands
Cellular manufacturing
Cellular network, cellular radio networks
U.S. Cellular Field, also known as "The Cell", a baseball stadium in Chicago
U.S. Cellular Arena, an arena in Milwaukee, Wisconsin
Terms such as cellular organization, cellular structure, cellular system, and so on may refer to:
Cell biology, the evaluation of how cells work and more
Cellular communication networks, systems for allowing communication through mobile phones and other mobile devices
Cellular organizational structures, methods of human organization in social groups
Clandestine cell organizations, entities organized to commit crimes, acts of terror, or other malicious activities
See also
Cell (disambiguation)
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https://en.wikipedia.org/wiki/County%20statistics%20of%20the%20United%20States
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In 45 of the 50 states of the United States, the county is used for the level of local government immediately below the state itself. Louisiana uses parishes, and Alaska uses boroughs. In Connecticut, Massachusetts, and Rhode Island, some or all counties within states have no governments of their own; the counties continue to exist as legal entities, however, and are used by states for some administrative functions and by the United States Census bureau for statistical analysis. There are 3,242 counties and county equivalent administrative units in total, including the District of Columbia and 100 county-equivalents in the U.S. territories.
There are 41 independent cities in the United States. In Virginia, any municipality that is incorporated as a city legally becomes independent of any county. Where indicated, the statistics below do not include Virginia's 38 independent cities.
In Alaska, most of the land area of the state has no county-level government. Those parts of the state are divided by the United States Census Bureau into census areas, which are not the same as boroughs. The state's largest statistical division by area is the Yukon–Koyukuk Census Area, which is larger than any of the state's boroughs. Although Anchorage is called a municipality, it is considered a consolidated city and borough.
There are 100 county-equivalents in the territories of the United States: they are the 3 districts and 2 atolls of American Samoa, all of Guam (Guam as one single county-equivalent), the 4 municipalities in the Northern Mariana Islands, the 78 municipalities of Puerto Rico, the 3 main islands of the U.S. Virgin Islands, and the 9 islands in the U.S. Minor Outlying Islands. All of these territorial county-equivalents are defined by the U.S. Census Bureau.
Count
This is the number of counties and county-equivalents for each state, the District of Columbia, the 5 inhabited territories of the United States, and the U.S. Minor Outlying Islands.
Lists of counties and county equivalents by number per political division:
Total (50 states and District of Columbia): 3,143 (3,007 counties and 136 county equivalents)
Total (50 states, District of Columbia and territories): 3,243 (3,007 counties and 236 county equivalents)
Average number of counties per state (not including D.C. and the territories): 62.84
Average number of counties per state (including D.C. and the territories): 56.8947368421
Population
Nationwide population extremes
These rankings include county equivalents.
The following is a list of the least populous counties and county-equivalents in all U.S. territory. Note that the only entity on this list with a permanent human population is Swains Island, American Samoa. The first 8 counties (county-equivalents) are uninhabited, while the 10th on the list (Palmyra Atoll) has a small non-permanent human population whose maximum capacity is 20 people.
Population per state or territory
Area
Nationwide land area extremes
The largest
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https://en.wikipedia.org/wiki/Census%20division%20statistics%20of%20Canada
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In some of Canada's provinces census divisions are equivalent to counties. They may also be known by different names in different provinces, or in different parts of provinces. The below table shows the largest and smallest census division in Canada and the provinces and territories by area and by population.
By area
By population
10 fastest growing population (2006–2011)
Division No. 16, Alberta 27.2%
La Jacques-Cartier RCM, Quebec 24.0%
Mirabel RCM, Quebec 21.2%
Division No. 2, Manitoba 17.0%
Les Moulins RCM, Quebec 15.8%
York RM, Ontario 15.7%
Vaudreuil-Soulanges RCM, Quebec 15.7%
Division No. 3, Manitoba 14.4%
Halton RM, Ontario 14.2%
Les Moulins RCM, Quebec 13.8%
10 fastest shrinking population (2006–2011)
Kenora, Ontario −10.6%
Guysborough County, Nova Scotia −10.1%
Northern Rockies Regional Municipality, British Columbia −9.3%
Division No. 3, Newfoundland and Labrador −7.8%
Division No. 9, Newfoundland and Labrador −7.2%
Shelburne, Nova Scotia −6.7%
Victoria County, Nova Scotia −6.3%
La Haute-Côte-Nord RCM, Quebec −6.2%
Inverness, Nova Scotia −5.7%
Témiscouata RCM, Quebec −5.6%
References
Census divisions of Canada
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https://en.wikipedia.org/wiki/Vector%20operator
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A vector operator is a differential operator used in vector calculus. Vector operators include the gradient, divergence, and curl:
Gradient is a vector operator that operates on a scalar field, producing a vector field.
Divergence is a vector operator that operates on a vector field, producing a scalar field.
Curl is a vector operator that operates on a vector field, producing a vector field.
Defined in terms of del:
The Laplacian operates on a scalar field, producing a scalar field:
Vector operators must always come right before the scalar field or vector field on which they operate, in order to produce a result. E.g.
yields the gradient of f, but
is just another vector operator, which is not operating on anything.
A vector operator can operate on another vector operator, to produce a compound vector operator, as seen above in the case of the Laplacian.
See also
del
d'Alembertian operator
Further reading
H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, .
Vector calculus
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https://en.wikipedia.org/wiki/Heesch
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Heesch can refer to:
Heesch, Netherlands, a town in the Bernheze municipality
Heinrich Heesch (1906–1995), German mathematician
Heesch's problem in mathematics
Surnames of German origin
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https://en.wikipedia.org/wiki/Survival%20analysis
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Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival?
To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of degree, or not otherwise localized in time. Even in biological problems, some events (for example, heart attack or other organ failure) may have the same ambiguity. The theory outlined below assumes well-defined events at specific times; other cases may be better treated by models which explicitly account for ambiguous events.
More generally, survival analysis involves the modelling of time to event data; in this context, death or failure is considered an "event" in the survival analysis literature – traditionally only a single event occurs for each subject, after which the organism or mechanism is dead or broken. Recurring event or repeated event models relax that assumption. The study of recurring events is relevant in systems reliability, and in many areas of social sciences and medical research.
Introduction to survival analysis
Survival analysis is used in several ways:
To describe the survival times of members of a group
Life tables
Kaplan–Meier curves
Survival function
Hazard function
To compare the survival times of two or more groups
Log-rank test
To describe the effect of categorical or quantitative variables on survival
Cox proportional hazards regression
Parametric survival models
Survival trees
Survival random forests
Definitions of common terms in survival analysis
The following terms are commonly used in survival analyses:
Event: Death, disease occurrence, disease recurrence, recovery, or other experience of interest
Time: The time from the beginning of an observation period (such as surgery or beginning treatment) to (i) an event, or (ii) end of the study, or (iii) loss of contact or withdrawal from the study.
Censoring / Censored observation: Censoring occurs when we have some information about individual survival time, but we do not know the survival time exactly. The subject is censored in the sense that nothing is observed or known about that subject after the time of censoring. A censored subject may or may not have an event after the end
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https://en.wikipedia.org/wiki/Solenoidal%20vector%20field
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In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
A common way of expressing this property is to say that the field has no sources or sinks.
Properties
The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
where is the outward normal to each surface element.
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:
automatically results in the identity (as can be shown, for example, using Cartesian coordinates):
The converse also holds: for any solenoidal v there exists a vector potential A such that (Strictly speaking, this holds subject to certain technical conditions on v, see Helmholtz decomposition.)
Etymology
Solenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.
Examples
The magnetic field B (see Gauss's law for magnetism)
The velocity field of an incompressible fluid flow
The vorticity field
The electric field E in neutral regions ();
The current density J where the charge density is unvarying, .
The magnetic vector potential A in Coulomb gauge
See also
Longitudinal and transverse vector fields
Stream function
Conservative vector field
Notes
References
Vector calculus
Fluid dynamics
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https://en.wikipedia.org/wiki/Universality
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Universality most commonly refers to:
Universality (philosophy)
Universality (dynamical systems)
Universality principle may refer to:
In statistics, universality principle, a property of systems that can be modeled by random matrices
In law, as a synonym for universal jurisdiction
In moral philosophy, the first formulation of Kant's categorical imperative.
Universality may also refer to several concepts that are also known as "universality"
Background independence, a concept of universality in physical science
Turing-complete, a concept of universality in computation
Universal property, a mathematical concept
Universal jurisdiction, in international law
Lepton universality in the Standard Model of particle physics.
Universality of the Church, a theological concept in Christian ecclesiology
See also
Universal (disambiguation)
Universalism (disambiguation)
Universality probability
Universalization
Universalizability
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https://en.wikipedia.org/wiki/Great%20rhombicosidodecahedron
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In geometry, this name may refer to:
Truncated icosidodecahedron - An Archimedean solid, with Schläfli symbol t0,1,2{5,3}.
Nonconvex great rhombicosidodecahedron - a nonconvex uniform polyhedron, with Schläfli symbol t0,2{5/3,3}.
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https://en.wikipedia.org/wiki/Great%20rhombicuboctahedron
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In geometry, this may refer to:
Truncated cuboctahedron - an Archimedean solid, with Schläfli symbol tr{4,3}, and Coxeter diagram .
Nonconvex great rhombicuboctahedron - a uniform star polyhedron, with Schläfli symbol r{4,3/2}, and Coxeter diagram .
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https://en.wikipedia.org/wiki/Calculus%20of%20communicating%20systems
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The calculus of communicating systems (CCS) is a process calculus introduced by Robin Milner around 1980 and the title of a book describing the calculus. Its actions model indivisible communications between exactly two participants. The formal language includes primitives for describing parallel composition, choice between actions and scope restriction. CCS is useful for evaluating the qualitative correctness of properties of a system such as deadlock or livelock.
According to Milner, "There is nothing canonical about the choice of the basic combinators, even though they were chosen with great attention to economy. What characterises our calculus is not the exact choice of combinators, but rather the choice of interpretation and of mathematical framework".
The expressions of the language are interpreted as a labelled transition system. Between these models, bisimilarity is used as a semantic equivalence.
Syntax
Given a set of action names, the set of CCS processes is defined by the following BNF grammar:
The parts of the syntax are, in the order given above
inactive process the inactive process is a valid CCS process
action the process can perform an action and continue as the process
process identifier write to use the identifier to refer to the process (which may contain the identifier itself, i.e., recursive definitions are allowed)
summation the process can proceed either as the process or the process
parallel composition tells that processes and exist simultaneously
renaming is the process with all actions named renamed as
restriction is the process without action
Related calculi, models, and languages
Communicating sequential processes (CSP), developed by Tony Hoare, is a formal language that arose at a similar time to CCS.
The Algebra of Communicating Processes (ACP) was developed by Jan Bergstra and Jan Willem Klop in 1982, and uses an axiomatic approach (in the style of Universal algebra) to reason about a similar class of processes as CCS.
The pi-calculus, developed by Robin Milner, Joachim Parrow, and David Walker in the late 80's extends CCS with mobility of communication links, by allowing processes to communicate the names of communication channels themselves.
PEPA, developed by Jane Hillston introduces activity timing in terms of exponentially distributed rates and probabilistic choice, allowing performance metrics to be evaluated.
Reversible Communicating Concurrent Systems (RCCS) introduced by Vincent Danos, Jean Krivine, and others, introduces (partial) reversibility in the execution of CCS processes.
Some other languages based on CCS:
Calculus of broadcasting systems
Language Of Temporal Ordering Specification (LOTOS)
Process Calculus for Spatially-Explicit Ecological Models (PALPS) is an extension of CCS with probabilistic choice, locations and attributes for locations
Java Orchestration Language Interpreter Engine (Jolie)
Models that have been used in the study of CCS-like s
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https://en.wikipedia.org/wiki/%CE%A0-calculus
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In theoretical computer science, the -calculus (or pi-calculus) is a process calculus. The -calculus allows channel names to be communicated along the channels themselves, and in this way it is able to describe concurrent computations whose network configuration may change during the computation.
The -calculus has few terms and is a small, yet expressive language (see ). Functional programs can be encoded into the -calculus, and the encoding emphasises the dialogue nature of computation, drawing connections with game semantics. Extensions of the -calculus, such as the spi calculus and applied , have been successful in reasoning about cryptographic protocols. Beside the original use in describing concurrent systems, the -calculus has also been used to reason about business processes and molecular biology.
Informal definition
The -calculus belongs to the family of process calculi, mathematical formalisms for describing and analyzing properties of concurrent computation. In fact, the -calculus, like the λ-calculus, is so minimal that it does not contain primitives such as numbers, booleans, data structures, variables, functions, or even the usual control flow statements (such as if-then-else, while).
Process constructs
Central to the -calculus is the notion of name. The simplicity of the calculus lies in the dual role that names play as communication channels and variables.
The process constructs available in the calculus are the following (a precise definition is given in the following section):
concurrency, written , where and are two processes or threads executed concurrently.
communication, where
input prefixing is a process waiting for a message that was sent on a communication channel named before proceeding as binding the name received to the name Typically, this models either a process expecting a communication from the network or a label c usable only once by a goto c operation.
output prefixing describes that the name is emitted on channel before proceeding as Typically, this models either sending a message on the network or a goto c operation.
replication, written , which may be seen as a process which can always create a new copy of Typically, this models either a network service or a label c waiting for any number of goto c operations.
creation of a new name, written , which may be seen as a process allocating a new constant within The constants of are defined by their names only and are always communication channels. Creation of a new name in a process is also called restriction.
the nil process, written , is a process whose execution is complete and has stopped.
Although the minimalism of the -calculus prevents us from writing programs in the normal sense, it is easy to extend the calculus. In particular, it is easy to define both control structures such as recursion, loops and sequential composition and datatypes such as first-order functions, truth values, lists and integers. Moreover, extensions of th
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https://en.wikipedia.org/wiki/Calculus%20of%20broadcasting%20systems
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Calculus of broadcasting systems (CBS) is a CCS-like calculus where processes speak one at a time and each is heard instantaneously by all others. Speech is autonomous, contention between speakers being resolved nondeterministically, but hearing only happens when someone else speaks. Observationally meaningful laws differ from those of CCS. The handshake communication of CCS is changed to broadcast communication in CBS. This allows several additional features:
Priority, which attaches only to autonomous actions, is simply added to CBS in contrast to CCS, where such actions are the result of communication.
A CBS simulator runs a process by returning a list of values it broadcasts. This permits a powerful combination, CBS with the host language. It yields several elegant algorithms. Only processes with a unique response to each input are needed in practice, so weak bi simulation is a congruence.
CBS subsystems are interfaced by translators; by mapping messages to silence, these can restrict hearing and hide speech. Reversing a translator turns its scope inside out. This permits a new specification for a communication link – the environment of each user should behave like the other user.
See also
Alternating bit protocol
Bisimulation
Calculus of communicating systems (CCS)
Communicating sequential processes (CSP)
Pi-calculus
References
K. V. S. Prasad: A Calculus of Broadcasting Systems, Science of Computer Programming, 25, 1995.
K. V. S. Prasad: Programming with broadcasts, Lecture Notes in Computer Science, Vol. 715, CONCUR, 1993, Springer-Verlag.
K. V. S. Prasad: Broadcasting in time, Lecture Notes in Computer Science, Vol. 1061, COORDINATION, 1996, Springer-Verlag.
External links
Citations from CiteSeer
A TCBS-Implementation on C++ - A Laboratory for the Course "Parallelism"
Process calculi
Parallel computing
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https://en.wikipedia.org/wiki/Discrete%20optimization
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Discrete optimization is a branch of optimization in applied mathematics and computer science.
Scope
As opposed to continuous optimization, some or all of the variables used in a discrete optimization problem are restricted to be discrete variables—that is, to assume only a discrete set of values, such as the integers.
Branches
Three notable branches of discrete optimization are:
combinatorial optimization, which refers to problems on graphs, matroids and other discrete structures
integer programming
constraint programming
These branches are all closely intertwined however, since many combinatorial optimization problems
can be modeled as integer programs (e.g. shortest path) or constraint programs,
any constraint program can be formulated as an integer program and vice versa,
and constraint and integer programs can often be given a combinatorial interpretation.
See also
Diophantine equation
References
Mathematical optimization
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https://en.wikipedia.org/wiki/Continuous%20optimization
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Continuous optimization is a branch of optimization in applied mathematics.
As opposed to discrete optimization, the variables used in the objective function are required to be continuous variables—that is, to be chosen from a set of real values between which there are no gaps (values from intervals of the real line). Because of this continuity assumption, continuous optimization allows the use of calculus techniques.
References
Mathematical optimization
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https://en.wikipedia.org/wiki/Axioms%20%28journal%29
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Axioms is a peer-reviewed open access scientific journal that focuses on all aspects of mathematics, mathematical logic and mathematical physics. It was established in June 2012 and is published quarterly by MDPI.
In September 2021 the journal was among the initial 13 journals included in the official Norwegian list of possibly predatory journals, known as level X. In February 2022, it was removed from the list. According to the Journal Citation Reports, the journal has a 2021 impact factor of 1.824.
The editor-in-chief is Humberto Bustince (Public University of Navarre).
Abstracting and indexing
The journal is abstracted and indexed in:
Science Citation Index Expanded
Scopus
, it is not indexed in MathSciNet or zbMATH.
References
External links
Mathematics journals
Open access journals
Academic journals established in 2012
MDPI academic journals
Quarterly journals
English-language journals
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https://en.wikipedia.org/wiki/Incidence%20matrix
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In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related (called incident in this context) and 0 if they are not. There are variations; see below.
Graph theory
Incidence matrix is a common graph representation in graph theory. It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs.
Undirected and directed graphs
In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.
The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a matrix B, where n and m are the numbers of vertices and edges respectively, such that
For example, the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, ):
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
The incidence matrix of a directed graph is a matrix B where n and m are the number of vertices and edges respectively, such that
(Many authors use the opposite sign convention.)
The oriented incidence matrix of an undirected graph is the incidence matrix, in the sense of directed graphs, of any orientation of the graph. That is, in the column of edge e, there is one 1 in the row corresponding to one vertex of e and one −1 in the row corresponding to the other vertex of e, and all other rows have 0. The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:
where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and Im is the identity matrix of dimension m.
The discrete Laplacian (or Kirchhoff matrix) is obtained from the oriented incidence matrix B(G) by the formula
The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.
Signed and bidirected graphs
The incidence matrix of a signed graph is a generalization of the oriented incidence matrix. It is the incidence matrix of any bidirected graph that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an
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https://en.wikipedia.org/wiki/2003%20Georgian%20parliamentary%20election
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Parliamentary elections were held in Georgia on 2 November 2003 alongside a constitutional referendum. According to statistics released by the Georgian Election Commission, the elections were won by a combination of parties supporting President Eduard Shevardnadze.
However, the results were annulled by the Georgia Supreme Court after the Rose Revolution on 25 November, following allegations of widespread electoral fraud and large public protests which led to the resignation of Shevardnadze. Fresh elections were held on 28 March 2004.
Parties
"For a New Georgia" was the electoral bloc that supported President Eduard Shevardnadze. The Revival Party was an ally of Shevardnadze. The National Movement (NM) was the party of opposition leader Mikhail Saakashvili.
Conduct
Reports of violence, voter intimidation and ballot box stuffing began coming in shortly after the polling stations opened. The biggest problem, however, was the voter lists prepared by the Georgian government.
Mikhail Saakashvili was among tens of thousands who were denied the right to vote. His name, along with names of many thousands across the country, was missing from the voter list prepared by the Georgian government. Entire neighborhoods were mysteriously removed from the voter list in the areas where opposition was likely to do well.
Georgian analysts described the vote as "the messiest and most chaotic election" the country has ever had.
"The government did everything to make this election chaotic. I think there were also (those in) government (who) did not want this election to be orderly because they knew they would lose it," said Ghia Nodia of the Caucasus Institute for Democracy and Development.
An international mission from the Organization for Security and Cooperation in Europe (OSCE) declared that the election fell short of international standards.
"These elections have, regrettably, been insufficient to enhance the credibility of either the electoral or the democratic process," said Bruce George, special co-ordinator of the OSCE chairman-in-office.
Some 450 international observers from 43 countries monitored the polls in one of the largest and longest election observation missions in the OSCE's history.
Supporting the allegations of electoral fraud were also exit polls conducted by an American company, Global Strategy, which showed that the opposition had won by a large margin, with the National Movement coming first with 20% and the government block polling only 14% of the vote.
Results
References
Annulled elections
Parliamentary elections in Georgia (country)
Rose Revolution
Georgia
2003 in Georgia (country)
Election and referendum articles with incomplete results
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https://en.wikipedia.org/wiki/Prenex%20normal%20form
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A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic (e.g. disjunctive normal form or conjunctive normal form), it provides a canonical normal form useful in automated theorem proving.
Every formula in classical logic is logically equivalent to a formula in prenex normal form. For example, if , , and are quantifier-free formulas with the free variables shown then
is in prenex normal form with matrix , while
is logically equivalent but not in prenex normal form.
Conversion to prenex form
Every first-order formula is logically equivalent (in classical logic) to some formula in prenex normal form. There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connectives appear in the formula.
Conjunction and disjunction
The rules for conjunction and disjunction say that
is equivalent to under (mild) additional condition , or, equivalently, (meaning that at least one individual exists),
is equivalent to ;
and
is equivalent to ,
is equivalent to under additional condition .
The equivalences are valid when does not appear as a free variable of ; if does appear free in , one can rename the bound in and obtain the equivalent .
For example, in the language of rings,
is equivalent to ,
but
is not equivalent to
because the formula on the left is true in any ring when the free variable x is equal to 0, while the formula on the right has no free variables and is false in any nontrivial ring. So will be first rewritten as and then put in prenex normal form .
Negation
The rules for negation say that
is equivalent to
and
is equivalent to .
Implication
There are four rules for implication: two that remove quantifiers from the antecedent and two that remove quantifiers from the consequent. These rules can be derived by rewriting the implication
as and applying the rules for disjunction and negation above. As with the rules for disjunction, these rules require that the variable quantified in one subformula does not appear free in the other subformula.
The rules for removing quantifiers from the antecedent are (note the change of quantifiers):
is equivalent to (under the assumption that ),
is equivalent to .
The rules for removing quantifiers from the consequent are:
is equivalent to (under the assumption that ),
is equivalent to .
For example, when the range of quantification is the non-negative natural number (viz. ), the statement
is logically equivalent to the statement
The former statement says that if x is less than any natural number, then x is less than zero. The latter statement says that there exists some natural number n such that if x is less than n, then x is less than zero. Both statements are true. The former s
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https://en.wikipedia.org/wiki/Axiom%20of%20countable%20choice
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The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
Overview
The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice . ACω holds in the Solovay model.
ZF+ACω suffices to prove that the union of countably many countable sets is countable. The converse statement "assuming ZF, 'every countable union of countable sets is countable' implies ACω" does not hold, as witnessed by Cohen's First Model. ZF+ACω also suffices to prove that every infinite set is Dedekind-infinite (equivalently: has a countably infinite subset).
ACω is particularly useful for the development of analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S ⊆ R is the limit of some sequence of elements of S \ {x}, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω. For other statements equivalent to ACω, see and .
A common misconception is that countable choice has an inductive nature and is therefore provable as a theorem (in ZF, or similar, or even weaker systems) by induction. However, this is not the case; this misconception is the result of confusing countable choice with finite choice for a finite set of size n (for arbitrary n), and it is this latter result (which is an elementary theorem in combinatorics) that is provable by induction. However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example, Vω − {Ø} has a choice function, where Vω is the set of hereditarily finite sets, i.e. the first set in the Von Neumann universe of non-finite rank. The choice function is (trivially) the least element in the well-ordering. Another example is the set of proper and bounded open intervals of real numbers with rational endpoints.
Use
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:
Let X be infinite. For each natural number n, let An be the set of all 2n-element subsets of X. Since X is infinite, each An is non-empty. The first application of ACω yields a sequence (Bn : n = 0,1,2,3,...) where each Bn is a subset of X with 2n elements.
The sets Bn are not necessarily disjoint, but we can define
C0 = B0
Cn = the difference be
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https://en.wikipedia.org/wiki/Ratio%20test
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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
The test
The usual form of the test makes use of the limit
The ratio test states that:
if L < 1 then the series converges absolutely;
if L > 1 then the series diverges;
if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let
.
Then the ratio test states that:
if R < 1, the series converges absolutely;
if r > 1, the series diverges; or equivalently if for all large n (regardless of the value of r), the series also diverges; this is because is nonzero and increasing and hence does not approach zero;
the test is otherwise inconclusive.
If the limit L in () exists, we must have L = R = r. So the original ratio test is a weaker version of the refined one.
Examples
Convergent because L < 1
Consider the series
Applying the ratio test, one computes the limit
Since this limit is less than 1, the series converges.
Divergent because L > 1
Consider the series
Putting this into the ratio test:
Thus the series diverges.
Inconclusive because L = 1
Consider the three series
The first series (1 + 1 + 1 + 1 + ⋯) diverges, the second one (the one central to the Basel problem) converges absolutely and the third one (the alternating harmonic series) converges conditionally. However, the term-by-term magnitude ratios of the three series are respectively and . So, in all three cases, one has that the limit is equal to 1. This illustrates that when L = 1, the series may converge or diverge, and hence the original ratio test is inconclusive. In such cases, more refined tests are required to determine convergence or divergence.
Proof
Below is a proof of the validity of the original ratio test.
Suppose that . We can then show that the series converges absolutely by showing that its terms will eventually become less than those of a certain convergent geometric series. To do this, consider a real number r such that . This implies that for sufficiently large n; say, for all n greater than N. Hence for each n > N and i > 0, and so
That is, the series converges absolutely.
On the other hand, if L > 1, then for sufficiently large n, so that the limit of the summands is non-zero. Hence the series diverges.
Extensions for L = 1
As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes al
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https://en.wikipedia.org/wiki/Non-analytic%20smooth%20function
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In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.
The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case.
The functions below are generally used to build up partitions of unity on differentiable manifolds.
An example function
Definition of the function
Consider the function
defined for every real number x.
The function is smooth
The function f has continuous derivatives of all orders at every point x of the real line. The formula for these derivatives is
where pn(x) is a polynomial of degree n − 1 given recursively by p1(x) = 1 and
for any positive integer n. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the one-sided limit
for any nonnegative integer m.
By the power series representation of the exponential function, we have for every natural number (including zero)
because all the positive terms for are added. Therefore, dividing this inequality by and taking the limit from above,
We now prove the formula for the nth derivative of f by mathematical induction. Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p1(x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0.
It remains to show that the right-hand side derivative of f at x = 0 is zero. Using the above limit, we see that
The induction step from n to n + 1 is similar. For x > 0 we get for the derivative
where pn+1(x) is a polynomial of degree n = (n + 1) − 1. Of course, the (n + 1)st derivative of f is zero for x < 0. For the right-hand side derivative of f (n) at x = 0 we obtain with the above limit
The function is not analytic
As seen earlier, the function f is smooth, and all its derivatives at the origin are 0. Therefore, the Taylor series of f at the origin converges everywhere to the zero function,
and so the Taylor series does not equal f(x) for x > 0. Consequently, f is not analytic at the origin.
Smooth transition functions
The function
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and
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https://en.wikipedia.org/wiki/130%20%28number%29
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130 (one hundred [and] thirty) is the natural number following 129 and preceding 131.
In mathematics
130 is a sphenic number. It is a noncototient since there is no answer to the equation x - φ(x) = 130.
130 is the only integer that is the sum of the squares of its first four divisors, including 1: 12 + 22 + 52 + 102 = 130.
130 is the largest number that cannot be written as the sum of four hexagonal numbers.
130 equals both 27 + 2 and 53 + 5 and is therefore a doubly strictly number.
In religion
The Book of Genesis states Adam had Seth at the age of 130.
The Second Book of Chronicles says that Jehoiada died at the age of 130.
In other fields
One hundred [and] thirty is also:
The year AD 130 or 130 BC
The 130 nanometer process is a semiconductor process technology by semiconductor companies
A 130-30 fund or a ratio up to 150/50 is a type of collective investment vehicle
The C130 Hercules aircraft
References
See also
List of highways numbered 130
United Nations Security Council Resolution 130
130 Liberty Street, New York City
Integers
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https://en.wikipedia.org/wiki/140%20%28number%29
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140 (one hundred [and] forty) is the natural number following 139 and preceding 141.
In mathematics
140 is an abundant number and a harmonic divisor number. It is the sum of the squares of the first seven integers, which makes it a square pyramidal number.
140 is an odious number because it has an odd number of ones in its binary representation. The sum of Euler's totient function φ(x) over the first twenty-one integers is 140.
140 is a repdigit in bases 13, 19, 27, 34, 69, and 139.
In other fields
140 is also:
The number of varieties of ashes from different varieties of pipe, cigar, and cigarette tobacco included in the Sherlock Holmes monograph.
The former Twitter entry-character limit, a well-known characteristic of the service (based on the text messaging limit)
A film, based on the Twitter entry-character limit, created and edited by Frank Kelly of Ireland
The age at which Job died
The atomic number of unquadnilium, a temporary chemical element
PRO 140 antibody found on T lymphocytes of the human immune system
Telephone directory assistance in Egypt
A video game developed by Jeppe Carlsen
The BPM (tempo) of the music genre Dubstep
See also
List of highways numbered 140
United Nations Security Council Resolution 140
United States Supreme Court cases, Volume 140
References
External links
The Natural Number 140
The Number 140 at The Database of Number Correlations
Integers
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https://en.wikipedia.org/wiki/150%20%28number%29
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150 (one hundred [and] fifty) is the natural number following 149 and preceding 151.
In mathematics
150 is the sum of eight consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31). Given 150, the Mertens function returns 0.
150 is conjectured to be the only minimal difference greater than 1 of any increasing arithmetic progression of n primes (in this case, n = 7) that is not a primorial (a product of the first m primes).
The sum of Euler's totient function φ(x) over the first twenty-two integers is 150.
150 is a Harshad number and an abundant number.
150 degrees is the measure of the internal angle of a regular dodecagon.
In the Bible
The last numbered Psalm in the Bible, Psalm 150, considered the one most often set to music.
The number of sons of Ulam, who were combat archers, in the Census of the men of Israel upon return from exile (I Chronicles 8:40)
In the Book of Genesis, the number of days the waters from the Great Flood persisted on the Earth before subsiding.
Manuscripts
Uncial 0150
Minuscule 150
Lectionary 150
In sports
In Round 20 of the 2011 AFL season, inflicted the worst ever defeat on the Gold Coast Suns by 150 points.
In other fields
150 is also:
The number of degrees in the quincunx astrological aspect explored by Johannes Kepler.
The approximate value for Dunbar's number, a theoretical value with implications in sociology and anthropology
The total number of Power Stars in Super Mario 64 DS for the Nintendo DS.
The total number of dragon eggs in Spyro: Year of the Dragon.
See also
List of highways numbered 150
United Nations Security Council Resolution 150
United States Supreme Court cases, Volume 150
References
Integers
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https://en.wikipedia.org/wiki/Indiana%20Academy%20for%20Science%2C%20Mathematics%2C%20and%20Humanities
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The Indiana Academy for Science, Mathematics, and Humanities (The Indiana Academy) is a nationally ranked public high school located on the campus of Ball State University in Muncie, Indiana. The Academy offers both residential and non-residential (commuter) options for juniors and seniors. As of the 2022-2023 academic year, a non-residential only pilot program for high school sophomores has been added, though it remains to be seen if it will persist. Admission is open to high ability, gifted, and talented high school students living anywhere in Indiana.
The Indiana Academy was founded in 1988 by the Indiana General Assembly. and is a part of Ball State's Teachers College. The first group of Academy juniors started in the fall of 1990.
Nearly 100% of graduates will attend 4-year colleges or other post-secondary education. For the Class of 2020, the average SAT score was 1336 vs the Indiana average of 1085 and the National average of 1059. The Indiana Academy offers courses in twelve AP subjects.
Athletics
The Academy's athletes participate in sports with the teams of Burris Laboratory School.
Alumni
Alumni include prominent doctors, surgeons, lawyers, teachers, professors, writers, and engineers.
References
External links
Official website
Public high schools in Indiana
Charter schools in Indiana
Gifted education
Schools in Delaware County, Indiana
Educational institutions established in 1988
Ball State University
Buildings and structures in Muncie, Indiana
Education in Muncie, Indiana
Public boarding schools in the United States
Boarding schools in Indiana
1988 establishments in Indiana
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https://en.wikipedia.org/wiki/Equivariant%20map
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In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.
Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant.
In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and equivariant stable homotopy theory.
Examples
Elementary geometry
In the geometry of triangles, the area and perimeter of a triangle are invariants: translating or rotating a triangle does not change its area or perimeter. However, triangle centers such as the centroid, circumcenter, incenter and orthocenter are not invariant, because moving a triangle will also cause its centers to move. Instead, these centers are equivariant: applying any Euclidean congruence (a combination of a translation and rotation) to a triangle, and then constructing its center, produces the same point as constructing the center first, and then applying the same congruence to the center. More generally, all triangle centers are also equivariant under similarity transformations (combinations of translation, rotation, and scaling),
and the centroid is equivariant under affine transformations.
The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However, these changes happen in a predictable way: if a triangle is scaled by a factor of , the perimeter also scales by and the area scales by . In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.
Statistics
Another class of simple examples comes from statistical estimation. The mean of a sample (a set of real numbers) is commonly used as a central tendency of the sample. It is equivariant under linear transformations of the real numbers, so for instance it is unaffected by the choice of units used to represent the numbers. By contrast, the mean is not equivariant with respect to nonlinear transformations such as exponentials.
The median of a sample is equivariant for a muc
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https://en.wikipedia.org/wiki/Primorial
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In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.
Definition for prime numbers
For the th prime number , the primorial is defined as the product of the first primes:
,
where is the th prime number. For instance, signifies the product of the first 5 primes:
The first five primorials are:
2, 6, 30, 210, 2310 .
The sequence also includes as empty product. Asymptotically, primorials grow according to:
where is Little O notation.
Definition for natural numbers
In general, for a positive integer , its primorial, , is the product of the primes that are not greater than ; that is,
,
where is the prime-counting function , which gives the number of primes ≤ . This is equivalent to:
For example, 12# represents the product of those primes ≤ 12:
Since , this can be calculated as:
Consider the first 12 values of :
1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
We see that for composite every term simply duplicates the preceding term , as given in the definition. In the above example we have since 12 is a composite number.
Primorials are related to the first Chebyshev function, written according to:
Since asymptotically approaches for large values of , primorials therefore grow according to:
The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, where it is used to derive the existence of another prime.
Characteristics
Let and be two adjacent prime numbers. Given any , where :
For the Primorial, the following approximation is known:
.
Notes:
Using elementary methods, mathematician Denis Hanson showed that
Using more advanced methods, Rosser and Schoenfeld showed that
Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for ,
Furthermore:
For , the values are smaller than , but for larger , the values of the function exceed the limit and oscillate infinitely around later on.
Let be the -th prime, then has exactly divisors. For example, has 2 divisors, has 4 divisors, has 8 divisors and already has divisors, as 97 is the 25th prime.
The sum of the reciprocal values of the primorial converges towards a constant
The Engel expansion of this number results in the sequence of the prime numbers (See )
According to Euclid's theorem, is used to prove the infinitude of the prime numbers.
Applications and properties
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance,
+ 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with . 23# is also the common difference in arithmetic
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https://en.wikipedia.org/wiki/Primitive%20root
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In mathematics, a primitive root may mean:
Primitive root modulo n in modular arithmetic
Primitive nth root of unity amongst the solutions of zn = 1 in a field
See also
Primitive element (disambiguation)
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https://en.wikipedia.org/wiki/Vector%20potential
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In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a vector potential is a vector field A such that
Consequence
If a vector field v admits a vector potential A, then from the equality
(divergence of the curl is zero) one obtains
which implies that v must be a solenoidal vector field.
Theorem
Let
be a solenoidal vector field which is twice continuously differentiable. Assume that decreases at least as fast as for .
Define
Then, A is a vector potential for , that is,
Here, is curl for variable y.
Substituting curl[v] for the current density j of the retarded potential, you will get this formula. In other words, v corresponds to the H-field.
You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;
A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
By analogy with Biot-Savart's law, the following is also qualify as a vector potential for v.
Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.
Let and let the Ω be a star domain centered on the p then,
translating Poincaré's lemma for differential forms into vector fields world, the following is also a vector potential for the
Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , then so is
where is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.
See also
Fundamental theorem of vector calculus
Magnetic vector potential
Solenoid
Closed and Exact Differential Forms
References
Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.
Concepts in physics
Potentials
Vector calculus
Vector physical quantities
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https://en.wikipedia.org/wiki/Interval%20class
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In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
Use of interval classes
The concept of interval class accounts for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage:
(To hear a MIDI realization, click the following:
In the example above, all four labeled pitch-pairs, or dyads, share a common "intervallic color." In atonal theory, this similarity is denoted by interval class—ic 5, in this case. Tonal theory, however, classifies the four intervals differently: interval 1 as perfect fifth; 2, perfect twelfth; 3, diminished sixth; and 4, perfect fourth.
Notation of interval classes
The unordered pitch class interval i(a, b) may be defined as
where i is an ordered pitch-class interval .
While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert , prefer to use braces, as in i{a, b}. Both notations are considered acceptable.
Table of interval class equivalencies
See also
Pitch interval
Similarity relation
Sources
Further reading
Friedmann, Michael (1990). Ear Training for Twentieth-Century Music. New Haven: Yale University Press. (cloth) (pbk)
Musical set theory
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https://en.wikipedia.org/wiki/Vitali%20set
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In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model).
Measurable sets
Certain sets have a definite 'length' or 'mass'. For instance, the interval [0, 1] is deemed to have length 1; more generally, an interval [a, b], a ≤ b, is deemed to have length b − a. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.
There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of rational numbers between 0 and 1, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any value between and including 0 and 1 may appear reasonable.
However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem) does not make it obvious whether non-measurable sets exist. The answer to that question involves the axiom of choice.
Construction and proof
A Vitali set is a subset of the interval of real numbers such that, for each real number , there is exactly one number such that is a rational number. Vitali sets exist because the rational numbers form a normal subgroup of the real numbers under addition, and this allows the construction of the additive quotient group of these two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbers under addition. This group consists of disjoint "shifted copies" of in the sense that each element of this quotient group is a set of the form for some in . The uncountably many elements of partition into disjoint sets, and each element is dense in . Each element of intersects , and the axiom of choice guarantees the existence of a subset of containing exactly one representative out of each element of . A set formed this way is called a Vitali set.
Every Vitali set is uncountable, and is irrational for any .
Non-measurability
A Vitali set is non-measurable. To show this, we
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https://en.wikipedia.org/wiki/Rotational%20invariance
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In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument.
Mathematics
Functions
For example, the function
is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle θ
the function, after some cancellation of terms, takes exactly the same form
The rotation of coordinates can be expressed using matrix form using the rotation matrix,
or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is
In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices.
The concept also extends to a vector-valued function f of one or more variables;
In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.
Operators
For a function
which maps elements from a subset X of the real line ℝ to itself, rotational invariance may also mean that the function commutes with rotations of elements in X. This also applies for an operator that acts on such functions. An example is the two-dimensional Laplace operator
which acts on a function f to obtain another function ∇2f. This operator is invariant under rotations.
If g is the function g(p) = f(R(p)), where R is any rotation, then (∇2g)(p) = (∇2f )(R(p)); that is, rotating a function merely rotates its Laplacian.
Physics
In physics, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.
Application to quantum mechanics
In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is
for any rotation R. Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [R, H] = 0.
For infinitesimal rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the (infinitesimal) rotation operator is
then
thus
in other words angular momentum is conserved.
See also
Axial symmetry
Invariant measure
Isotropy
Maxwell's theorem
Rotational symmetry
References
Stenger, Victor J. (2000). Timeless Reality. Prometheus Books. Especially chpt. 12. Nontechnical.
Rotational symmetry
Conservation laws
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https://en.wikipedia.org/wiki/Einstein%20field%20equations
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In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor).
Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.
As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light.
Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.
Mathematical form
The Einstein field equations (EFE) may be written in the form:
where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
The Einstein tensor is defined as
where is the Ricci curvature tensor, and is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives.
The Einstein gravitational constant is defined as
where is the Newtonian constant of gravitation and is the speed of light in vacuum.
The EFE can thus also be written as
In standard units, each term on the left has units of 1/length2.
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.
These equations, together with the geodesic equation, which dictates how freely falling matter moves through spacetime, form th
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https://en.wikipedia.org/wiki/Seventh
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Seventh is the ordinal form of the number seven.
Seventh may refer to:
Seventh Amendment to the United States Constitution
A fraction (mathematics), , equal to one of seven equal parts
Film and television
"The Seventh", a second-season episode of Star Trek: Enterprise
Music
A seventh (interval), the difference between two pitches
Diminished seventh, a chromatically reduced minor seventh interval
Major seventh, the larger of two commonly occurring musical intervals that span seven diatonic scale degrees
Minor seventh, the smaller of two commonly occurring musical intervals that span seven diatonic scale degrees
Harmonic seventh, the interval of exactly 4:7, whose approximation to the minor seventh in equal temperament explains the "sweetness" of the dominant seventh chord in a major key
Augmented seventh, an interval
Leading-tone or subtonic, the seventh degree and the chord built on the seventh degree
Seventh chord, a chord consisting of a triad plus a note forming an interval of a seventh above the chord root
Seventh (chord), a factor of a chord
Seventh, an Australian independent band who provided music for the 2001 video game Operation Flashpoint: Cold War Crisis
The Seventh, stage name of American football player and singer-songwriter Chris Paul (American football)
See also
Seventh Army (disambiguation)
Seventh Avenue (disambiguation)
Seventh day (disambiguation)
Seventh Generation (disambiguation)
Seventh Heaven (disambiguation)
Seventh Sea (disambiguation)
Seventh son of a seventh son (folk concept)
Seventh Son of a Seventh Son (music album)
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https://en.wikipedia.org/wiki/Palindromic%20prime
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In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:
2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, …
Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. The largest known is
101888529 - 10944264 - 1.
which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov. On the other hand, it is known that, for any base, almost all palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than n tends to 1.
Other bases
In binary, the palindromic primes include the Mersenne primes and the Fermat primes. All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are divisible by 3. The sequence of binary palindromic primes begins (in binary):
11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ...
The palindromic primes in base 12 are: (using A and B for ten and eleven, respectively)
2, 3, 5, 7, B, 11, 111, 131, 141, 171, 181, 1B1, 535, 545, 565, 575, 585, 5B5, 727, 737, 747, 767, 797, B1B, B2B, B6B, ...
The palindromic prime numbers can also be generated based on Smarandache function (Kempner function) using prime number algorithm.
Property
Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after Belphegor, one of the seven princes of Hell. Belphegor's Prime consists of the number 666, on either side enclosed by thirteen zeroes and a one. Belphegor's Prime is an example of a beastly palindromic prime in which a prime p is palindromic with 666 in the center. Another beastly palindromic prime is 700666007.
Ribenboim defines a triply palindromic prime as a prime p for which: p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime. For example, p = 1011310 + 4661664 + 1, which has q = 11311 digits, and 11311 has r = 5 digits. The first (base-10) triply palindromic prime is the 11-digit number 10000500001. It is possible that a triply palindromic prime in base 10 may also be palindromic in another base, such as base 2, but it would be highly remarkable if it were also a triply palindromic prime in that base as well.
Palindromic Prime in Decimal Expansion of Pi
On June 8, 2022 Google cloud announced that they have calculated 100 Trillion digits of pi using y-cruncher on their cloud pl
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