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https://en.wikipedia.org/wiki/Applications%20of%20randomness
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Randomness has many uses in science, art, statistics, cryptography, gaming, gambling, and other fields. For example, random assignment in randomized controlled trials helps scientists to test hypotheses, and random numbers or pseudorandom numbers help video games such as video poker.
These uses have different levels of requirements, which leads to the use of different methods. Mathematically, there are distinctions between randomization, pseudorandomization, and quasirandomization, as well as between random number generators and pseudorandom number generators. For example, applications in cryptography usually have strict requirements, whereas other uses (such as generating a "quote of the day") can use a looser standard of pseudorandomness.
Early uses
Games
Unpredictable (by the humans involved) numbers (usually taken to be random numbers) were first investigated in the context of gambling developing, sometimes, pathological forms like apophenia. Many randomizing devices such as dice, shuffling playing cards, and roulette wheels, seem to have been developed for use in games of chance. Electronic gambling equipment cannot use these and so theoretical problems are less easy to avoid; methods of creating them are sometimes regulated by governmental gaming commissions.
Modern electronic casino games contain often one or more random number generators which decide the outcome of a trial in the game. Even in modern slot machines, where mechanical reels seem to spin on the screen, the reels are actually spinning for entertainment value only. They eventually stop exactly where the machine's software decided they would stop when the handle was first pulled. It has been alleged that some gaming machines' software is deliberately biased to prevent true randomness, in the interests of maximizing their owners' revenue; the history of biased machines in the gambling industry is the reason government inspectors attempt to supervise the machines—electronic equipment has extended the range of supervision. Some thefts from casinos have used clever modifications of internal software to bias the outcomes of the machines—at least in those which have been discovered. Gambling establishments keep close track of machine payouts in an attempt to detect such alterations.
Random draws are often used to make a decision where no rational or fair basis exists for making a deterministic decision, or to make unpredictable moves.
Political use
Athenian democracy
Fifth century BC Athenian democracy developed out of a notion of isonomia (equality of political rights), and random selection was a principal way of achieving this fairness. Greek democracy (literally meaning "rule by the people") was actually run by the people: administration was in the hands of committees allotted from the people and regularly changed. Although it may seem strange to those used to modern liberal democracy, the Athenian Greeks considered elections to be essentially undemocratic. This was because
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https://en.wikipedia.org/wiki/Eric%20Hehner
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Eric "Rick" C. R. Hehner (born 16 September 1947) is a Canadian computer scientist. He was born in Ottawa. He studied mathematics and physics at Carleton University, graduating with a Bachelor of Science (B.Sc.) in 1969. He studied computer science at the University of Toronto, graduating with a Master of Science (M.Sc.) in 1970, and a Doctor of Philosophy (Ph.D.) in 1974. He then joined the faculty there, becoming a full professor in 1983. He became the Bell University Chair in software engineering in 2001, and retired in 2012.
Hehner's main research area is formal methods of software design. His method, initially called predicative programming, later called Practical Theory of Programming, is to consider each specification to be a binary (boolean) expression, and each programming construct to be a binary expression specifying the effect of executing the programming construct. Refinement is just implication. This is the simplest formal method, and the most general, applying to sequential, parallel, stand-alone, communicating, terminating, nonterminating, natural-time, real-time, deterministic, and probabilistic programs, and includes time and space bounds. This idea has influenced other computer science researchers, including Tony Hoare.
Hehner's other research areas include probabilistic programming, unified algebra, and high-level circuit design. In 1979, Hehner invented a generalization of radix complement called quote notation, which is a representation of the rational numbers that allows easier arithmetic and precludes roundoff error.
He was involved with developing international standards in programming and informatics, as a member of the International Federation for Information Processing (IFIP) IFIP Working Group 2.1 on Algorithmic Languages and Calculi, which specified, maintains, and supports the programming languages ALGOL 60 and ALGOL 68. and of IFIP Working Group 2.3 on Programming Methodology.
References
External links
DBLP publications
A Practical Theory of Programming, Professor Hehner's book, available free in PDF
Eric Hehner archival papers held at the University of Toronto Archives and Records Management Services
1947 births
Canadian computer scientists
Formal methods people
Living people
Carleton University alumni
University of Toronto alumni
Academic staff of the University of Toronto
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https://en.wikipedia.org/wiki/Prime%20signature
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In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having prime factorization is the multiset .
For example, all prime numbers have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of } and the products of a square of a prime and a different prime (e.g. 12, 18, 20, ...) have a prime signature of }.
Properties
The divisor function τ(n), the Möbius function μ(n), the number of distinct prime divisors ω(n) of n, the number of prime divisors Ω(n) of n, the indicator function of the squarefree integers, and many other important functions in number theory, are functions of the prime signature of n.
In particular, τ(n) equals the product of the incremented by 1 exponents from the prime signature of n.
For example, 20 has prime signature {2,1} and so the number of divisors is (2+1) × (1+1) = 6. Indeed, there are six divisors: 1, 2, 4, 5, 10 and 20.
The smallest number of each prime signature is a product of primorials. The first few are:
1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ... .
A number cannot divide another unless its prime signature is included in the other numbers prime signature in the Young's lattice.
Numbers with same prime signature
Sequences defined by their prime signature
Given a number with prime signature S, it is
A prime number if S = {1},
A square if gcd S is even,
A cube if gcd S is divisible by 3,
A square-free integer if max S = 1,
A cube-free integer if max S ≤ 2,
A powerful number if min S ≥ 2,
A perfect power if gcd S > 1,
An Achilles number if min S ≥ 2 and gcd S = 1,
k-almost prime if sum S = k.
See also
Canonical representation of a positive integer
References
External links
List of the first 400 prime signatures
Iterative Mapping of Prime Signatures
Number theory
Prime numbers
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https://en.wikipedia.org/wiki/Robin%20Wilson%20%28mathematician%29
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Robin James Wilson (born 5 December 1943) is an emeritus professor in the Department of Mathematics at the Open University, having previously been Head of the Pure Mathematics Department and Dean of the Faculty. He was a stipendiary lecturer at Pembroke College, Oxford and, , Gresham Professor of Geometry at Gresham College, London, where he has also been a visiting professor. On occasion, he teaches at Colorado College in the United States. He is also a long standing fellow of Keble College, Oxford.
Professor Wilson is a son of former British Prime Minister Harold Wilson and his wife, Mary.
Early life and education
Wilson was born in 1943 to the politician Harold Wilson, who later became Prime Minister, and his wife the poet Mary Wilson (née Baldwin). He has a younger brother, Giles, who in his 50s gave up a career as a teacher to be a train driver. Wilson attended University College School in Hampstead, North London. He achieved a BA First Class Honours in Mathematics from Balliol College, Oxford, an MA from the University of Pennsylvania, a PhD from the University of Pennsylvania (1965–1968) and a BA First Class Honours in Humanities with Music from the Open University. In a Guardian interview in 2008, Wilson spoke of the fact he grew up known to everyone primarily as a son of the Labour Party leader and Prime Minister Harold Wilson: "I hated the attention and I still dislike being introduced as Harold Wilson's son. I feel uncomfortable talking about it to strangers even now."
Mathematics career
Wilson's academic interests lie in graph theory, particularly in colouring problems, e.g. the four colour problem, and algebraic properties of graphs. He also researches the history of mathematics, particularly British mathematics and mathematics in the 17th century and the period 1860 to 1940, and the history of graph theory and combinatorics.
In 1974, he won the Lester R. Ford Award from the Mathematical Association of America for his expository article An introduction to matroid theory. Due to his collaboration on a 1977 paper with the Hungarian mathematician Paul Erdős, Wilson has an Erdős number of 1.
In July 2008, he published a study of the mathematical work of Lewis Carroll, the creator of Alice's Adventures in Wonderland and Through the Looking-Glass — Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life (Allen Lane, 2008. ). From January 1999 to September 2003, Wilson was editor-in-chief of the European Mathematical Society Newsletter. He is past President of the British Society for the History of Mathematics.
Other interests
He has strong interests in music, including the operas of Gilbert and Sullivan, and is the co-author (with Frederic Lloyd) of Gilbert and Sullivan: The Official D'Oyly Carte Picture History. In 2007, he was a guest on Private Passions, the biographical music discussion programme on BBC Radio 3.
Personal life
Wilson is married and has twin daughters.
Publications
Wilson has written or edited ab
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https://en.wikipedia.org/wiki/Primefree%20sequence
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In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor is equal to 1, and such that for there are no primes in the sequence of numbers calculated from the formula
.
The first primefree sequence of this type was published by Ronald Graham in 1964.
Wilf's sequence
A primefree sequence found by Herbert Wilf has initial terms
The proof that every term of this sequence is composite relies on the periodicity of Fibonacci-like number sequences modulo the members of a finite set of primes. For each prime , the positions in the sequence where the numbers are divisible by repeat in a periodic pattern, and different primes in the set have overlapping patterns that result in a covering set for the whole sequence.
Nontriviality
The requirement that the initial terms of a primefree sequence be coprime is necessary for the question to be non-trivial. If the initial terms share a prime factor (e.g., set and for some and both greater than 1), due to the distributive property of multiplication and more generally all subsequent values in the sequence will be multiples of . In this case, all the numbers in the sequence will be composite, but for a trivial reason.
The order of the initial terms is also important. In Paul Hoffman's biography of Paul Erdős, The man who loved only numbers, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime .
Other sequences
Several other primefree sequences are known:
(sequence A083104 in the OEIS; Graham 1964),
(sequence A083105 in the OEIS; Knuth 1990), and
(sequence A082411 in the OEIS; Nicol 1999).
The sequence of this type with the smallest known initial terms has
(sequence A221286 in the OEIS; Vsemirnov 2004).
Notes
References
External links
Problem 31. Fibonacci- all composites sequence. The prime puzzles and problems connection.
Integer sequences
Number theory
Recurrence relations
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https://en.wikipedia.org/wiki/Rectified%20600-cell
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In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.
Containing the cell realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron.
The vertex figure of the rectified 600-cell is a uniform pentagonal prism.
Semiregular polytope
It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a octicosahedric for being made of octahedron and icosahedron cells.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC600.
Alternate names
octicosahedric (Thorold Gosset)
Icosahedral hexacosihecatonicosachoron
Rectified 600-cell (Norman W. Johnson)
Rectified hexacosichoron
Rectified polytetrahedron
Rox (Jonathan Bowers)
Images
Related polytopes
Diminished rectified 600-cell
A related vertex-transitive polytope can be constructed with equal edge lengths removes 120 vertices from the rectified 600-cell, but isn't uniform because it contains square pyramid cells, discovered by George Olshevsky, calling it a swirlprismatodiminished rectified hexacosichoron, with 840 cells (600 square pyramids, 120 pentagonal prisms, and 120 pentagonal antiprisms), 2640 faces (1800 triangles, 600 square, and 240 pentagons), 2400 edges, and 600 vertices. It has a chiral bi-diminished pentagonal prism vertex figure.
Each removed vertex creates a pentagonal prism cell, and diminishes two neighboring icosahedra into pentagonal antiprisms, and each octahedron into a square pyramid.
This polytope can be partitioned into 12 rings of alternating 10 pentagonal prisms and 10 antiprisms, and 30 rings of square pyramids.
Net
H4 family
Pentagonal prism vertex figures
References
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation
External links
Archimedisches Polychor Nr. 45 (rectified 600-cel
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https://en.wikipedia.org/wiki/Computational%20epidemiology
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Computational epidemiology is a multidisciplinary field that uses techniques from computer science, mathematics, geographic information science and public health to better understand issues central to epidemiology such as the spread of diseases or the effectiveness of a public health intervention. Computational epidemiology traces its origins to mathematical epidemiology, but began to experience significant growth with the rise of big data and the democratization of high-performance computing through cloud computing.
Introduction
In contrast with traditional epidemiology, computational epidemiology looks for patterns in unstructured sources of data, such as social media. It can be thought of as the hypothesis-generating antecedent to hypothesis-testing methods such as national surveys and randomized controlled trials.
A mathematical model is developed which describes the observed behavior of the viruses, based on the available data. Then simulations of the model are performed to understand the possible outcomes given the model used. These simulations produce as results projections which can then be used to make predictions or verify the facts and then be used to plan interventions and meters towards the control of the disease's spread.
References
External links
Sax Institute - Decision Analytics
Computational science
Epidemiology
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https://en.wikipedia.org/wiki/Rectified%205-cell
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In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
Topologically, under its highest symmetry, [3,3,3], there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron.
The vertex figure of the rectified 5-cell is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends.
Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual because the vertex figure (a uniform triangular prism) is not a dual of the polychoron's cells.
Wythoff construction
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Structure
Together with the simplex and 24-cell, this shape and its dual (a polytope with ten vertices and ten triangular bipyramid facets) was one of the first 2-simple 2-simplicial 4-polytopes known. This means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5-cells together; since then, infinitely many 2-simple 2-simplicial polytopes have been constructed.
Semiregular polytope
It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5.
Alternate names
Tetroctahedric (Thorold Gosset)
Dispentachoron
Rectified 5-cell (Norman W. Johnson)
Rectified 4-simplex
Fully truncated 4-simplex
Rectified pentachoron (Acronym: rap) (Jonathan Bowers)
Ambopentachoron (Neil Sloane & John Horton Conway)
(5,2)-hypersimplex (the convex hull of five-dimensional (0,1)-vectors with exactly two ones)
Images
Coordinates
The Cartesian coordinates of the vertices of an origin-centered rectified 5-cell having edge length 2 are:
More simply, the vertices of the rectified 5-cell can be positioned on a hyperplane in 5-space as permutations of (0,0,0,1,1) or (0,0,1,1,1). These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively.
Related 4-polytopes
The rectified 5-cell is the vertex figure of th
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https://en.wikipedia.org/wiki/Evan%20Esar
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Evan Esar (1899–1995) was an American humorist who wrote Esar's Comic Dictionary in 1943, Humorous English in 1961, and 20,000 Quips and Quotes in 1968.
He is known for quotes like "Statistics — the only science that enables different experts using the same figures to draw different conclusions." He also wrote The Legend of Joe Miller, which was privately printed for members of the Roxburghe Club of San Francisco by the Grabhorn Press in 1957.
His quotes are commonly found in Crossword puzzles.
References
External links
Author details
Esar's New York Times Obituary
Esar's AncientFaces profile
American humorists
1899 births
1995 deaths
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https://en.wikipedia.org/wiki/Machin-like%20formula
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In mathematics, Machin-like formulae are a popular technique for computing (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706:
which he used to compute to 100 decimal places.
Machin-like formulas have the form
where is a positive integer, are signed non-zero integers, and and are positive integers such that .
These formulas are used in conjunction with Gregory's series, the Taylor series expansion for arctangent:
Derivation
The angle addition formula for arctangent asserts that
if
All of the Machin-like formulas can be derived by repeated application of equation . As an example, we show the derivation of Machin's original formula one has:
and consequently
Therefore also
and so finally
An insightful way to visualize equation is to picture what happens when two complex numbers are multiplied together:
The angle associated with a complex number is given by:
Thus, in equation , the angle associated with the product is:
Note that this is the same expression as occurs in equation . Thus equation can be interpreted as saying that multiplying two complex numbers means adding their associated angles (see multiplication of complex numbers).
The expression:
is the angle associated with:
Equation can be re-written as:
Here is an arbitrary constant that accounts for the difference in magnitude between the vectors on the two sides of the equation. The magnitudes can be ignored, only the angles are significant.
Using complex numbers
Other formulas may be generated using complex numbers. For example, the angle of a complex number is given by and, when one multiplies complex numbers, one adds their angles. If then is 45 degrees or radians. This means that if the real part and complex part are equal then the arctangent will equal . Since the arctangent of one has a very slow convergence rate if we find two complex numbers that when multiplied will result in the same real and imaginary part we will have a Machin-like formula. An example is and . If we multiply these out we will get . Therefore, .
If you want to use complex numbers to show that you first must know that when multiplying angles you put the complex number to the power of the number that you are multiplying by. So and since the real part and imaginary part are equal then,
Lehmer's measure
One of the most important parameters that characterize computational efficiency of a Machin-like formula is the Lehmer's measure, defined as
.
In order to obtain the Lehmer's measure as small as possible, it is necessary to decrease the ratio of positive integers in the arctangent arguments and to minimize the number of the terms in the Machin-like formula. Nowadays at the smallest known Lehmer's measure is due to H. Chien-Lih (1997), whose Machin-like formula is shown below. It is very common in the Machin-like formulas when all numerators
Two-term formulas
In the special
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https://en.wikipedia.org/wiki/Dudeney%20number
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In number theory, a Dudeney number in a given number base is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his puzzles, Root Extraction, where a professor in retirement at Colney Hatch postulates this as a general method for root extraction.
Mathematical definition
Let be a natural number. We define the Dudeney function for base and power to be the following:
where is the times the number of digits in the number in base .
A natural number is a Dudeney root if it is a fixed point for , which occurs if . The natural number is a generalised Dudeney number, and for , the numbers are known as Dudeney numbers. and are trivial Dudeney numbers for all and , all other trivial Dudeney numbers are nontrivial trivial Dudeney numbers.
For and , there are exactly six such integers :
A natural number is a sociable Dudeney root if it is a periodic point for , where for a positive integer , and forms a cycle of period . A Dudeney root is a sociable Dudeney root with , and a amicable Dudeney root is a sociable Dudeney root with . Sociable Dudeney numbers and amicable Dudeney numbers are the powers of their respective roots.
The number of iterations needed for to reach a fixed point is the Dudeney function's persistence of , and undefined if it never reaches a fixed point.
It can be shown that given a number base and power , the maximum Dudeney root has to satisfy this bound:
implying a finite number of Dudeney roots and Dudeney numbers for each order and base .
is the digit sum. The only Dudeney numbers are the single-digit numbers in base , and there are no periodic points with prime period greater than 1.
Dudeney numbers, roots, and cycles of Fp,b for specific p and b
All numbers are represented in base .
Extension to negative integers
Dudeney numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
Programming example
The example below implements the Dudeney function described in the definition above to search for Dudeney roots, numbers and cycles in Python.
def dudeneyf(x: int, p: int, b: int) -> int:
"""Dudeney function."""
y = pow(x, p)
total = 0
while y > 0:
total = total + y % b
y = y // b
return total
def dudeneyf_cycle(x: int, p: int, b: int) -> List:
seen = []
while x not in seen:
seen.append(x)
x = dudeneyf(x, p, b)
cycle = []
while x not in cycle:
cycle.append(x)
x = dudeneyf(x, p, b)
return cycle
See also
Arithmetic dynamics
Factorion
Happy number
Kaprekar's constant
Kaprekar number
Meertens number
Narcissistic number
Perfect digit-to-digit invariant
Perfect digital invariant
Sum-product number
References
H. E. Dudeney, 536 Puzzles & Curious Problems, Souvenir Press
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https://en.wikipedia.org/wiki/Multivector
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In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (also known as decomposable -vectors or -blades) of the form
where are in .
A -vector is such a linear combination that is homogeneous of degree (all terms are -blades for the same ). Depending on the authors, a "multivector" may be either a -vector or any element of the exterior algebra (any linear combination of -blades with potentially differing values of ).
In differential geometry, a -vector is a vector in the exterior algebra of the tangent vector space; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of tangent vectors, for some integer . A differential -form is a -vector in the exterior algebra of the dual of the tangent space, which is also the dual of the exterior algebra of the tangent space.
For and , -vectors are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms.
Exterior product
The exterior product (also called the wedge product) used to construct multivectors is multilinear (linear in each input), associative and alternating. This means for vectors u, v and w in a vector space V and for scalars α, β, the exterior product has the properties:
Linear in an input:
Associative:
Alternating:
The exterior product of k vectors or a sum of such products (for a single k) is called a grade k multivector, or a k-vector. The maximum grade of a multivector is the dimension of the vector space V.
Linearity in either input together with the alternating property implies linearity in the other input. The multilinearity of the exterior product allows a multivector to be expressed as a linear combination of exterior products of basis vectors of V. The exterior product of k basis vectors of V is the standard way of constructing each basis element for the space of k-vectors, which has dimension () in the exterior algebra of an n-dimensional vector space.
Area and volume
The k-vector obtained from the exterior product of k separate vectors in an n-dimensional space has components that define the projected -volumes of the k-parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the k-parallelotope.
The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped.
It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.
Multivectors in R2
Properties of multivectors can be seen
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https://en.wikipedia.org/wiki/Primary%20field
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In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. All other operators in a representation are called descendants; they can be obtained by acting on the primary with the raising generators.
History of the concept
Primary fields in a D-dimensional conformal field theory were introduced in 1969 by Mack and Salam where they were called interpolating fields. They were then studied by Ferrara, Gatto, and Grillo who called them irreducible conformal tensors, and by Mack who called them lowest weights. Polyakov used an equivalent definition as fields which cannot be represented as derivatives of other fields.
The modern terms primary fields and descendants were introduced by Belavin, Polyakov and Zamolodchikov in the context of two-dimensional conformal field theory. This terminology is now used both for D=2 and D>2.
Conformal field theory in D>2 spacetime dimensions
In dimensions conformal primary fields can be defined in two equivalent ways. Campos Delgado provided a pedagogical proof of the equivalence.
First definition
Let be the generator of dilations and let be the generator of special conformal transformations. A conformal primary field , in the representation of the Lorentz group and with conformal dimension satisfies the following conditions at :
;
.
Second definition
A conformal primary field , in the representation of the Lorentz group and with conformal dimension , transforms under a conformal transformation as
where and implements the action of in the representation of .
Conformal field theory in D2 dimensions
In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators . Primaries are defined as the operators annihilated by all with n>0, which are the lowering generators. Descendants are obtained from the primaries by acting with with n<0.
The Virasoro algebra has a finite dimensional subalgebra generated by . Operators annihilated by are called quasi-primaries. Each primary field is a quasi-primary, but the converse is not true; in fact each primary has infinitely many quasi-primary descendants.
Quasi-primary fields in two-dimensional conformal field theory are the direct analogues of the primary fields in the D>2 dimensional case.
Superconformal field theory
In dimensions, conformal algebra allows graded extensions containing fermionic generators. Quantum field theories invariant with respect to such extended algebras are called superconformal. In superconformal field theories, one considers superconformal primary operators.
In dimensions, superconformal primaries are annihilated by and by the fermionic generator
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https://en.wikipedia.org/wiki/Resummation
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In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) function in which the individual terms defining the original function are re-scaled, and an integral transformation of this new function to obtain the original function. Borel resummation is probably the most well-known example.
The simplest method is an extension of a variational approach to higher order based on a paper by R.P. Feynman and H. Kleinert.
In quantum mechanics it was extended to any order here, and in quantum field theory here.
See also Chapters 16–20 in the textbook cited below.
See also
Perturbation theory
Perturbation theory (quantum mechanics)
References
Books
Hagen Kleinert, Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback (also available online) (together with V. Schulte-Frohlinde).
Quantum field theory
Summability methods
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https://en.wikipedia.org/wiki/2000%20Costa%20Rican%20census
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The 2000 Costa Rican census was undertaken by the National Institute of Statistics and Census (Instituto Nacional de Estadística y Censos (INEC)).
Description
According to this census, Costa Rica had 3,810,179 inhabitants in 2000, a population density of 74.6/km², and 59% of the people lived in urban areas.
Results by canton
References
Censuses in Costa Rica
2000 in Costa Rica
2000 censuses
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https://en.wikipedia.org/wiki/Resolvent
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In mathematics, resolvent meaning "that which resolves" may refer to:
Resolvent formalism in operator theory
Resolvent set in operator theory, the set of points where an operator is "well-behaved"
in probability theory
Resolvent (Galois theory) of an equation for a permutation group, in particular:
Resolvent quadratic of a cubic equation
Resolvent cubic of a quartic equation
In logic:
Resolvent (logic), the clause produced by a resolution
In the consensus theorem, the term produced by a consensus in Boolean logic
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https://en.wikipedia.org/wiki/INEC
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INEC may refer to:
Ilocos Norte Electric Cooperative
Independent National Electoral Commission, Nigeria
National Institute of Statistics and Census (disambiguation) (Portuguese and Spanish abbreviation: INEC)
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https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20and%20Census%20of%20Costa%20Rica
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The National Institute of Statistics and Census of Costa Rica (Instituto Nacional de Estadística y Censos de Costa Rica, or INEC, in Spanish) is the governmental institution entrusted with the running of censuses and official surveys in the country. Its main office is located in San José.
History
INEC was first called Oficina Central de Estadística about its foundation in 1861. Later, in 1951, it was called Dirección General de Estadística y Censos, until 1998, when INEC was its legal name. INEC ran its first census in 1864, and the latest was the 10th population and the 6th dwellings census, held in June 2011.
Censuses in Costa Rica
1864. First Population Census.
1883. Second Population Census.
1892. Third Population Census.
1927. Fourth Population Census.
1950. Fifth Population Census.
1963. Sixth Population Census.
1973. Seventh Population Census.
1984. Eight Population Census.
2000. Ninth Population Census.
2011. Tenth Population Census.
2022. Eleventh Population Census.
References
External links
INEC website (in spanish only)
Central American Population Center website (in spanish only). Centro Centroamericano de Población (CCP), organization in charge of demographic studies at University of Costa Rica, has information about the historical census that can be accessed online.
First Population Census.
Second Population Census.
Third Population Census.
Fourth Population Census.
Fifth Population Census.
Sixth Population Census.
Costa Rica
Demographics of Costa Rica
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https://en.wikipedia.org/wiki/Napoleon%27s%20theorem
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In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.
The triangle thus formed is called the inner or outer Napoleon triangle. The difference in the areas of the outer and inner Napoleon triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may date back to W. Rutherford's 1825 question published in The Ladies' Diary, four years after the French emperor's death, but the result is covered in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820, whereas Napoleon died the following May.
Proofs
In the figure above, is the original triangle. are equilateral triangles constructed on its sides' exteriors, and points are the centroids of those triangles. The theorem for outer triangles states that triangle (green) is equilateral.
A quick way to see that is equilateral is to observe that becomes under a clockwise rotation of 30° around and a homothety of ratio with the same center, and that also becomes after a counterclockwise rotation of 30° around and a homothety of ratio with the same center. The respective spiral similarities are That implies and the angle between them must be 60°.
There are in fact many proofs of the theorem's statement, including a synthetic (coordinate-free) one, a trigonometric one, a symmetry-based approach, and proofs using complex numbers.
Background
The theorem has frequently been attributed to Napoleon, but several papers have been written concerning this issue which cast doubt upon this assertion (see ).
The following entry appeared on page 47 in the Ladies' Diary of 1825 (so in late 1824, a year or so after the compilation of Dublin examination papers). This is an early appearance of Napoleon's theorem in print, and Napoleon's name is not mentioned.
VII. Quest.(1439); by Mr. W. Rutherford, Woodburn.
"Describe equilateral triangles (the vertices being either all outward or all inward) upon the three sides of any triangle : then the lines which join the centres of gravity of those three equilateral triangles will constitute an equilateral triangle. Required a demonstration."
Since William Rutherford was a very capable mathematician, his motive for requesting a proof of a theorem that he could certainly have proved himself is unknown. Maybe he posed the question as a challenge to his peers, or perhaps he hoped that the responses would yield a more elegant solution. However, it is clear from reading successive issues of the Ladies' Diary in
the 1820s, that the Editor aimed to include a varied set of questions each year, with some suited for the exercise of beginners.
Plainly there is no reference to Napoleon in either the question or the published responses, which
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https://en.wikipedia.org/wiki/Tensor%20product%20of%20Hilbert%20spaces
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In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.
Definition
Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arises naturally from those of the factors. Let and be two Hilbert spaces with inner products and respectively. Construct the tensor product of and as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining
and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of and
Explicit construction
The tensor product can also be defined without appealing to the metric space completion. If and are two Hilbert spaces, one associates to every simple tensor product the rank one operator from to that maps a given as
This extends to a linear identification between and the space of finite rank operators from to The finite rank operators are embedded in the Hilbert space of Hilbert–Schmidt operators from to The scalar product in is given by
where is an arbitrary orthonormal basis of
Under the preceding identification, one can define the Hilbertian tensor product of and that is isometrically and linearly isomorphic to
Universal property
The Hilbert tensor product is characterized by the following universal property :
A weakly Hilbert-Schmidt mapping is defined as a bilinear map for which a real number exists, such that
for all and one (hence all) orthonormal bases of and of
As with any universal property, this characterizes the tensor product H uniquely, up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.
Infinite tensor products
Two different definitions have historically been proposed for the tensor product of an arbitrary-sized collection of Hilbert spaces. Von Neumann's traditional definition simply takes the "obvious" tensor product: to compute , first collect all simple tensors of the f
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https://en.wikipedia.org/wiki/Glide%20plane
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In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged.
Glide planes are noted in the Hermann–Mauguin notation by a, b or c, depending on which axis the glide is along. (The orientation of the plane is determined by the position of the symbol in the Hermann–Mauguin designation.) If the axis is not defined, then the glide plane may be noted by g. When the glide plane is parallel to the screen, these planes may be indicated by a bent arrow in which the arrowhead indicates the direction of the glide. When the glide plane is perpendicular to the screen, these planes can be represented either by dashed lines when the glide is parallel to the plane of the screen or dotted lines when the glide is perpendicular to the plane of the screen. Additionally, a centered lattice can cause a glide plane to exist in two directions at the same time. This type of glide plane may be indicated by a bent arrow with an arrowhead on both sides when the glide plan is parallel to the plane of the screen or a dashed and double-dotted line when the glide plane is perpendicular to the plane of the screen. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a fourth of either a face or space diagonal of the unit cell . The latter is often called the diamond glide plane as it features in the diamond structure. The n glide plane may be indicated by diagonal arrow when it is parallel to the plane of the screen or a dashed-dotted line when the glide plane is perpendicular to the plane of the screen. A d glide plane may be indicated by a diagonal half-arrow if the glide plane is parallel to the plane of the screen or a dashed-dotted line with arrows if the glide plane is perpendicular to the plane of the screen. If a d glide plane is present in a crystal system, then that crystal must have a centered lattice.
In today's version of Hermann–Mauguin notation, the symbol e is used in cases where there are two possible ways of designating the glide direction because both are true. For example if a crystal has a base-centered Bravais lattice centered on the C face, then a glide of half a cell unit in the a direction gives the same result as a glide of half a cell unit in the b direction.
Formal treatment
In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector.
The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane. However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that. Thus
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https://en.wikipedia.org/wiki/Matrix%20Template%20Library
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The Matrix Template Library (MTL) is a linear algebra library for C++ programs.
The MTL uses template programming, which considerably reduces the code length. All matrices and vectors are available in all classical numerical formats: float, double, complex<float> or complex<double>.
Furthermore, generic programming allows the usage of arbitrary types as long as they provide the necessary operations. For instance one can use arbitrary integer formats (e.g. unsigned short), types for interval arithmetic (e.g. boost::interval) from the Boost C++ Libraries, quaternions (e.g. boost::quaternion), types of higher precision (e.g. GNU Multi-Precision Library) and appropriate user-defined types.
The MTL supports several implementations of dense matrices and sparse matrices. MTL2 has been developed by Jeremy Siek and Andrew Lumsdaine.
The latest version, MTL4, is developed by Peter Gottschling and Andrew Lumsdaine. It contains most of MTL2's functionality and adds new optimization techniques as meta-tuning, e.g. loop unrolling of dynamically sized containers can be specified in the function call. Platform-independent performance scalability is reached by recursive data structures and algorithms.
Generic applications can be written in a natural notation, e.g. v += A*q - w;, while the library dispatches to the appropriate algorithms: matrix vector products vs. matrix products vs. vector scalar products etcetera. The goal is to encapsulate performance issues inside the library and provide scientists an intuitive interface. MTL4 is used in different finite element and finite volume packages, e.g. the FEniCS Project.
See also
List of numerical libraries
References
External links
MTL homepage
MTL4 homepage
C++ numerical libraries
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https://en.wikipedia.org/wiki/Laning%20and%20Zierler%20system
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The Laning and Zierler system (sometimes called "George" by its users) was the first operating algebraic compiler, that is, a system capable of accepting mathematical formulas in algebraic notation and producing equivalent machine code (the term compiler had not yet been invented and the system was referred to as "an interpretive program"). It was implemented in 1952 for the MIT WHIRLWIND by J. Halcombe Laning and Neal Zierler. It is preceded by non-algebraic compilers such as the UNIVAC A-0.
Description
The system accepted formulas in a more or less algebraic notation. It respected the standard rules for operator precedence, allowed nested parentheses, and used superscripts to indicate exponents. It was among the first programming systems to allow symbolic variable names and allocate storage automatically.
The system also automated the following tasks: floating point computation, linkage to subroutines for the basic functions of analysis (sine, etc.) and printing, and arrays and indexing.
The system accepted input on punched tape produced by a Friden Flexowriter. The character set in use at the Whirlwind installation included "upper-case" (superscript) digits and a hyphen, which were used to indicate array indices, function codes, and (integer) exponents. Like other programming notations of its time, the system accepted only single-letter variable names and multiplication was indicated by juxtaposition of operands. A raised dot was available to indicate multiplication explicitly (the character was created by filing off the lower half of a colon!) The system also included support for solution of linear differential equations via the Runge–Kutta method.
The system was described in an 18-page typewritten manual written for people familiar with mathematics but perhaps unfamiliar with computers. It contains almost nothing in the way of an introduction to computer hardware.
Sample program
The following example, taken from page 11 of the system's manual, evaluates for using the Taylor series expansion. The implementation is not terribly efficient, and the system already includes in its subroutine library, but the example serves to give a flavor of the system's syntax. Note that division in the system is evaluated after multiplication:
x = 0,
1 z = 1 - x2/2 + x4/2·3·4 - x6/2·3·4·5·6
+ x8/2·3·4·5·6·7·8 - x10/2·3·4·5·6·7·8·9·10,
PRINT x, z.
x = x + .1,
e = x - 1.05,
CP 1,
STOP
Applications
Few applications were written for the system. One documented application, authored by Laning and Zierler themselves, involved a problem in aeronautics. The problem required seven systems of differential equations to express, and had been given to the Whirlwind because it was too large for MIT's Differential Analyzer to handle. The authors, exploiting the Runge-Kutta feature of their programming system, produced a 97-statement program in two and half hours. The program ran successfully the first time.
Influence on FORTRAN
Som
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https://en.wikipedia.org/wiki/GeoGebra
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GeoGebra (a portmanteau of geometry and algebra) is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. GeoGebra is available on multiple platforms, with apps for desktops (Windows, macOS and Linux), tablets (Android, iPad and Windows) and web. It is presently owned by Indian edutech firm Byju's.
History
GeoGebra's creator, Markus Hohenwarter, started the project in 2001 as part of his master's thesis at the University of Salzburg. After a successful Kickstarter campaign, GeoGebra expanded its offering to include an iPad, an Android and a Windows Store app version. In 2013, GeoGebra incorporated Xcas into its CAS view. The project is now freeware (with open-source portions) and multi-lingual, and Hohenwarter continues to lead its development at the University of Linz.
GeoGebra includes both commercial and not-for-profit entities that work together from the head office in Linz, Austria, to expand the software and cloud services available to users.
In December 2021, GeoGebra was acquired by edtech conglomerate Byju's for approximately $100 million USD.
Features
GeoGebra is an interactive mathematics software suite for learning and teaching science, technology, engineering, and mathematics from primary school up to the university level. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions, all of which can be edited dynamically later. Elements can be entered and modified using mouse and touch controls, or through an input bar. GeoGebra can store variables for numbers, vectors and points, calculate derivatives and integrals of functions, and has a full complement of commands like Root or Extremum. Teachers and students can use GeoGebra as an aid in formulating and proving geometric conjectures.
GeoGebra's main features are:
Interactive geometry environment (2D and 3D)
Built-in spreadsheet
Built-in computer algebra system (CAS)
Built-in statistics and calculus tools
Scripting hooks
Large number of interactive learning and teaching resources at GeoGebra Materials.
GeoGebra Materials Platform
The GeoGebra Materials platform is a cloud service that allows users to upload and share GeoGebra applets with others. GeoGebra Materials was originally launched as GeoGebraTube in June 2011, and was renamed in 2016. As of April 2016 the service hosts more than 1 million resources, 400,000+ of which are public. "Materials" include interactive worksheets, simulations, games and e-books created using GeoGebraBook.
GeoGebra Materials can be also exported in several formats, including SVG, Animated GIF, Windows Metafile, PNG, PDF and EPS, as well as copied directly to the clipboard. GeoGebra can also generate code for use in LaTeX files.
Licensing
Before version 4.2, GeoGebra was published under the GNU General Public License (GPL-3.0-or-later).
Af
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https://en.wikipedia.org/wiki/Constrained%20geometry%20complex
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In organometallic chemistry, a "constrained geometry complex" (CGC) is a kind of catalyst used for the production of polyolefins such as polyethylene and polypropylene. The catalyst was one of the first major departures from metallocene-based catalysts and ushered in much innovation in the development of new plastics.
Structure
CGC complexes feature a pi-bonded moiety (e.g. cyclopentadienyl) linked to one of the other ligands on the same metal centre in such a way that the angle at this metal between the centroid of the pi-system and the additional ligand is smaller than in comparable unbridged complexes. More specifically, the term CGC was used for ansa-bridged cyclopentadienyl amido complexes, although the definition goes far beyond this class of compounds. The term CGC is frequently used in connection with other more or less related ligand systems that may or may not be isolobal and/or isoelectronic with the ansa-bridged cyclopentadienyl amido ligand system. Furthermore, the term is frequently used for related complexes with long ansa-bridges that induce no strain. Ansa-bridged cyclopentadienyl amido complexes are known for the Group 3, 4, 5, 6 and some Group 8 metals, with the Group 4 congeners being the most studied ones.
Applications
Like Group 4 metallocenes, suitable Group 4 CGCs may be activated for the polymerisation of ethylene and alpha-olefins by reaction with co-catalysts, e.g. methylaluminoxane (MAO), tris(pentafluorophenyl)boranes, and trityl borates. The catalytic systems based on CGCs, however, display incorporation of alpha-olefin comonomers to a greater extend than comparable metallocene based systems. This superiority of CGCs in copolymerisation reactions is ascribed to (i) a high accessibility of the reactive centre and (ii) a low tendency of the bulk polymer chain to undergo chain transfer reactions. CGC derived polymers are currently marketed by The Dow Chemical Company as part of their INSITE technology.
Beyond the use of CGCs for polymerisation reactions, a number of other transformations catalysed by CGCs (both of Group 3 and 4 metals) have been reported from academic laboratories. These include the application of CGCs as catalysts for hydrogenation of imines, hydroboration of alkenes, carboalumination of alkenes, hydrosilylation of alkenes, hydroamination/cyclisation of alpha, omega-aminoalkenes and dimerisation of terminal alkynes.
History
The first CGC was reported by Shapiro and Bercaw for a scandium complex. The following year patents were issued to The Dow Chemical Company and Exxon for applications in alkene polymerization. and today are made at the billion pound scale.
References
Organometallic chemistry
Metal amides
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https://en.wikipedia.org/wiki/Cantor%27s%20paradox
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In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.
This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and 1897). Like a number of "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.
Statements and proofs
In order to state the paradox it is necessary to understand that the cardinal numbers are totally ordered, so that one can speak about one being greater or less than another. Then Cantor's paradox is:
This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set.
Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result.
Discussion and consequences
Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".
Historical notes
While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901.
References
External links
An Historical Account of Set-Theoretic An
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https://en.wikipedia.org/wiki/Fr%C3%A9chet%20derivative
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In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.
Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative.
The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.
Definition
Let and be normed vector spaces, and be an open subset of A function is called Fréchet differentiable at if there exists a bounded linear operator such that
The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces), using and as the two metric spaces, and the above expression as the function of argument in As a consequence, it must exist for all sequences of non-zero elements of that converge to the zero vector Equivalently, the first-order expansion holds, in Landau notation
If there exists such an operator it is unique, so we write and call it the Fréchet derivative of at
A function that is Fréchet differentiable for any point of is said to be C1 if the function
is continuous ( denotes the space of all bounded linear operators from to ). Note that this is not the same as requiring that the map be continuous for each value of (which is assumed; bounded and continuous are equivalent).
This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers since the linear maps from to are just multiplication by a real number. In this case, is the function
Properties
A function differentiable at a point is continuous at that point.
Differentiation is a linear operation in the following sense: if and are two maps which are differentiable at and is a scalar (a real or complex number), then the Fréchet derivative obeys the following properties:
The chain rule is also valid in this context: if is differentiable at and is differentiable at then the composition is differentiable in and the derivative is the composition of the derivatives:
Finite dimensions
The Fréchet derivative in finite-dimensional spaces is the usual derivative. In particular, it is represented in coordinates by the Jacobian matrix.
Suppose that is a map, with an open set. If is Fréchet differentiable at a point then its derivative is
where denotes the Jacobian matrix of at
Furthermore, the partial derivatives of are given by
where is the canonical bas
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https://en.wikipedia.org/wiki/Psychographics
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Psychographics is defined as "market research or statistics classifying population groups according to psychological variables" The term psychographics is derived from the words “psychological” and “demographics” Two common approaches to psychographics include analysis of consumers' activities, interests, and opinions (AIO variables), and values and lifestyles (VALS).
Psychographics have been applied to the study of personality, values, opinions, attitudes, interests, and lifestyles. Psychographic segmentation is a technique for grouping populations into sub-groups according to similar psychological variables.
Psychographic studies of individuals or communities can be valuable in the fields of marketing, demographics, opinion research, prediction, and social research in general. Psychographic attributes can be contrasted with demographic variables (such as age and gender), behavioral variables (such as purchase data or usage rate), and organizational descriptors (sometimes called firmographic variables), such as industry, number of employees, and functional area.
Psychographic methods gained prominence in the 2016 US presidential election and the opposing campaigns of Hillary Clinton and Donald Trump, with the latter using them extensively in microtargeting advertisements to narrow constituencies.
Uses
Psychographics is utilized in the field of marketing and advertising to understand the preferences of consumers and to predict behavior. Private research companies conduct psychographic research using proprietary techniques. For example, VALS is a proprietary framework created by Strategic Business Insights that separates US adults into eight distinct types by evaluating their motivations and resources to understand anticipated consumer behavior. Psychographics is often used for market segmentation and improved target marketing.
Psychographic segmentation is also applied to other fields and across cultures in order to understand motivations and behavior including in healthcare, politics, tourism and lifestyle choices.
Psychographic profiling
Psychographics are applied to the study of cognitive attributes such as attitudes, interests, opinions, and belief, as well as the study of overt behavior (e.g., activities). A "psychographic profile" consists of a relatively complete profile of a person or group's psychographic make-up. These profiles are used in market segmentation as well as in advertising. Some categories of psychographic factors used in market segmentation include:
activity, interest, opinion (AIOs)
attitudes
values
behavior
expressions
gesture
Comparison to demographics
Psychographics is often confused with demographics, in which historical generations may be defined both by demographics, such as the years in which a particular generation is born or even the fertility rates of that generation's parents, but also by psychographic variables like attitudes, personality formation, and cultural touchstones. For example, the t
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https://en.wikipedia.org/wiki/Bicorn
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In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation
It has two cusps and is symmetric about the y-axis.
History
In 1864, James Joseph Sylvester studied the curve
in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.
Properties
The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at x=0, z=0. If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain
This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at and .
The parametric equations of a bicorn curve are and with .
See also
List of curves
References
External links
Plane curves
Algebraic curves
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https://en.wikipedia.org/wiki/Bullet-nose%20curve
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In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation
The bullet curve has three double points in the real projective plane, at and , and , and and , and is therefore a unicursal (rational) curve of genus zero.
If
then
are the two branches of the bullet curve at the origin.
References
Plane curves
Algebraic curves
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https://en.wikipedia.org/wiki/Cl%C3%A9lie
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In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:
If the surface of a sphere is described as usual by the longitude (angle ) and the colatitude (angle ) then
.
The curve was named by Luigi Guido Grandi after Clelia Borromeo.
Viviani's curve and spherical spirals are special cases of Clelia curves. In practice Clelia curves occur as polar orbits of satellites with circular orbits, whose traces on the earth include the poles. If the orbit is a geosynchronous one, then and the trace is a Viviani's curve.
Parametric representation
If the sphere is parametrized by
and the angles are linearly connected by , then one gets a parametric representation of a Clelia curve:
Examples
Any Clelia curve meets the poles at least once.
Spherical spirals:
A spherical spiral usually starts at the south pole and ends at the north pole (or vice versa).
Viviani's curve:
Trace of a polar orbit of a satellite:
In case of the curve is periodic, if is rational (see rose). For example: In case of the period is . If is a non rational number, the curve is not periodic.
The table (second diagram) shows the floor plans of Clelia curves. The lower four curves are spherical spirals. The upper four are polar orbits. In case of the lower arcs are hidden exactly by the upper arcs. The picture in the middle (circle) shows the floor plan of a Viviani's curve. The typical 8-shaped appearance can only achieved by the projection along the x-axis.
References
H. A. Pierer: Universal-Lexikon der Gegenwart und Vergangenheit oder neuestes encyclopädisches Wörterbuch der Wissenschaften, Künste und Gewerbe. Verlag H. A. Pierer, 1844, p. 82.
External links
Clelia., Mathcurve.com..
Spherical curves
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https://en.wikipedia.org/wiki/Cochleoid
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In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation
the Cartesian equation
or the parametric equations
The cochleoid is the inverse curve of Hippias' quadratrix.
Notes
References
Cochleoid in the Encyclopedia of Mathematics
Liliana Luca, Iulian Popescu: A Special Spiral: The Cochleoid. Fiabilitate si Durabilitate - Fiability & Durability no 1(7)/ 2011, Editura "Academica Brâncuşi" , Târgu Jiu,
Roscoe Woods: The Cochlioid. The American Mathematical Monthly, Vol. 31, No. 5 (May, 1924), pp. 222–227 (JSTOR)
Howard Eves: A Graphometer. The Mathematics Teacher, Vol. 41, No. 7 (November 1948), pp. 311-313 (JSTOR)
External links
cochleoid at 2dcurves.com
Plane curves
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https://en.wikipedia.org/wiki/Folium%20of%20Descartes
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In geometry, the folium of Descartes (; named for René Descartes) is an algebraic curve defined by the implicit equation
History
The curve was first proposed and studied by René Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.
Graphing the curve
The folium of Descartes can be expressed in polar coordinates as
which is plotted on the left. This is equivalent to
Another technique is to write and solve for and in terms of . This yields the rational parametric equations:
.
We can see that the parameter is related to the position on the curve as follows:
corresponds to , : the right, lower, "wing".
corresponds to , : the left, upper "wing".
corresponds to , : the loop of the curve.
Another way of plotting the function can be derived from symmetry over . The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis.
This operation is equivalent to a substitution:
and yields
Plotting in the Cartesian system of gives the folium rotated by 45° and therefore symmetric by -axis.
Properties
It forms a loop in the first quadrant with a double point at the origin and asymptote
It is symmetrical about the line . As such, the two intersect at the origin and at the point .
Implicit differentiation gives the formula for the slope of the tangent line to this curve to beUsing either one of the polar representations above, the area of the interior of the loop is found to be . Moreover, the area between the "wings" of the curve and its slanted asymptote is also .
Relationship to the trisectrix of Maclaurin
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation
and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting
In the plane the equation is
If we stretch the curve in the direction by a factor of this becomes
which is the equation of the trisectrix of Maclaurin.
Notes
References
J. Dennis Lawrence: A catalog of special plane curves, 1972, Dover Publications. , pp. 106–108
George F. Simmons: Calculus Gems: Brief Lives and Memorable Mathematics, New York 1992, McGraw-Hill, xiv,355. ; new edition 2007, The Mathematical Association of America (MAA)
External links
"Folium of Descartes" at MacTutor's Famous Curves Index
"Cartesian Folium" at MathCurve
René Descartes
Cubic curves
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https://en.wikipedia.org/wiki/Hessian%20form%20of%20an%20elliptic%20curve
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In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse.
This curve was suggested for application in elliptic curve cryptography, because arithmetic in this curve representation is faster and needs less memory than arithmetic in standard Weierstrass form.
Definition
Let be a field and consider an elliptic curve in the following special case of Weierstrass form over :
where the curve has discriminant
Then the point has order 3.
To prove that has order 3, note that the tangent to at is the line which intersects with multiplicity 3 at .
Conversely, given a point of order 3 on an elliptic curve both defined over a field one can put the curve into Weierstrass form with so that the tangent at is the line . Then the equation of the curve is with .
To obtain the Hessian curve, it is necessary to do the following transformation:
First let denote a root of the polynomial
Then
Note that if has a finite field of order , then every element of has a unique cube root; in general, lies in an extension field of K.
Now by defining the following value another curve, C, is obtained, that is birationally equivalent to E:
which is called cubic Hessian form (in projective coordinates)
in the affine plane (satisfying and ).
Furthermore, (otherwise, the curve would be singular).
Starting from the Hessian curve, a birationally equivalent Weierstrass equation is given by
under the transformations:
and
where:
and
Group law
It is interesting to analyze the group law of the elliptic curve, defining the addition and doubling formulas (because the SPA and DPA attacks are based on the running time of these operations). Furthermore, in this case, we only need to use the same procedure to compute the addition, doubling or subtraction of points to get efficient results, as said above.
In general, the group law is defined in the following way: if three points lie in the same line then they sum up to zero. So, by this property, the group laws are different for every curve.
In this case, the correct way is to use the Cauchy-Desboves´ formulas, obtaining the point at infinity , that is, the neutral element (the inverse of is again).
Let be a point on the curve. The line contains the point and the point at infinity .
Therefore, is the third point of the intersection of this line with the curve. Intersecting the elliptic curve with the line, the following condition is obtained
Since is non zero (because is distinct to 1), the -coordinate of is and the -coordinate of is , i.e., or in projective coordinates .
In some application of elliptic curve cryptography and the elliptic curve method of factorization (ECM) it is necessary to compute the scalar multiplications of , say for some integer , and they are based on the double-and-add method; these operations need the addition and doubling formulas.
Doubling
Now, if is a poi
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https://en.wikipedia.org/wiki/Hippopede
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In geometry, a hippopede () is a plane curve determined by an equation of the form
where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular, rational, algebraic curves of degree 4 and symmetric with respect to both the and axes.
Special cases
When d > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after 19th-century mathematician James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For , the hippopede corresponds to the lemniscate of Bernoulli.
Definition as spiric sections
Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in turn is a type of toric section.
If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting hippopede in polar coordinates
or in Cartesian coordinates
.
Note that when a > b the torus intersects itself, so it does not resemble the usual picture of a torus.
See also
List of curves
References
Lawrence JD. (1972) Catalog of Special Plane Curves, Dover Publications. Pp. 145–146.
Booth J. A Treatise on Some New Geometrical Methods, Longmans, Green, Reader, and Dyer, London, Vol. I (1873) and Vol. II (1877).
"Hippopede" at 2dcurves.com
"Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables
External links
"The Hippopede of Proclus" at The National Curve Bank
Algebraic curves
Spiric sections
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https://en.wikipedia.org/wiki/Strophoid
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In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from is the same as the distance from to (i.e. ). The locus of such points and is then the strophoid of with respect to the pole and fixed point . Note that and are at right angles in this construction.
In the special case where is a line, lies on , and is not on , then the curve is called an oblique strophoid. If, in addition, is perpendicular to then the curve is called a right strophoid, or simply strophoid by some authors. The right strophoid is also called the logocyclic curve or foliate.
Equations
Polar coordinates
Let the curve be given by where the origin is taken to be . Let be the point . If is a point on the curve the distance from to is
The points on the line have polar angle , and the points at distance from on this line are distance from the origin. Therefore, the equation of the strophoid is given by
Cartesian coordinates
Let be given parametrically by . Let be the point and let be the point . Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:
where
An alternative polar formula
The complex nature of the formulas given above limits their usefulness in specific cases. There is an alternative form which is sometimes simpler to apply. This is particularly useful when is a sectrix of Maclaurin with poles and .
Let be the origin and be the point . Let be a point on the curve, the angle between and the -axis, and the angle between and the -axis. Suppose can be given as a function , say Let be the angle at so We can determine in terms of using the law of sines. Since
Let and be the points on that are distance from , numbering so that and is isosceles with vertex angle , so the remaining angles, and are The angle between and the -axis is then
By a similar argument, or simply using the fact that and are at right angles, the angle between and the -axis is then
The polar equation for the strophoid can now be derived from and from the formula above:
is a sectrix of Maclaurin with poles and when is of the form in that case and will have the same form so the strophoid is either another sectrix of Maclaurin or a pair of such curves. In this case there is also a simple polar equation for the polar equation if the origin is shifted to the right by .
Specific cases
Oblique strophoids
Let be a line through . Then, in the notation used above, where is a constant. Then and The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at are then
and
It's easy to check that these equations describe the same curve.
Moving the origin to (again, see Sectrix of Maclaurin) and replacing with produces
and rotating by in turn produces
In rectangular
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https://en.wikipedia.org/wiki/Trident%20curve
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In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula:
Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y = 1, z = 0; if we substitute x = and y = into the equation of the trident curve, we get
which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero.
References
External links
Cubic curves
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https://en.wikipedia.org/wiki/Trochoid
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In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval.
Basic description
As a circle of radius rolls without slipping along a line , the center moves parallel to , and every other point in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let . Parametric equations of the trochoid for which is the -axis are
where is the variable angle through which the circle rolls.
Curtate, common, prolate
If lies inside the circle (), on its circumference (), or outside (), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively. A curtate trochoid is traced by a pedal (relative to the ground) when a normally geared bicycle is pedaled along a straight line. A prolate trochoid is traced by the tip of a paddle (relative to the water's surface) when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where touches the line .
General description
A more general approach would define a trochoid as the locus of a point orbiting at a constant rate around an axis located at ,
which axis is being translated in the x-y-plane at a constant rate in either a straight line,
or a circular path (another orbit) around (the hypotrochoid/epitrochoid case),
The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, , is a rational number, say , where & are coprime, in which case, one period consists of orbits around the moving axis and orbits of the moving axis around the point . The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius while it is rolled on the perimeter of a stationary circle of radius , have the following properties:
where is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".
See also
Aristotle's wheel paradox
Brachistochrone
Cyclogon
Cycloid
Epitrochoid
Hypotrochoid
List of periodic functions
Roulette (curve)
Spirograph
Trochoidal wave
References
External links
Online experiment
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https://en.wikipedia.org/wiki/Viviani%27s%20curve
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In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles (a diameter) of the sphere (see diagram). Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.
The orthographic projection of Viviani's curve onto a plane perpendicular to the line through the crossing point and the sphere center is the lemniscate of Gerono, while the stereographic projection is a hyperbola or the lemniscate of Bernoulli, depending on which point on the same line is used to project.
In 1692 Viviani solved the following task: Cut out of a half sphere (radius ) two windows, such that the remaining surface (of the half sphere) can be squared, i.e. a square with the same area can be constructed using only compasses and ruler. His solution has an area of (see below).
Equations
In order to keep the proof for squaring simple,
the sphere has the equation
and
the cylinder is upright with equation .
The cylinder has radius and is tangent to the sphere at point
Properties of the curve
Floor plan, elevation and side plan
Elimination of , , respectively yields:
The orthogonal projection of the intersection curve onto the
--plane is the circle with equation
--plane the parabola with equation
--plane the algebraic curve with the equation
Parametric representation
Representing the sphere by
and setting yields the curve
One easily checks that the spherical curve fulfills the equation of the cylinder. But the boundaries allow only the red part (see diagram) of Viviani's curve. The missing second half (green) has the property
With help of this parametric representation it is easy to prove the statement: The area of the half sphere (containing Viviani's curve) minus the area of the two windows is . The area of the upper right part of Viviani's window (see diagram) can be calculated by an integration:
Hence the total area of the spherical surface included by Viviani's curve is and the area of the half sphere () minus the area of Viviani's window is , the area of a square with the sphere's diameter as the length of an edge.
Rational Bezier representation
The quarter of Viviani's curve that lies in the all-positive quadrant of 3D space cannot be represented exactly by a regular Bezier curve of any degree.
However, it can be represented exactly by a 3D rational Bezier segment of degree 4, and there is an infinite family of rational Bezier control points generating that segment.
One possible solution is given by the following five control points:
The corresponding rational parametrization is:
Relation to other curves
The 8-shaped elevation (see above) is a Lemniscate of Gerono.
Viviani's curve is a special Clelia curve. For a Clelia curve the relation between the angles is
Subtracting 2× the cylinder equatio
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https://en.wikipedia.org/wiki/Watt%27s%20curve
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In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0)). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate partially back and forth or completely around. It arose in connection with James Watt's pioneering work on the steam engine.
The equation of the curve can be given in polar coordinates as
Derivation
Polar coordinates
The polar equation for the curve can be derived as follows:
Working in the complex plane, let the centers of the circles be at a and −a, and the connecting segment have endpoints at −a+bei λ and a+bei ρ. Let the angle of inclination of the segment be ψ with its midpoint at rei θ. Then the endpoints are also given by rei θ ± cei ψ. Setting expressions for the same points equal to each other gives
Add these and divide by two to get
Comparing radii and arguments gives
Similarly, subtracting the first two equations and dividing by 2 gives
Write
Then
Cartesian coordinates
Expanding the polar equation gives
Letting d 2=a2+b2–c2 simplifies this to
Form of the curve
The construction requires a quadrilateral with sides 2a, b, 2c, b. Any side must be less than the sum of the remaining sides, so the curve is empty (at least in the real plane) unless a<b+c and c<b+a.
The curve has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b<a+c. If b=a+c then two branches of the curve meet at the origin with a common vertical tangent, making it a quadruple point.
Given b<a+c, the shape of the curve is determined by the relative magnitude of b and d. If d is imaginary, that is if a2+b2 <c2 then the curve has the form of a figure eight. If d is 0 then the curve is a figure eight with two branches of the curve having a common horizontal tangent at the origin. If 0<d<b then the curve has two additional double points at ±d and the curve crosses itself at these points. The overall shape of the curve is pretzel-like in this case. If d=b then a=c and the curve decomposes into a circle of radius b and a lemniscate of Booth, a figure eight shaped curve. A special case of this is a=c, b=√2c which produces the lemniscate of Bernoulli. Finally, if d>b then the points ±d are still solutions to the Cartesian equation of the curve, but the curve does not cross these points and they are acnodes. The curve again has a figure eight shape though the shape is distorted if d is close to b.
Given b>a+c, the shape of the curve is determined by the relative sizes of a and c. If a<c then the curve has the form of two loops that cross each other at ±d. If a=c then the curve decomposes into a circle of radius b and an oval of Booth. If a>c then the curve does not cross the x-axis at all and consists of two flattened ov
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https://en.wikipedia.org/wiki/Trisectrix%20of%20Maclaurin
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In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.
Equations
Let two lines rotate about the points and so that when the line rotating about has angle with the x axis, the rotating about has angle . Let be the point of intersection, then the angle formed by the lines at is . By the law of sines,
so the equation in polar coordinates is (up to translation and rotation)
.
The curve is therefore a member of the Conchoid of de Sluze family.
In Cartesian coordinates the equation of this curve is
.
If the origin is moved to (a, 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes
making it an example of a limacon with a loop.
The trisection property
Given an angle , draw a ray from whose angle with the -axis is . Draw a ray from the origin to the point where the first ray intersects the curve. Then, by the construction of the curve, the angle between the second ray and the -axis is .
Notable points and features
The curve has an x-intercept at and a double point at the origin. The vertical line is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at . As a nodal cubic, it is of genus zero.
Relationship to other curves
The trisectrix of Maclaurin can be defined from conic sections in three ways. Specifically:
It is the inverse with respect to the unit circle of the hyperbola
.
It is cissoid of the circle
and the line relative to the origin.
It is the pedal with respect to the origin of the parabola
.
In addition:
The inverse with respect to the point is the Limaçon trisectrix.
The trisectrix of Maclaurin is related to the Folium of Descartes by affine transformation.
References
"Trisectrix of Maclaurin" at MacTutor's Famous Curves Index
Maclaurin Trisectrix at mathcurve.com
"Trisectrix of Maclaurin" at Visual Dictionary Of Special Plane Curves
External links
Loy, Jim "Trisection of an Angle", Part VI
Cubic curves
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https://en.wikipedia.org/wiki/Primitive%20permutation%20group
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In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into either a single set or into |X| singleton sets. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.
While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-transitive action, either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case all nontrivial partitions of X (which exists for |X| ≥ 3) are preserved by G.
This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive.
Properties
In the same letter in which he introduced the term "primitive", Galois stated the following theorem:If G is a primitive solvable group acting on a finite set X, then the order of X is a power of a prime number p. Further, X may be identified with an affine space over the finite field with p elements, and G acts on X as a subgroup of the affine group.If the set X on which G acts is finite, its cardinality is called the degree of G.
A corollary of this result of Galois is that, if is an odd prime number, then the order of a solvable transitive group of degree is a divisor of In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by must be a divisor of ), and is the cardinality of the affine group of an affine space with elements.
It follows that, if is a prime number greater than 3, the symmetric group and the alternating group of degree are not solvable, since their order are greater than Abel–Ruffini theorem results from this and the fact that there are polynomials with a symmetric Galois group.
An equivalent definition of primitivity relies on the fact that every transitive action of a group G is isomorphic to an action arising from the canonical action of G on the set G/H of cosets for H a subgroup of G. A group action is primitive if it is isomorphic to G/H for a maximal subgroup H of G, and imprimitive otherwise (that is, if there is a proper subgroup K of G of which H is a proper subgroup). These imprimitive actions are examples of induced representations.
The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:
There are a large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group o
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https://en.wikipedia.org/wiki/Osculating%20plane
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In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold.
The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
See also
Normal plane (geometry)
Osculating circle
References
Differential geometry
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https://en.wikipedia.org/wiki/Hom%20functor
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In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
Formal definition
Let C be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes).
For all objects A and B in C we define two functors to the category of sets as follows:
{| class=wikitable
|-
! Hom(A, –) : C → Set
! Hom(–, B) : C → Set
|-
| This is a covariant functor given by:
Hom(A, –) maps each object X in C to the set of morphisms, Hom(A, X)
Hom(A, –) maps each morphism f : X → Y to the function
Hom(A, f) : Hom(A, X) → Hom(A, Y) given by
for each g in Hom(A, X).
| This is a contravariant functor given by:
Hom(–, B) maps each object X in C to the set of morphisms, Hom(X, B)
Hom(–, B) maps each morphism h : X → Y to the function
Hom(h, B) : Hom(Y, B) → Hom(X, B) given by
for each g in Hom(Y, B).
|}
The functor Hom(–, B) is also called the functor of points of the object B.
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.
The pair of functors Hom(A, –) and Hom(–, B) are related in a natural manner. For any pair of morphisms f : B → B′ and h : A′ → A the following diagram commutes:
Both paths send g : A → B to f∘g∘h : A′ → B′.
The commutativity of the above diagram implies that Hom(–, –) is a bifunctor from C × C to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–, –) is a bifunctor
Hom(–, –) : Cop × C → Set
where Cop is the opposite category to C. The notation HomC(–, –) is sometimes used for Hom(–, –) in order to emphasize the category forming the domain.
Yoneda's lemma
Referring to the above commutative diagram, one observes that every morphism
h : A′ → A
gives rise to a natural transformation
Hom(h, –) : Hom(A, –) → Hom(A′, –)
and every morphism
f : B → B′
gives rise to a natural transformation
Hom(–, f) : Hom(–, B) → Hom(–, B′)
Yoneda's lemma implies that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category SetCop (covariant or contravariant depending on which Hom functor is used).
Internal Hom functor
Some categories may possess a functor that behaves like a Hom functor, but takes values in the category C itself, rather than Set. Such a functor is referred to as the internal Hom functor, and is often written as
to emphasize its product-like nature, or as
to emphasize its functorial nature, or sometimes merely in lower-case:
For examples, see Category of relations.
Categories that possess an internal Hom f
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https://en.wikipedia.org/wiki/Mathematical%20Association
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The Mathematical Association is a professional society concerned with mathematics education in the UK.
History
It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1894. It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of the death of Sir Isaac Newton, attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal, Sir Horace Lamb, and G. H. Hardy.
In 1951, Mary Cartwright became the first female president of the Mathematical Association.
In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university, a teaching award that was examined was the Diploma of the Mathematical Association, later known as the Diploma in Mathematical Education of the Mathematical Association.
Function
It exists to "bring about improvements in the teaching of mathematics and its applications, and to provide a means of communication among students and teachers of mathematics". Since 1894 it has published The Mathematical Gazette. It is one of the participating bodies in the quadrennial British Congress of Mathematics Education, organised by the Joint Mathematical Council, and it holds its annual general meeting as part of the Congress.
Structure
It is based in the south-east of Leicester on London Road (A6), just south of the Charles Frears campus of De Montfort University.
Aside from the Council, it has seven other specialist committees.
Regions
Its branches are sometimes shared with the Association of Teachers of Mathematics (ATM):
Birmingham
Cambridge
East Midlands
Exeter
Gloucester
Liverpool
London
Greater Manchester
Meridian
Stoke and Staffordshire
Sheffield
Sussex
Yorkshire
Past presidents
Past presidents of The Association for the Improvement of Geometrical Teaching included:
1871 Thomas Archer Hirst
1878 Robert Baldwin Hayward MA, FRS
1889 G M Minchin MA, FRS
1891 James Joseph Sylvester
1892 The Reverend C Taylor DD
1893 R Wormell MA, DSc
1895 Joseph Larmor
Past presidents of The Mathematical Association have included:
1897 Alfred Lodge
1899–1900 Robert Stawell Ball
1901 John Fletcher Moulton, Baron Moulton
1903 Andrew Forsyth
1905 George Ballard Mathews
1907 George H. Bryan
1909–1910 Herbert Hall Turner
1911–1912 E. W. Hobson
1913–1914 Alfred George Greenhill
1915–1916 Alfred North Whitehead
1918–1919 Percy Nunn
1920 E. T. Whittaker
1921 James Wilson
1922–1923 Thomas Little Heath
1924–1925 G. H. Hardy
1926–1927 Micaiah John Muller Hill
1928–1929 William Fleetwood Sheppard
1930–1931 Arthur Eddington
1932–1933 G. N. Watson
1934 Eric Harold Neville
1935 A W Siddons
1936 Andrew Forsyth
1937 Louis Napoleon George Filon
1938 W Hope-Jones
1939 W C Fletcher
1944 C O Tuckey MA
1945 Sydney Chapman
1946 Warin Foster Bushell
1947
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https://en.wikipedia.org/wiki/Naum%20Idelson
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Naum Ilyich Idelson () (March 1(13), 1885, Saint Petersburg - July 14, 1951, Leningrad) was a Soviet theoretical astronomer and expert in history of physics and mathematics.
The crater Idelson on the Moon is named after him.
References
Further reading
Russian astronomers
1885 births
1951 deaths
Scientists from Saint Petersburg
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https://en.wikipedia.org/wiki/Quadrifolium
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The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation:
with corresponding algebraic equation
Rotated counter-clockwise by 45°, this becomes
with corresponding algebraic equation
In either form, it is a plane algebraic curve of genus zero.
The dual curve to the quadrifolium is
The area inside the quadrifolium is , which is exactly half of the area of the circumcircle of the quadrifolium. The perimeter of the quadrifolium is
where is the complete elliptic integral of the second kind with modulus , is the arithmetic–geometric mean and denotes the derivative with respect to the second variable.
Notes
References
External links
Interactive example with JSXGraph
Sextic curves
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https://en.wikipedia.org/wiki/Quasi-invariant%20measure
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In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T.
To express this idea more formally in measure theory terms, the idea is that the Radon–Nikodym derivative of the transformed measure μ′ with respect to μ should exist everywhere; or that the two measures should be equivalent (i.e. mutually absolutely continuous):
That means, in other words, that T preserves the concept of a set of measure zero. Considering the whole equivalence class of measures ν, equivalent to μ, it is also the same to say that T preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as invariant measure class.
In general, the 'freedom' of moving within a measure class by multiplication gives rise to cocycles, when transformations are composed.
As an example, Gaussian measure on Euclidean space Rn is not invariant under translation (like Lebesgue measure is), but is quasi-invariant under all translations.
It can be shown that if E is a separable Banach space and μ is a locally finite Borel measure on E that is quasi-invariant under all translations by elements of E, then either dim(E) < +∞ or μ is the trivial measure μ ≡ 0.
See also
References
Measures (measure theory)
Dynamical systems
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https://en.wikipedia.org/wiki/Conchospiral
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In mathematics, a conchospiral a specific type of space spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral.
Conchospirals are used in biology for modelling snail shells, and flight paths of insects and in electrical engineering for the construction of antennas.
Parameterization
In cylindrical coordinates, the conchospiral is described by the parametric equations:
The projection of a conchospiral on the plane is a logarithmic spiral.
The parameter controls the opening angle of the projected spiral, while the parameter controls the slope of the cone on which the curve lies.
History
The name "conchospiral" was given to these curves by 19th-century German mineralogist Georg Amadeus Carl Friedrich Naumann, in his study of the shapes of sea shells.
Applications
The conchospiral has been used in the design for radio antennas. In this application, it has the advantage of producing a radio beam in a single direction, towards the apex of the cone.
References
External links
Spirals
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https://en.wikipedia.org/wiki/Crossed%20product
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In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product
is a basic method of constructing a new von Neumann algebra from
a von Neumann algebra acted on by a group. It is related to
the semidirect product construction for groups. (Roughly speaking, crossed product is the expected structure for a group ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they appear in functional analysis.)
Motivation
Recall that if we have two finite groups and N with an action of G on N we can form the semidirect product . This contains N
as a normal subgroup, and the action of G on N is given by conjugation in the semidirect product. We can replace N by its complex group algebra C[N], and again form a product in a similar way; this algebra is a sum of subspaces gC[N] as g runs through the elements of G, and is the group algebra of .
We can generalize this construction further by replacing C[N]
by any algebra A acted on by G to get a crossed product
, which is the sum of subspaces
gA and where the action of G on A is given by conjugation in the crossed product.
The crossed product of a von Neumann algebra by a group G acting on it is similar except that we have to be more careful about topologies, and need to construct a Hilbert space acted on by the crossed product. (Note that the von Neumann algebra crossed product is usually larger than the algebraic crossed product discussed above; in fact it is some sort of completion of the algebraic crossed product.)
In physics, this structure appears in presence of the so called gauge group of the first kind. G is the gauge group, and N the "field" algebra. The observables are then defined as the fixed points of N under the action of G. A result by Doplicher, Haag and Roberts says that under some assumptions the crossed product can be recovered from the algebra of observables.
Construction
Suppose that A is a von Neumann algebra of operators acting on a Hilbert space H and G is a discrete group acting on A. We let K be the Hilbert space of all square summable H-valued functions on G. There is an action of A on K
given by
a(k)(g) = g−1(a)k(g)
for k in K, g, h in G, and a in A,
and there is an action of G on K given by
g(k)(h) = k(g−1h).
The crossed product is the von Neumann algebra acting on K generated by the actions of A and G on K. It does not depend (up to isomorphism) on the choice of the Hilbert space H.
This construction can be extended to work for any locally compact group G acting on any von Neumann algebra A. When is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann.
Properties
We let G be an infinite countable discrete group acting on the abelian von Neumann algebra A. The action is called free if
A has no non-zero projections p such that some nontrivial g fixes
all elements of pAp. The action is cal
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https://en.wikipedia.org/wiki/Stability%20%28probability%29
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In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the stable distribution describes this family together with some of the properties of these distributions.
The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables.
Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution. For details see stable distribution.
Definition
There are several basic definitions for what is meant by stability. Some are based on summations of random variables and others on properties of characteristic functions.
Definition via distribution functions
Feller makes the following basic definition. A random variable X is called stable (has a stable distribution) if, for n independent copies Xi of X, there exist constants cn > 0 and dn such that
where this equality refers to equality of distributions. A conclusion drawn from this starting point is that the sequence of constants cn must be of the form
for
A further conclusion is that it is enough for the above distributional identity to hold for n=2 and n=3 only.
Stability in probability theory
There are a number of mathematical results that can be derived for distributions which have the stability property. That is, all possible families of distributions which have the property of being closed under convolution are being considered. It is convenient here to call these stable distributions, without meaning specifically the distribution described in the article named stable distribution, or to say that a distribution is stable if it is assumed that it has the stability property. The following results can be obtained for univariate distributions which are stable.
Stable distributions are always infinitely divisible.
All stable distributions are absolutely continuous.
All stable distributions are unimodal.
Other types of stability
The above concept of stability is based on the idea of a class of distributions being closed under a given set of operations on random variables, where the operation is "summation" or "averaging". Other operations that have been considered include:
geometric stability: here the operation is to take the sum of a random number of random variables, where the number has a geometric distribution. The counterpart of the stable distribution in this case is the geometric stable distribution
Max-stability: here the operation is to take the maximum
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https://en.wikipedia.org/wiki/Multiple%20integral
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In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
Introduction
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
Multiple integration of a function in variables: over a domain is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:
Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.
Mathematical definition
For , consider a so-called "half-open" -dimensional hyperrectangular domain , defined as:
Partition each interval into a finite family of non-overlapping subintervals , with each subinterval closed at the left end, and open at the right end.
Then the finite family of subrectangles given by
is a partition of ; that is, the subrectangles are non-overlapping and their union is .
Let be a function defined on . Consider a partition of as defined above, such that is a family of subrectangles and
We can approximate the total -dimensional volume bounded below by the -dimensional hyperrectangle and above by the -dimensional graph of with the following Riemann sum:
where is a point in and is the product of the lengths of the intervals whose Cartesian product is , also known as the measure of .
The diameter of a subrectangle is the largest of the lengths of the intervals whose Cartesian product is . The diameter of a given partition of is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition is restricted smaller and smaller, the number of subrectangles gets larger, and the measure of each subrectangle grows smaller. The function is said to be Riemann integrable if the
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https://en.wikipedia.org/wiki/Oricon
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, established in 1999, is the holding company at the head of a Japanese corporate group that supplies statistics and information on music and the music industry in Japan and Western music. It started as, which was founded by Sōkō Koike in November 1967 and became known for its music charts. Oricon Inc. was originally set up as a subsidiary of Original Confidence and took over the latter's Oricon record charts in April 2002.
The charts are compiled from data drawn from some 39,700 retail outlets (as of April 2011) and provide sales rankings of music CDs, DVDs, electronic games, and other entertainment products based on weekly tabulations. Results are announced every Tuesday and published in Oricon Style by subsidiary Oricon Entertainment Inc. The group also lists panel survey-based popularity ratings for television commercials on its official website.
Oricon started publishing Combined Chart, which includes CD sales, digital sales, and streaming together, on December 19, 2018.
History
Original Confidence Inc., the original Oricon company, was founded by the former Snow Brand Milk Products promoter Sōkō Koike in 1967. That November, the company began publishing a singles chart on an experimental basis.
Entitled , this went official on January 4, 1968.
Like the preceding Japanese music charts provided by Tokushin Music Report which was started in 1962, early Original Confidence was an exclusive information magazine only for the people who worked in the music industry. In the 1970s, Koike advertised his company's charts to make its existence prevail among the Japanese public. Thanks to his intensive promotional efforts through multiple media including television programs, the hit parade became known by its abbreviation "Oricon" by the late 1970s.
The company shortened its name to Oricon in 1992 and was split into a holding company and several subsidiaries in 1999. Since Sōkō Koike's death, Oricon has been managed by the founder's relatives.
Policy
Oricon monitors and reports on sales of CDs, DVDs, video games, and entertainment content in several other formats; manga and book sales were also formerly covered. Charts are published every Tuesday in Oricon Style and on Oricon's official website. Every Monday, Oricon receives data from outlets, but data on merchandise sold through certain channels does not make it into the charts. For example, the debut single of NEWS, a pop group, was released only through 7-Eleven stores, which are not covered by Oricon, and its sales were not reflected in the Oricon charts. Oricon's rankings of record sales are therefore not completely accurate. Before data was collected electronically, the charts were compiled on the basis of faxes that were sent from record shops.
Controversy
In 2006, Oricon sued journalist Hiro Ugaya when he was quoted in a Saizo (or Cyso) magazine article as suggesting that Oricon was manipulating its statistics to benefit certain management companies and labels, specifically Johnny and
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https://en.wikipedia.org/wiki/Combinatorial%20design
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Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.
Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.
Example
Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n.
This has a solution only if n has the form q2 + q + 1. It is less simple to prove that a solution exists if q is a prime power. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the Bruck–Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms.
When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect. When q = 2, the projective plane is called the Fano plane.
History
Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square. One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square.
Combinatorial designs developed along with the general growth of combinatorics from the 18th century, for example with Latin squares in the 18th century and Steiner systems in the 19th century. Designs have also been popular in recreational mathematics, such as Kirkman's schoolgirl problem (1850), and in practical problems, such as the scheduling of round-robin tournaments (solution published 1880s). In the 20th century designs were applied to the design of experiments, notably Latin squares, finite geometry, and association schemes, yielding the field of algebraic statistics.
Fundamental combinatorial designs
The classi
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https://en.wikipedia.org/wiki/Five%20Equations%20That%20Changed%20the%20World
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Five Equations That Changed the World: The Power and Poetry of Mathematics is a book by Michael Guillen, published in 1995.
It is divided into five chapters that talk about five different equations in physics and the people who have developed them.
The scientists and their equations are:
Isaac Newton (Universal Law of Gravity)
Daniel Bernoulli (Law of Hydrodynamic Pressure)
Michael Faraday (Law of Electromagnetic Induction)
Rudolf Clausius (Second Law of Thermodynamics)
Albert Einstein (Theory of Special Relativity)
The book is a light study in science and history, portraying the preludes to and times and settings of discoveries that have been the basis of further development, including space travel, flight and nuclear power. Each chapter of the book is divided into sections titled Veni, Vidi, Vici.
The reviews of the book have been mixed. Publishers Weekly called it "wholly accessible, beautifully written", Kirkus Reviews wrote that it is a "crowd-pleasing kind of book designed to make the science as palatable as possible", and Frank Mahnke wrote that Guillen "has a nice touch for the history of mathematics and physics and their impact on the world". However, in contrast, Charles Stephens panned "the superficiality of the author's treatment of scientific ideas", and the editors of The Capital Times called the book a "miserable failure" at its goal of helping the public appreciate the beauty of mathematics.
References
1995 non-fiction books
Popular physics books
Mathematical physics
Popular mathematics books
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https://en.wikipedia.org/wiki/State%20of%20Origin%20results%20and%20statistics
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State of Origin results and statistics have been accumulating since the 1980 State of Origin game. Every game played under State of Origin selection rules, including the additional 1987 exhibition match and the matches played between New South Wales and Queensland for the Super League Tri-series are detailed below unless stated otherwise.
Results
Series matches
Queensland have won 24 series. NSW have won 16. On 2 occasion have both teams drawn a series (1999, 2002). (As of 2023)
One off matches
The matches in 1980 and 1981 were one off experimental matches after New South Wales had already won the interstate series in both years. Both games count toward official statistics, but are not considered series.
After Queensland had won the 1987 State of Origin series 2–1, a further game was played in Long Beach, California to showcase rugby league to the American public. On 15 July 2003 the Australian Rugby League announced that this game was to be classified as an "official match" and that the match would count towards the players' individual statistics and overall match win–loss–draw records. However, the match does not count towards the series win–loss–draw record and the 1987 series still remains a 2–1 win to Queensland.
Super League
New South Wales and Queensland played two matches against each other under State of Origin selection rules using players from the Super League (Australia) competition. These matches were not sanctioned by the Australian Rugby League and are not counted as official State of Origin series matches. The Tri-series also included both sides playing a game against New Zealand.
Statistics
Series
Earliest start: 3 May (1993)
Latest finish: 18 November (2020)
Largest aggregate crowd: 224,135 (2015)
With 2 games in Queensland: 187,374 (2005)
Smallest Crowd: 67,003 (1982)
Series Won by QLD: 24
Series Won by NSW: 16
Series Drawn: 2
Matches
Largest crowd: 91,513 at Melbourne Cricket Ground (Match 2, 2015)
Smallest crowd: 16,559 at Lang Park (match 3, 1984), 12,439 at Veteran's Stadium, Los Angeles, USA (1987)
Most points scored: 72, New South Wales d. Queensland 56–16 (match 3, 2000)
Fewest points scored: 2, Queensland d. New South Wales 2–0 (match 1, 1995)
Total points scored: 1940 Queensland, 1836 New South Wales
Most consecutive wins: 8, Queensland (match 2, 1987 – match 3, 1989)
New South Wales wins: 55
Queensland wins: 67
Drawn matches: 2
Largest winning margin: 46, Queensland d. New South Wales 52–6 (Match 3, 2015)
Highest score: 56, New South Wales d. Queensland 56–16 (Match 3, 2000)
Grounds
Since 1988 either New South Wales or Queensland usually hosts two of the three matches on a rotational basis. Prior to this Queensland hosted two matches every year. In 1990, 1994, 1995, 1997, 2006, 2009, 2012, 2015 and 2018 one of the matches was played in Melbourne. The following venues have hosted State of Origin matches since 1980.
Players
Individual records
Most tries in a match: 3 – Chris Anderson (
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https://en.wikipedia.org/wiki/The%20Geometer%27s%20Sketchpad
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The Geometer's Sketchpad is a commercial interactive geometry software program for exploring Euclidean geometry, algebra, calculus, and other areas of mathematics. It was created as part of the NSF-funded Visual Geometry Project led by Eugene Klotz and Doris Schattschneider from 1986 to 1991 at Swarthmore College. Nicholas Jackiw, a student at the time, was the original designer and programmer of the software, and inventor of its trademarked "Dynamic Geometry" approach; he later moved to Key Curriculum Press, KCP Technologies, and McGraw-Hill Education to continue ongoing design and implementation of the software over multiple major releases and hardware platforms. Present versions run Microsoft Windows and MacOS Ventura. It also runs on Linux under Wine with a few bugs. There was also a version developed for the TI-89 and TI-92 series of Calculators. In June 2019, McGraw-Hill announced that it would no longer sell new licenses. Nonetheless, a license-free 64-bit version of Mac Sketchpad that is compatible with the latest Apple silicon chips is available. A license-free Windows version of the software is also available. The Sketchpad Repository contains over 200 videos, with Sketchpad and Web Sketchpad tutorials as well as an archive of Sketchpad webinars that were offered by Key Curriculum Press.
Features
The Geometer's Sketchpad includes the traditional Euclidean tools of classical geometric constructions. It also can perform transformations (translations, rotations, reflections, dilations) of geometric figures drawn or constructed on screen. It is far more than a construction or transformation tool, however. It can manipulate constructed objects "dynamically" by stretching or dragging while maintaining all constraints of the construction so that a seemingly infinite number of cases of a constructed figure can be viewed. Thus it is a natural tool for making or testing conjectures about geometric figures. Its accurate drawings of geometric figures make it a useful tool for illustrating mathematical articles for publication. If a figure (such as the pentadecagon) can be constructed with the compass and straightedge method, it can also be constructed in the program. The program allows "cheat" transformations to create figures impossible to construct under the compass and straightedge rules (such as the regular nonagon). Objects can be animated.
The program allows the creation of numerous objects which can be measured, and potentially used to solve hard math problems. The program allows the determination of the midpoints and midsegments of objects. The Geometer's Sketchpad can measure lengths of segments, measures of angles, area, perimeter, etc. Some of the tools include a construct function, which allows the user to create objects in relation to selected objects. The transform function allows the user to create points in relation to objects, which include distance, angle, ratio, and others.
Web Sketchpad
Built on the foundation of The Ge
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https://en.wikipedia.org/wiki/Mary%20Everest%20Boole
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Mary Everest Boole (11 March 1832 in Wickwar, Gloucestershire – 17 May 1916 in Middlesex, England) was a self-taught mathematician who is best known as an author of didactic works on mathematics, such as Philosophy and Fun of Algebra, and as the wife of fellow mathematician George Boole. Her progressive ideas on education, as expounded in The Preparation of the Child for Science, included encouraging children to explore mathematics through playful activities such as curve stitching. Her life is of interest to feminists as an example of how women made careers in an academic system that did not welcome them.
Life
She was born in England, the daughter of Reverend Thomas Roupell Everest, Rector of Wickwar, and Mary nee Ryall. Her uncle was George Everest, the surveyor and geographer after whom Mount Everest was named. She spent the first part of her life in France where she received an education in mathematics from a private tutor. On returning to England at the age of 11, she continued to pursue her interest in mathematics through self-instruction. Self-taught mathematician George Boole tutored her, and she visited him in Ireland where he held the position of professor of mathematics at Queen's College Cork. Upon the death of her father in 1855, they married and she moved to Cork. Mary greatly contributed as an editor to Boole's The Laws of Thought, a work on algebraic logic. She had five daughters with him.
She was widowed in 1864, at the age of 32, and returned to England, where she was offered a post as a librarian at Queen's College on Harley Street, London. In August 1865, her address was listed as 68 Harley Street in a Deed of Assignment in which she disposed of her husband's former house in Ireland, acting as the Executrix of his will. The deed was witnessed by "John Knights, Porter at Queens College, Harley Street, London and Jane White, Housekeeper at 68 Harley Street, London". As well as working as a librarian, she also tutored privately in mathematics and developed a philosophy of teaching that involved the use of natural materials and physical activities to encourage an imaginative conception of the subject. Her interest extended beyond mathematics to Darwinian theory, philosophy and psychology and she organised discussion groups on these subjects among others. At Queen's College, against the approval of the authorities, she organised discussion groups of students with the unconventional James Hinton, a promulgator of polygamy. This in part led to her mental breakdown and the dispersal of her children.
In later life, she belonged to the circle of the Tolstoyan pacifist publisher, C. W. Daniel; she chose the name The Crank for his magazine because, she said, 'a crank was a little thing that made revolutions'.
Mary took an active interest in politics, introducing her daughter Ethel to the Russian anti-tsarist cause under Sergei Stepniak. After the Boer war 1899-1902 she became more outspoken in her writings against imperialism, organis
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https://en.wikipedia.org/wiki/Dirichlet%27s%20approximation%20theorem
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In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers and , with , there exist integers and such that and
Here represents the integer part of .
This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality
is satisfied by infinitely many integers p and q. This shows that any irrational number has irrationality measure at least 2. This corollary also shows that the Thue–Siegel–Roth theorem, a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of algebraic numbers cannot be improved by increasing the exponent beyond 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the golden ratio can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the irrationality measure.
Simultaneous version
The simultaneous version of the Dirichlet's approximation theorem states that given real numbers and a natural number then there are integers such that
Method of proof
Proof by the pigeonhole principle
This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. The method extends to simultaneous approximation.
Proof outline: Let be an irrational number and be an integer. For every we can write such that is an integer and .
One can divide the interval into smaller intervals of measure . Now, we have numbers and intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those such that . Now:
Dividing both sides by will result in:
And we proved the theorem.
Proof by Minkowski's theorem
Another simple proof of the Dirichlet's approximation theorem is based on Minkowski's theorem applied to the set
Since the volume of is greater than , Minkowski's theorem establishes the existence of a non-trivial point with integral coordinates. This proof extends naturally to simultaneous approximations by considering the set
See also
Dirichlet's theorem on arithmetic progressions
Hurwitz's theorem (number theory)
Heilbronn set
Kronecker's theorem (generalization of Dirichlet's theorem)
Notes
References
External links
Diophantine approximation
Theorems in number theory
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https://en.wikipedia.org/wiki/Artin%20L-function
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In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis.
Definition
Given , a representation of on a finite-dimensional complex vector space , where is the Galois group of the finite extension of number fields, the Artin -function: is defined by an Euler product. For each prime ideal in 's ring of integers, there is an Euler factor, which is easiest to define in the case where is unramified in (true for almost all ). In that case, the Frobenius element is defined as a conjugacy class in . Therefore, the characteristic polynomial of is well-defined. The Euler factor for is a slight modification of the characteristic polynomial, equally well-defined,
as rational function in t, evaluated at , with a complex variable in the usual Riemann zeta function notation. (Here N is the field norm of an ideal.)
When is ramified, and I is the inertia group which is a subgroup of G, a similar construction is applied, but to the subspace of V fixed (pointwise) by I.
The Artin L-function is then the infinite product over all prime ideals of these factors. As Artin reciprocity shows, when G is an abelian group these L-functions have a second description (as Dirichlet L-functions when K is the rational number field, and as Hecke L-functions in general). Novelty comes in with non-abelian G and their representations.
One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers. In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin L-functions, for each irreducible representation of G. For example, the simplest case is when G is the symmetric group on three letters. Since G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function (for the trivial representation) and an L-function of Dirichlet's type for the signature representation.
More precisely for a Galois extension of degree n, the factorization
follows from
where is the multiplicity of the irreducible representation in the regular representation, f is the order of and n is replaced by n/e at the ramified primes.
Since characters are an or
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https://en.wikipedia.org/wiki/Levi%20decomposition
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In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real{Change real Lie algebra to a Lie algebra over a field of characterisitic 0} Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra.
One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra.
When viewed as a factor-algebra of g, this semisimple Lie algebra is also called the Levi factor of g. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple.
Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form
where z is in the nilradical (Levi–Malcev theorem).
An analogous result is valid for associative algebras and is called the Wedderburn principal theorem.
Extensions of the results
In representation theory, Levi decomposition of parabolic subgroups of a reductive group is needed to construct a large family of the so-called parabolically induced representations. The Langlands decomposition is a slight refinement of the Levi decomposition for parabolic subgroups used in this context.
Analogous statements hold for simply connected Lie groups, and, as shown by George Mostow, for algebraic Lie algebras and simply connected algebraic groups over a field of characteristic zero.
There is no analogue of the Levi decomposition for most infinite-dimensional Lie algebras; for example affine Lie algebras have a radical consisting of their center, but cannot be written as a semidirect product of the center and another Lie algebra. The Levi decomposition also fails for finite-dimensional algebras over fields of positive characteristic.
See also
Lie group decompositions
References
Bibliography
Reprinted in: Opere Vol. 1, Edizione Cremonese, Rome (1959), p. 101.
.
External links
Lie algebras
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https://en.wikipedia.org/wiki/Quasinormal%20subgroup
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In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term quasinormal subgroup was introduced by Øystein Ore in 1937.
Two subgroups are said to permute (or commute) if any element from the first
subgroup, times an element of the second subgroup, can be written as an element of the second
subgroup, times an element of the first subgroup. That is, and
as subgroups of are said to commute if HK = KH, that is, any element of the form
with and can be written in the form
where and .
Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group. The converse is not true. For instance, any extension of a cyclic -group by another cyclic -group for the same (odd) prime has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal.
Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups. This follows from the modular property of groups. If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a modular group, although this latter term has other meanings.
In any group, every quasinormal subgroup is ascendant.
A conjugate permutable subgroup is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.
In finite groups
Every quasinormal subgroup of a finite group is a subnormal subgroup. This follows from the somewhat stronger statement that every conjugate permutable subgroup is subnormal, which in turn follows from the statement that every maximal conjugate permutable subgroup is normal. (The finiteness is used crucially in the proofs.)
In summary, a subgroup H of a finite group G is permutable in G if and only if H is both modular and subnormal in G.
PT-groups
Permutability is not a transitive relation in general. The groups in which permutability is transitive are called PT-groups, by analogy with T-groups in which normality is transitive.
See also
Central product
Semipermutable subgroup
References
Stewart E. Stonehewer, "Old, Recent and New Results on Quasinormal subgroups", Irish Math. Soc. Bulletin 56 (2005), 125–133
Tuval Foguel, "Conjugate-Permutable Subgroups", Journal of Algebra 191, 235-239 (1997)
Subgroup properties
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https://en.wikipedia.org/wiki/Subnormal%20subgroup
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In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.
In notation, is -subnormal in if there are subgroups
of such that is normal in for each .
A subnormal subgroup is a subgroup that is -subnormal for some positive integer .
Some facts about subnormal subgroups:
A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.
Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal.
Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.
Every 2-subnormal subgroup is a conjugate-permutable subgroup.
The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal
subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.
If every subnormal subgroup of G is normal in G, then G is called a T-group.
See also
Characteristic subgroup
Normal core
Normal closure
Ascendant subgroup
Descendant subgroup
Serial subgroup
References
Subgroup properties
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https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20and%20Census%20of%20Argentina
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The National Institute of Statistics and Censuses (; INDEC) is an Argentine decentralized public body that operates within the Ministry of Economy, which exercises the direction of all official statistical activities carried out in the country.
In February 2013, the International Monetary Fund censured Argentina for failing to report accurate inflation data. Political intervention in the INDEC figures ended, and the IMF declared in November 2016 that Argentine statistics were again in accordance with international standards.
Definition
The National Institute of Statistics and Census (, INDEC) is a public
deconcentrated body, of a technical nature, within the scope of Argentina's National Ministry of Economy, and which runs all the official statistical activities carried out throughout the country.
Its creation and operation are regulated by Law 176221, executive orders 3110/702 and 1831/933, and INDEC Provision 176/99.4 It is a deconcentrated entity within the scope of the Ministry of Treasury of Argentina.
INDEC produces statistical information on Argentina, which can be used by governments for public policy planning. It can also be used for research and projections in the academic and private fields.
Citizen and stakeholder cooperation and contribution of primary data are fundamental in statistical production. Individual data are confidential and are protected by statistical confidentiality established in Law 17622, and results are always published as statistics. Release dates are informed in the annual advance calendar, available on the Institute's web page.
Duties
INDEC's duties are established in Law 17622, section 5: to implement a statistical policy for the Argentine State; to give structure to National Statistical System (NSS) and lead it; to design statistical methodologies for statistical production; to organise and run statistical infrastructure operations; and to produce basic indicators and social, economic, demographic and geographic data.
Structure
INDEC has a pyramid hierarchy. Resolution 426-E/20176 "empowers the minister of Treasury, following the involvement of the Undersecretary of Public Employment Planning of the Ministry of Modernisation, to approve the lower organisation structure up to 28 directorates and 15 coordinating units."
In May 2017, the structure of the Institute was redesigned and confirmed by Administrative Resolution 3057 of the Office of the National Chief of Cabinet. The new organisation structure provided for the inclusion, standardisation, reassignment and recognition of the national and general directorates and their corresponding responsibilities and actions, and created the position of Managing Director.
The Management Directorate leads the National Directorate of the National Statistical System, the General Directorate of Administration and Operations, the General Directorate of Human Resources and Organisation, the Informatics Directorate, and the Legal Affairs Directorate.
The Technic
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https://en.wikipedia.org/wiki/Refinement%20calculus
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The refinement calculus is a formalized approach to stepwise refinement for program construction. The required behaviour of the final executable program is specified as an abstract and perhaps non-executable "program", which is then refined by a series of correctness-preserving transformations into an efficiently executable program.
Proponents include Ralph-Johan Back, who originated the approach in his 1978 PhD thesis On the Correctness of Refinement Steps in Program Development, and Carroll Morgan, especially with his book Programming from Specifications (Prentice Hall, 2nd edition, 1994, ). In the latter case, the motivation was to link Abrial's specification notation Z, via a rigorous relation of behaviour-preserving program refinement, to an executable programming notation based on Dijkstra's language of guarded commands. Behaviour-preserving in this case means that any Hoare triple satisfied by a program should also be satisfied by any refinement of it, which notion leads directly to specification statements as pre- and postconditions standing, on their own, for any program that could soundly be placed between them.
References
Formal methods
Formal specification languages
Logical calculi
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https://en.wikipedia.org/wiki/Ralph-Johan%20Back
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Ralph-Johan Back is a Finnish computer scientist. Back originated the refinement calculus, an important approach to the formal development of programs using stepwise refinement, in his 1978 PhD thesis at the University of Helsinki, On the Correctness of Refinement Steps in Program Development. He has undertaken much subsequent research in this area. He has held positions at CWI Amsterdam, the Academy of Finland and the University of Tampere.
Since 1983, he has been Professor of Computer Science at the Åbo Akademi University in Turku. For 2002–2007, he was an Academy Professor at the Academy of Finland. He is Director of CREST (Center for Reliable Software Technology) at Åbo Akademi.
Back is a member of Academia Europaea.
References
External links
Ralph-Johan Back home page
Year of birth missing (living people)
Living people
University of Helsinki alumni
Academic staff of the University of Tampere
Academic staff of Åbo Akademi University
Finnish computer scientists
Formal methods people
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https://en.wikipedia.org/wiki/Degree%20of%20an%20algebraic%20variety
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In mathematics, the degree of an affine or projective variety of dimension is the number of intersection points of the variety
with hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see ).
The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.
The degree of a hypersurface is equal to the total degree of its defining equation. A generalization of Bézout's theorem asserts that, if an intersection of projective hypersurfaces has codimension , then the degree of the intersection is the product of the degrees of the hypersurfaces.
The degree of a projective variety is the evaluation at of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations.
Definition
For V embedded in a projective space Pn and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, such that
Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example, the projective line has an (essentially unique) embedding of degree n in Pn.
Properties
The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it (granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem).
Other approaches
For a more sophisticated approach, the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections. The tautological line bundle on Pn pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of Pn, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.
Extending Bézout's theorem
The degree can be used to generalize Bézout's theorem in an expected way to intersections of n hypersurfaces in Pn.
Notes
Algebraic varieties
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https://en.wikipedia.org/wiki/Quaternion%20algebra
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In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over , and indeed the only one over apart from the 2 × 2 real matrix algebra, up to isomorphism. When , then the biquaternions form the quaternion algebra over F.
Structure
Quaternion algebra here means something more general than the algebra of Hamilton's quaternions. When the coefficient field F does not have characteristic 2, every quaternion algebra over F can be described as a 4-dimensional F-vector space with basis , with the following multiplication rules:
where a and b are any given nonzero elements of F. From these rules we get:
The classical instances where are Hamilton's quaternions (a = b = −1) and split-quaternions (a = −1, b = +1). In split-quaternions, and , differing from Hamilton's equations.
The algebra defined in this way is denoted (a,b)F or simply (a,b). When F has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the definition of a quaternion algebra over F as a 4-dimensional central simple algebra over F applies uniformly in all characteristics.
A quaternion algebra (a,b)F is either a division algebra or isomorphic to the matrix algebra of 2 × 2 matrices over F; the latter case is termed split. The norm form
defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic C(a,b) defined by
has a point (x,y,z) with coordinates in F in the split case.
Application
Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of order two in the Brauer group of F. For some fields, including algebraic number fields, every element of order 2 in its Brauer group is represented by a quaternion algebra. A theorem of Alexander Merkurjev implies that each element of order 2 in the Brauer group of any field is represented by a tensor product of quaternion algebras. In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.
Classification
It is a theorem of Frobenius that there are only two real quaternion algebras: 2 × 2 matrices over the reals and Hamilton's real quaternions.
In a similar way, over any local field F there are exactly two quaternion algebras: the 2 × 2 matrices over F and a division algebra.
But the quaternion division algebra over a local field is usually not Hamilton
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https://en.wikipedia.org/wiki/Dense%20graph
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In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem.
The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges.
For undirected simple graphs, the graph density is:
For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is:
where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is , so the maximal density is 1 (for complete graphs) and the minimal density is 0 .
For families of graphs of increasing size, one often calls them sparse if as . Sometimes, in computer science, a more restrictive definition of sparse is used like or even .
Upper density
Upper density is an extension of the concept of graph density defined above from finite graphs to infinite graphs. Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper density. Formally, the upper density of a graph is the infimum of the values α such that the finite subgraphs of with density α have a bounded number of vertices. It can be shown using the Erdős–Stone theorem that the upper density can only be 1 or one of the superparticular ratios (see, e.g., Diestel, edition 5, p. 189).
Sparse and tight graphs
and define a graph as being -sparse if every nonempty subgraph with vertices has at most edges, and -tight if it is -sparse and has exactly edges. Thus trees are exactly the -tight graphs, forests are exactly the -sparse graphs, and graphs with arboricity are exactly the -sparse graphs. Pseudoforests are exactly the -sparse graphs, and the Laman graphs arising in rigidity theory are exactly the -tight graphs.
Other graph families not characterized by their sparsity can also be described in this way. For instance the facts that any planar graph with vertices has at most edges (except for graphs with fewer than 3 vertices), and that any subgraph of a planar graph is planar, together imply that the planar graphs are -sparse. However, not every -sparse graph is planar. Similarly, outerplanar graphs are -sparse and planar bipartite graphs are -sparse.
Streinu and Theran show that testing -sparsity may be performed in polynomial time when and are integers and .
For a graph family, the existence of and such that the graphs in the family are all -sparse is equivalent to the graphs in the family having bo
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https://en.wikipedia.org/wiki/Extreme
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Extreme may refer to:
Science and mathematics
Mathematics
Extreme point, a point in a convex set which does not lie in any open line segment joining two points in the set
Maxima and minima, extremes on a mathematical function
Science
Extremophile, an organism which thrives in or requires "extreme"
Extremes on Earth
List of extrasolar planet extremes
Politics
Extremism, political ideologies or actions deemed outside the acceptable range
The Extreme (Italy) or Historical Far Left, a left-wing parliamentary group in Italy 1867–1904
Business
Extreme Networks, a California-based networking hardware company
Extreme Records, an Australia-based record label
Extreme Associates, a California-based adult film studio
Computer science
Xtreme Mod, a peer-to-peer file sharing client for Windows
Sports and entertainment
Sport
Extreme sport
Extreme Sports Channel A global sports and lifestyle brand dedicated to extreme sports and youth culture
Los Angeles Xtreme, a defunct XFL franchise
Buffalo eXtreme, an ABA franchise
Music
Extreme metal, an umbrella term for a group of related heavy metal subgenres
Extreme (band), an American band
Extreme (album), an album by Extreme
Xtreme (group), a bachata duo
Xtreme (album), an album by Xtreme
Extremes (album), an album by Collin Raye
X-Treme, a stage name of Italian singer and producer Agostino Carollo
Entertainment
Extreme Sports Channel, a global TV channel dedicated to extreme sports and youth culture
RTL CBS Extreme, a Southeast Asian TV channel simply known as "Extreme" prior to the rebranding as Blue Ant Extreme
Extreme (1995 TV series), a 1995 American action series that aired on ABC
Extreme (2009 TV series), a 2009 American television series that aired on the Travel Channel
"Extreme" (CSI: Miami), a season two episode of CSI: Miami
Literature
The Extreme (novel), a 1998 Animorphs novel by K. A. Applegate
Extremes (novel), by Kristine Kathryn Rusch
Extreme Studios, a forerunner of the American comic book studio Image Comics
Adam X the X-Treme, a character in the Marvel Comics universe
Extreme, an autobiography by Sharon Osbourne
Other uses
Chevrolet Extreme, a name for the Chevrolet S-10 pickup truck
Extrême, a pre-filled ice-cream cone brand by Nestlé
See also
Extremities (disambiguation)
Lunatic fringe (disambiguation)
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https://en.wikipedia.org/wiki/Spin%20tensor
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In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in
general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field theory.
The special Euclidean group SE(d) of direct isometries is generated by translations and rotations. Its Lie algebra is written .
This article uses Cartesian coordinates and tensor index notation.
Background on Noether currents
The Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum P. Conservation of four-momentum is given by the continuity equation:
where is the stress–energy tensor, and ∂ are partial derivatives that make up the four-gradient (in non-Cartesian coordinates this must be replaced by the covariant derivative). Integrating over space:
gives the four-momentum vector at time t.
The Noether current for a rotation about the point y is given by a tensor of 3rd order, denoted . Because of the Lie algebra relations
where the 0 subscript indicates the origin (unlike momentum, angular momentum depends on the origin), the integral:
gives the angular momentum tensor at time t.
Definition
The spin tensor is defined at a point x to be the value of the Noether current at x of a rotation about x,
The continuity equation
implies:
and therefore, the stress–energy tensor is not a symmetric tensor.
The quantity S is the density of spin angular momentum (spin in this case is not only for a point-like particle, but also for an extended body), and M is the density of orbital angular momentum. The total angular momentum is always the sum of spin and orbital contributions.
The relation:
gives the torque density showing the rate of conversion between the orbital angular momentum and spin.
Examples
Examples of materials with a nonzero spin density are molecular fluids, the electromagnetic field and turbulent fluids. For molecular fluids, the individual molecules may be spinning. The electromagnetic field can have circularly polarized light. For turbulent fluids, we may arbitrarily make a distinction between long wavelength phenomena and short wavelength phenomena. A long wavelength vorticity may be converted via turbulence into tinier and tinier vortices transporting the angular momentum into smaller and smaller wavelengths while simultaneously reducing the vorticity. This can be approximated by the eddy viscosity.
See also
Belinfante–Rosenfeld stress–energy tensor
Poincaré group
Lorentz group
Relativistic angular momentum
Center of mass (relativistic)
Mathisson–Papapetrou–Dixon equations
Pauli–Lubanski pseudovector
References
External links
Tensors
Special relativity
General relativity
Quantum mechanics
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https://en.wikipedia.org/wiki/Subfactor
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In the theory of von Neumann algebras, a subfactor of a factor is a subalgebra that is a factor and contains . The theory of subfactors led to the discovery of the
Jones polynomial in knot theory.
Index of a subfactor
Usually is taken to be a factor of type , so that it has a finite trace.
In this case every Hilbert space module has a dimension which is a non-negative real number or .
The index of a subfactor is defined to be . Here is the representation
of obtained from the GNS construction of the trace of .
Jones index theorem
This states that if is a subfactor of (both of type ) then the index is either of the form for , or is at least . All these values occur.
The first few values of are
Basic construction
Suppose that is a subfactor of , and that both are finite von Neumann algebras.
The GNS construction produces a Hilbert space acted on by
with a cyclic vector . Let be the projection onto the subspace . Then and generate a new von Neumann algebra acting on , containing as a subfactor. The passage from the inclusion of in to the inclusion of in is called the basic construction.
If and are both factors of type and has finite index in then is also of type .
Moreover the inclusions have the same index: and .
Jones tower
Suppose that is an inclusion of type factors of finite index. By iterating the basic construction we get a tower of inclusions
where and , and each is generated by the previous algebra and a projection. The union of all these algebras has a tracial state whose restriction to each is the tracial state, and so the closure of the union is another type von Neumann algebra .
The algebra contains a sequence of projections which satisfy the Temperley–Lieb relations at parameter . Moreover, the algebra generated by the is a -algebra in which the are self-adjoint, and such that when is in the algebra generated by up to . Whenever these extra conditions are satisfied, the algebra is called a Temperly–Lieb–Jones algebra at parameter . It can be shown to be unique up to -isomorphism. It exists only when takes on those special values for , or the values larger than .
Standard invariant
Suppose that is an inclusion of type factors of finite index. Let the higher relative commutants be and .
The standard invariant of the subfactor is the following grid:
which is a complete invariant in the amenable case. A diagrammatic axiomatization of the standard invariant is given by the notion of planar algebra.
Principal graphs
A subfactor of finite index is said to be irreducible if either of the following equivalent conditions is satisfied:
is irreducible as an bimodule;
the relative commutant is .
In this case defines a bimodule as well as its conjugate bimodule . The relative tensor product, described in and often called Connes fusion after a prior definition for general von Neumann algebras of Alain Connes, can be used to define new bimodul
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https://en.wikipedia.org/wiki/Arunas%20Rudvalis
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Arunas Rudvalis (born June 8, 1945) is an Emeritus Professor of Mathematics at the University of Massachusetts Amherst. He is best known for the Rudvalis group.
Rudvalis went to the Harvey Mudd College and received his Ph.D. degree in Dartmouth College under direction of Ernst Snapper.
External links
Arunas Rudvalis's Web Page
1945 births
20th-century American mathematicians
21st-century American mathematicians
Harvey Mudd College alumni
Dartmouth College alumni
University of Massachusetts Amherst faculty
Living people
Place of birth missing (living people)
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https://en.wikipedia.org/wiki/Ergodic%20flow
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In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S1 = G / AN and G / A = S1 × S1 \ diag S1. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III0 is an ergodic flow on a measure space.
Hedlund's theorem: ergodicity of geodesic and horocycle flows
The method using representation theory relies on the following two results:
If = acts unitarily on a Hilbert space and is a unit vector fixed by the subgroup of upper unitriangular matrices, then is fixed by .
If = acts unitarily on a Hilbert space and is a unit vector fixed by the subgroup of diagonal matrices of determinant , then is fixed by .
(1) As a topological space, the homogeneous space = can be identified with } with the standard action of as matrices. The subgroup of has two kinds of orbits: orbits parallel to the -axis with ; and points on the -axis. A continuous function on that is constant on -orbits must therefore be constant on the real axis with the origin removed. Thus the matrix coefficient = satisfies = for in . By unitarity, |||| = = , so that = for all in = = . Now let be the matrix . Then, as is easily verified, the double coset is dense in ; this is a special case of the Bruhat decomposition. Since is fixed by , the matrix coefficient is constant on . By density, = for all in . The same argument as above shows that = for all in .
(2) Suppose that is fixed by . For the unitary 1-parameter group ≅ , let be the spectral subspace corresponding to the interval . Let be the diagonal matrix with entries and for || . Then . As || tends to infinity the latter projections tend to 0 in the strong operator topology if or . Since = , it follows = in either case. By the spectral theorem, it follows that is in the spectral subspace ; in other words is fixed by . But then, by the first result, must be fixed by .
The classical theorems of Gustav Hedlund from the early 1930s assert the ergodicity of the geodesic and horocycle flows corresponding to compact Riemann surfaces of constant negative curvature. Hedlund's theorem can b
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https://en.wikipedia.org/wiki/Rigidity%20theory
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Rigidity theory may refer to
Study of the concept of rigidity (mathematics)
Mathematical theory of structural rigidity
Rigidity theory (physics), or topological constraints theory, describes or predicts the mechanical properties of glass
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https://en.wikipedia.org/wiki/Mostow%20rigidity%20theorem
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In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by and extended to finite volume manifolds by in 3 dimensions, and by in all dimensions at least 3. gave an alternate proof using the Gromov norm. gave the simplest available proof.
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic -manifold (for ) is a point, for a hyperbolic surface of genus there is a moduli space of dimension that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory. There is also a rich theory of deformation spaces of hyperbolic structures on infinite volume manifolds in three dimensions.
The theorem
The theorem can be given in a geometric formulation (pertaining to finite-volume, complete manifolds), and in an algebraic formulation (pertaining to lattices in Lie groups).
Geometric form
Let be the -dimensional hyperbolic space. A complete hyperbolic manifold can be defined as a quotient of by a group of isometries acting freely and properly discontinuously (it is equivalent to define it as a Riemannian manifold with sectional curvature -1 which is complete). It is of finite volume if the integral of a volume form is finite (which is the case, for example, if it is compact). The Mostow rigidity theorem may be stated as:
Suppose and are complete finite-volume hyperbolic manifolds of dimension . If there exists an isomorphism then it is induced by a unique isometry from to .
Here is the fundamental group of a manifold . If is an hyperbolic manifold obtained as the quotient of by a group then .
An equivalent statement is that any homotopy equivalence from to can be homotoped to a unique isometry. The proof actually shows that if has greater dimension than then there can be no homotopy equivalence between them.
Algebraic form
The group of isometries of hyperbolic space can be identified with the Lie group (the projective orthogonal group of a quadratic form of signature . Then the following statement is equivalent to the one above.
Let and and be two lattices in and suppose that there is a group isomorphism . Then and are conjugate in . That is, there exists a such that .
In greater generality
Mostow rigidity holds (in its geometric formulation) more generally for fundamental groups of all complete, finite volume, non-positively curved (without Euclidean factors) locally symmetric spaces of dimension at least three, or in its algebraic formulation for all lattices in simple Lie groups not locally isomorphic to .
Applications
It follows from the Mostow rigidity theorem that the group of isometries of a finite-volume hyperbolic n-manifold M (f
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https://en.wikipedia.org/wiki/Lefschetz%20pencil
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In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
Description
A pencil is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the projective line. This means that in the case of a complex algebraic variety V, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity.
The first point comes up if we assume that V is given as a projective variety, and the divisors on V are hyperplane sections. Suppose given hyperplanes H and H′, spanning the pencil — in other words, H is given by L = 0 and H′ by L′= 0 for linear forms L and L′, and the general hyperplane section is V intersected with
Then the intersection J of H with H′ has codimension two. There is a rational mapping
which is in fact well-defined only outside the points on the intersection of J with V. To make a well-defined mapping, some blowing up must be applied to V.
The second point is that the fibers may themselves 'degenerate' and acquire singular points (where Bertini's lemma applies, the general hyperplane section will be smooth). A Lefschetz pencil restricts the nature of the acquired singularities, so that the topology may be analysed by the vanishing cycle method. The fibres with singularities are required to have a unique quadratic singularity, only.
It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. It has also been shown that Lefschetz pencils exist in characteristic p for the étale topology.
Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them.
See also
Picard–Lefschetz theory
References
Notes
External links
Geometry of divisors
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https://en.wikipedia.org/wiki/Rational%20mapping
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In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
Definition
Formal definition
Formally, a rational map between two varieties is an equivalence class of pairs in which is a morphism of varieties from a non-empty open set to , and two such pairs and are considered equivalent if and coincide on the intersection (this is, in particular, vacuously true if the intersection is empty, but since is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma:
If two morphisms of varieties are equal on some non-empty open set, then they are equal.
is said to be birational if there exists a rational map which is its inverse, where the composition is taken in the above sense.
The importance of rational maps to algebraic geometry is in the connection between such maps and maps between the function fields of and . Even a cursory examination of the definitions reveals a similarity between that of rational map and that of rational function; in fact, a rational function is just a rational map whose range is the projective line. Composition of functions then allows us to "pull back" rational functions along a rational map, so that a single rational map induces a homomorphism of fields . In particular, the following theorem is central: the functor from the category of projective varieties with dominant rational maps (over a fixed base field, for example ) to the category of finitely generated field extensions of the base field with reverse inclusion of extensions as morphisms, which associates each variety to its function field and each map to the associated map of function fields, is an equivalence of categories.
Examples
Rational maps of projective spaces
There is a rational map sending a ratio . Since the point cannot have an image, this map is only rational, and not a morphism of varieties. More generally, there are rational maps sending for sending an -tuple to an -tuple by forgetting the last coordinates.
Inclusions of open subvarieties
On a connected variety , the inclusion of any open subvariety is a birational equivalence since the two varieties have equivalent function fields. That is, every rational function can be restricted to a rational function and conversely, a rational function defines a rational equivalence class on . An excellent example of this phenomenon is the birational equivalence of and , hence .
Covering spaces on open subsets
Covering spaces on open subsets of a variety give ample examples of rational maps which are not birational. For example, Belyi's theorem states that every algebraic curve admits a map which ramifies at three points. Then, there is an associated covering space which defines a dominant rational morphism which is not birational. Another class o
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https://en.wikipedia.org/wiki/Complex%20dimension
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In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a Cartesian product of the form for some , and the complex dimension is the exponent in this product. Because can in turn be modeled by , a space with complex dimension will have real dimension . That is, a smooth manifold of complex dimension has real dimension ; and a complex algebraic variety of complex dimension , away from any singular point, will also be a smooth manifold of real dimension .
However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points. The real dimension is not greater than the dimension, and equals it if the variety is irreducible and has real points that are nonsingular.
For example, the equation defines a variety of (complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.
The same considerations apply to codimension. For example a smooth complex hypersurface in complex projective space of dimension n will be a manifold of dimension 2(n − 1). A complex hyperplane does not separate a complex projective space into two components, because it has real codimension 2.
References
Complex manifolds
Algebraic geometry
Dimension
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https://en.wikipedia.org/wiki/Constant%20sheaf
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In mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets on whose stalks are all equal to . It is denoted by or . The constant presheaf with value is the presheaf that assigns to each open subset of the value , and all of whose restriction maps are the identity map . The constant sheaf associated to is the sheafification of the constant presheaf associated to . This sheaf identifies with the sheaf of locally constant -valued functions on .
In certain cases, the set may be replaced with an object in some category (e.g. when is the category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
Basics
Let be a topological space, and a set. The sections of the constant sheaf over an open set may be interpreted as the continuous functions , where is given the discrete topology. If is connected, then these locally constant functions are constant. If is the unique map to the one-point space and is considered as a sheaf on , then the inverse image is the constant sheaf on . The sheaf space of is the projection map (where is given the discrete topology).
A detailed example
Let be the topological space consisting of two points and with the discrete topology. has four open sets: . The five non-trivial inclusions of the open sets of are shown in the chart.
A presheaf on chooses a set for each of the four open sets of and a restriction map for each of the nine inclusions (five non-trivial inclusions and four trivial ones). The constant presheaf with value , which we will denote , is the presheaf that chooses all four sets to be , the integers, and all restriction maps to be the identity. is a functor, hence a presheaf, because it is constant. satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets: Vacuously, any two sections of over the empty set are equal when restricted to any set in the empty family. The local identity axiom would therefore imply that any two sections of over the empty set are equal, but this is not true.
A similar presheaf that satisfies the local identity axiom over the empty set is constructed as follows. Let , where 0 is a one-element set. On all non-empty sets, give the value . For each inclusion of open sets, returns either the unique map to 0, if the smaller set is empty, or the identity map on .
Notice that as a consequence of the local identity axiom for the empty set, all the restriction maps involving the empty set are boring. This is true for any presheaf satisfying the local identity axiom for the empty set, and in particular for any sheaf.
is a separated presheaf (that is, satisfies the local identity axiom), but unlike it fails the gluing axiom. is covered by the two open sets and , and these sets have empty intersection. A section
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https://en.wikipedia.org/wiki/Function%20field
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Function field may refer to:
Function field of an algebraic variety
Function field (scheme theory)
Algebraic function field
Function field sieve
Function field analogy
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https://en.wikipedia.org/wiki/Function%20field%20of%20an%20algebraic%20variety
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In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
Definition for complex manifolds
In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in .) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.
For the Riemann sphere, which is the variety over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions).
Construction in algebraic geometry
In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data as agree on the intersections of open affines. We may define the function field of V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
Generalization to arbitrary scheme
In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if is an integral scheme, then for every open affine subset of the ring of sections on is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the local ring of the generic point of . Thus the function field of is just the local ring of its generic point. This point of view is developed further in function field (scheme theory). See .
Geometry of the function field
If V is a variety defined over a field K, then the function field K(V) is a finitely generated field extension of the ground field K; its transcendence degree is equal to the dimension of the variety. All extensions of K that are finitely-generated as fields over K arise in this way from some algebraic variety. These field extensions are also known as algebraic function fields over K.
Properties of the variety V that depend only on the function field are studied in birational geometry.
Examples
The funct
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https://en.wikipedia.org/wiki/Noetherian%20scheme
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In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets , noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after Emmy Noether.
It can be shown that, in a locally noetherian scheme, if is an open affine subset, then A is a noetherian ring. In particular, is a noetherian scheme if and only if A is a noetherian ring. Let X be a locally noetherian scheme. Then the local rings are noetherian rings.
A noetherian scheme is a noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-noetherian valuation ring.
The definitions extend to formal schemes.
Properties and Noetherian hypotheses
Having a (locally) Noetherian hypothesis for a statement about schemes generally makes a lot of problems more accessible because they sufficiently rigidify many of its properties.
Dévissage
One of the most important structure theorems about Noetherian rings and Noetherian schemes is the Dévissage theorem. This theorem makes it possible to decompose arguments about coherent sheaves into inductive arguments. It is because given a short exact sequence of coherent sheavesproving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf and a sub-coherent sheaf , showing has some property can be reduced to looking at and . Since this process can only be applied a finite number of times in a non-trivial manner, this makes many induction arguments possible.
Number of irreducible components
Every Noetherian scheme can only have finitely many components.
Morphisms from Noetherian schemes are quasi-compact
Every morphism from a Noetherian scheme is quasi-compact.
Homological properties
There are many nice homological properties of Noetherian schemes.
Cech and sheaf cohomology
Cech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the sheaf cohomology of using Cech cohomology for the standard open cover.
Compatibility of colimits with cohomology
Given a direct system of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphismmeaning the functorspreserve direct limits and coproducts.
Derived direct image
Given a locally finite type morphism to a Noetherian scheme and a complex of sheaves with bounded coherent cohomology such that the sheaves have proper support over , then the derived pushforward has bounded coherent cohomology over , meaning it is an object in .
Examples
Many of the schemes found in the wild are Noetherian schemes.
Locally of finite type over a Noetherian base
Another class of examples of Noetherian schemes are families of schemes where the base is Noetherian and is of fi
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https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson%20theorem
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In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
Theorem
Given a differentiable real dynamical system defined on an open subset of the plane, every non-empty compact ω-limit set of an orbit, which contains only finitely many fixed points, is either
a fixed point,
a periodic orbit, or
a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these.
Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point.
A weaker version of the theorem was originally conceived by , although he lacked a complete proof which was later given by .
Discussion
The condition that the dynamical system be on the plane is necessary to the theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit.
In particular, chaotic behaviour can only arise in continuous dynamical systems whose phase space has three or more dimensions. However the theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems.
Applications
One important implication is that a two-dimensional continuous dynamical system cannot give rise to a strange attractor. If a strange attractor C did exist in such a system, then it could be enclosed in a closed and bounded subset of the phase space. By making this subset small enough, any nearby stationary points could be excluded. But then the Poincaré–Bendixson theorem says that C is not a strange attractor at all—it is either a limit cycle or it converges to a limit cycle.
See also
Rotation number
References
Theorems in dynamical systems
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https://en.wikipedia.org/wiki/Mixed%20boundary%20condition
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In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.
For example, given a solution to a partial differential equation on a domain with boundary , it is said to satisfy a mixed boundary condition if, consisting of two disjoint parts, and , such that , verifies the following equations:
and
where and are given functions defined on those portions of the boundary.
The mixed boundary condition differs from the Robin boundary condition in that the latter requires a linear combination, possibly with pointwise variable coefficients, of the Dirichlet and the Neumann boundary value conditions to be satisfied on the whole boundary of a given domain.
Historical note
The first boundary value problem satisfying a mixed boundary condition was solved by Stanisław Zaremba for the Laplace equation: according to himself, it was Wilhelm Wirtinger who suggested him to study this problem.
See also
Dirichlet boundary condition
Neumann boundary condition
Cauchy boundary condition
Robin boundary condition
Notes
References
. In the paper "Existential analysis of the solutions of mixed boundary value problems, related to second order elliptic equation and systems of equations, selfadjoint" (English translation of the title), Gaetano Fichera gives the first proofs of existence and uniqueness theorems for the mixed boundary value problem involving a general second order selfadjoint elliptic operators in fairly general domains.
.
.
, translated from the Italian by Zane C. Motteler.
, translated in Russian as .
Boundary conditions
Partial differential equations
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https://en.wikipedia.org/wiki/Port%20Saunders
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Port Saunders is a community of 674 located in Newfoundland and Labrador, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Port Saunders had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of cities and towns in Newfoundland and Labrador
References
External links
Port Saunders Town website
Port Saunders - Encyclopedia of Newfoundland and Labrador, vol. 4, p. 399-401.
Populated coastal places in Canada
Towns in Newfoundland and Labrador
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https://en.wikipedia.org/wiki/Monomial%20representation
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In the mathematical fields of representation theory and group theory, a linear representation (rho) of a group is a monomial representation if there is a finite-index subgroup and a one-dimensional linear representation of , such that is equivalent to the induced representation .
Alternatively, one may define it as a representation whose image is in the monomial matrices.
Here for example and may be finite groups, so that induced representation has a classical sense. The monomial representation is only a little more complicated than the permutation representation of on the cosets of . It is necessary only to keep track of scalars coming from applied to elements of .
Definition
To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple where is a finite-dimensional complex vector space, is a finite set and is a family of one-dimensional subspaces of such that .
Now Let be a group, the monomial representation of on is a group homomorphism such that for every element , permutes the 's, this means that induces an action by permutation of on .
References
Representation theory of groups
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https://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel%20lemma
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In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the
0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz, Richard Zippel, and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922.
Statement and proof of the lemma
Theorem 1 (Schwartz, Zippel). Let
be a non-zero polynomial of total degree over an integral domain R. Let S be a finite subset of R and let be selected at random independently and uniformly from S. Then
Equivalently, the Lemma states that for any finite subset S of R, if Z(P) is the zero set of P, then
Proof. The proof is by mathematical induction on n. For , as was mentioned before, P can have at most d roots. This gives us the base case.
Now, assume that the theorem holds for all polynomials in variables. We can then consider P to be a polynomial in x1 by writing it as
Since is not identically 0, there is some such that is not identically 0. Take the largest such . Then , since the degree of is at most d.
Now we randomly pick from . By the induction hypothesis,
If , then is of degree (and thus not identically zero) so
If we denote the event by , the event by , and the complement of by , we have
Applications
The importance of the Schwartz–Zippel Theorem and Testing Polynomial Identities follows
from algorithms which are obtained to problems that can be reduced to the problem
of polynomial identity testing.
Zero testing
For example, is
To solve this, we can multiply it out and check that all the coefficients are 0. However, this takes exponential time. In general, a polynomial can be algebraically represented by an arithmetic formula or circuit.
Comparison of two polynomials
Given a pair of polynomials and , is
?
This problem can be solved by reducing it to the problem of polynomial identity testing. It is equivalent to checking if
Hence if we can determine that
where
then we can determine whether the two polynomials are equivalent.
Comparison of polynomials has applications for branching programs (also called binary decision diagrams). A read-once branching program can be represented by a multilinear polynomial which computes (over any field) on {0,1}-inputs the same Boolean function as the branching program, and two branching programs compute the same function if and only if the corresponding polynomials are equal. Thus, identity of Boolean functions computed by read-once branching programs can be reduced to polynomial identity testing.
Comparison of two polynomials (and therefore testing polynomial identities) also has
applications in 2D-compression, where the problem of
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https://en.wikipedia.org/wiki/Paul%20Sally
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Paul Joseph Sally, Jr. (January 29, 1933 – December 30, 2013) was a professor of mathematics at the University of Chicago, where he was the director of undergraduate studies for 30 years. His research areas were p-adic analysis and representation theory.
He created several programs to improve the preparation of school mathematics teachers, and was seen by many as "a legendary math professor at the University of Chicago."
Life and education
Sally was born in the Roslindale neighborhood of Boston, Massachusetts on January 29, 1933. He was a star basketball player at Boston College High School. He received his BS and MS degrees from Boston College in 1954 and 1956.
After a short career in Boston area high schools and at Boston College he entered the first class of mathematics graduate students at Brandeis in 1957 and earned his PhD in 1965. During his graduate career he married Judith D. Sally and had three children in three years. David, the oldest, is a Visiting Associate Professor of Business Administration at Tuck School of Business at Dartmouth College, Stephen is a partner at Ropes & Gray, and Paul, the youngest, is Superintendent at New Trier High School.
Sally was diagnosed with type 1 diabetes in 1948. The condition resulted in his use of an eye patch and two prosthetic legs, which caused him to be widely referred to as "Professor Pirate," and "The Math Pirate" around the University of Chicago campus. He was known to detest cell phones in class and has destroyed several over the years by inviting students to stomp on them or by throwing them out of a window.
Career
Sally joined the University of Chicago faculty in 1965 and taught there until his death. He was a member of the Institute for Advanced Study from 1967–68, 1971–72, 1981–82, and 1983–84.
While at the IAS he collaborated with Joseph Shalika. In 1983, he became the first director of the University of Chicago School Mathematics Project, which is responsible for the Everyday Mathematics program (also called "Chicago math").
He founded Seminars for Elementary Specialists and Mathematics Educators (SESAME) in 1992. He co-founded the Young Scholars Program with Dr. Diane Herrmann in 1988, providing mathematical enrichment for gifted Chicago-area students in grades 7–12.
Death
Sally died December 30, 2013, aged 80, from congestive heart failure, at the University of Chicago Hospital.
Awards
Amoco Foundation Award for Long-Term Excellence in Undergraduate Teaching, 1995
American Mathematical Society Distinguished Service Award, 2000
Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics of the Mathematical Association of America, 2002
Fellow of the American Mathematical Society, 2012.
Selected publications
References
External links
1933 births
2013 deaths
20th-century American mathematicians
21st-century American mathematicians
Boston College alumni
Brandeis University alumni
University of Chicago faculty
Fellows of the American
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https://en.wikipedia.org/wiki/Lawrence%20C.%20Evans
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Lawrence Craig Evans (born November 1, 1949) is an American mathematician and Professor of Mathematics at the University of California, Berkeley.
His research is in the field of nonlinear partial differential equations, primarily elliptic equations. In 2004, he shared the Leroy P. Steele Prize for Seminal Contribution to Research with Nicolai V. Krylov for their proofs, found independently, that solutions of concave, fully nonlinear, uniformly elliptic equations are . Evans also made significant contributions to the development of the theory of viscosity solutions of nonlinear equations, to the understanding of the Hamilton–Jacobi–Bellman equation arising in stochastic optimal control theory, and to the theory of harmonic maps. He is also well known as the author of the textbook Partial Differential Equations, which is considered as a standard introduction to the theory at the graduate level. His textbook Measure theory and fine properties of functions (coauthored with Ronald Gariepy), an exposition on Hausdorff measure, capacity, Sobolev functions, and sets of finite perimeter, is also widely cited.
Evans is an ISI highly cited researcher.
Biography
Lawrence Evans was born November 1, 1949 in Atlanta, Georgia. He received a BA from Vanderbilt University in 1971 and a PhD, with thesis advisor Michael G. Crandall, from the University of California, Los Angeles in 1975. From 1975 to 1980, he worked at the University of Kentucky; from 1980 to 1989, at the University of Maryland; and since 1989, at the University of California, Berkeley.
Awards
2023 - Steele Prize for Mathematical Exposition
2014 - National Academy of Sciences
2013 - AMS Fellow
2004 - Steele Prize for Seminal Contribution to Research, with Nikolay V. Krylov
2003 - American Academy of Arts and Sciences
1979 - Sloan Fellow
Major publications
Evans, Lawrence C. Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363.
Crandall, M.G.; Evans, L.C.; Lions, P.-L. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502.
Evans, L.C.; Souganidis, P.E. Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 33 (1984), no. 5, 773–797.
Evans, Lawrence C. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986), no. 3, 227–252.
Evans, Lawrence C. The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, 359–375.
Evans, Lawrence C. Partial regularity for stationary harmonic maps into spheres. Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113.
Evans, L.C.; Spruck, J. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), no. 3, 635–681.
Evans, Lawrence C. Periodic homogenisation of certain fully nonlinear partial differential equations.
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https://en.wikipedia.org/wiki/Topple%20rate
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Topple rate is measure of how often the leading companies in a particular industry changes. It is defined as the probability that a firm in the industry, already in the top 20% in revenue, will drop out of this revenue leadership position within the next five years. The topple rate is one method of measuring overall competition in a particular industry; higher topple rates are often viewed as indicating a strong market with healthy economic activity. The term's invention is credited to McKinsey consultant Patrick Viguerie.
Modern industries have increasingly taken advantage of advances in technology and globalization to drive significant increases in productivity, but these changes also more easily enable new competitors and new innovations. These competitors often have lower costs, lower return requirements, or cheaper or imitation products. Topple rates across all industries doubled between 1972 and 2002, even when controlling for acquisitions of previous industry leaders; including them in the data showed that the overall rate actually tripled. At the peak of the Great Depression in 1937, companies listed on the S&P 500 had an average lifespan of 75 years. By 2011, that figure had fallen to 18 years, a reduction in lifespan of 76%.
In 2011, a study by Deloitte of 20,000 firms from 1965 to 2010 showed an overall doubling of the topple rate, though the effect occurred at different speeds across various industries. Those least affected by the increase in topple rate tended to be those more heavily regulated, such as aerospace, health care, and defense. However, rapid change can occur even in these industries if there are fundamental shifts in regulations or other disruptive forces.
References
History of business
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https://en.wikipedia.org/wiki/George%20Alfred%20Barnard
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George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control.
Early life and education
George Barnard was born in Walthamstow, London. His father was a cabinet maker and his mother had been a domestic servant. His sister Dorothy Wedderburn became a sociologist and eventually Principal of Royal Holloway, University of London. Barnard attended the local grammar school, the Monoux School, and from there he won a scholarship to St John's College, Cambridge, to read mathematics. In 1937 he went on to Princeton University to do graduate work on mathematical logic with Alonzo Church.
Career
Barnard was on holiday in Britain when the Second World War started and he never went back to Princeton to finish his PhD. The war made Barnard into a statistician as it did for many mathematicians of his generation. In 1940 he joined an engineering firm, Plessey, as a mathematical consultant. In 1942 he moved to the Ministry of Supply to apply quality control and sampling methods to the products for which they were responsible. It was there that Barnard began doing statistics. The group he was put in charge of included Peter Armitage, Dennis Lindley and Robin Plackett. Lindley recalls that they were like students working for a doctorate with Barnard as supervisor. Abraham Wald was in a similar group in the United States. Both groups developed sequential methods of sampling.
At the end of the war, Barnard went to Imperial College London, as a lecturer, becoming a reader in 1948 and professor of mathematical statistics in 1954. In 1961 he was elected as a Fellow of the American Statistical Association. In 1966 he moved to the newly created University of Essex, from which he retired in 1975. Barnard, however, kept on doing statistics until he died aged 86. Until 1981 he spent much of each year at the University of Waterloo, Canada, and after that he continued writing papers and corresponding with colleagues all over the world.
Barnard's best known contribution is probably his 1962 paper on likelihood inference but the paper he thought his best was the 1949 paper in which he first espoused the likelihood principle. He had originally described the principle in the context of optional stopping. A statement by Leonard Savage brings out how surprising the principle first seemed:
I learned the stopping rule principle from Professor Barnard in ... 1952. Frankly, I then thought it a scandal that anyone in the profession could advance an idea so patently wrong, even as today I can scarcely believe that some people resist an idea so patently right.
Political activism
In an interview Barnard recalled, "my main interest above everything was politics from about 1933 until 1956. Well, that’s not true – until the end of the war it would be fair to say." At school he proposed the motion to the school debating society that "Socialism is preferable to Capitalism." He join
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https://en.wikipedia.org/wiki/Madhava%20of%20Sangamagrama
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Mādhava of Sangamagrāma (Mādhavan) () was an Indian mathematician and astronomer who is considered as the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Late Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".
Biography
Little is known about Mādhava's life with certainty. However, from scattered references to Mādhava found in diverse manuscripts, historians of Kerala school have pieced together informations about the mathematician. In a manuscript preserved in the Oriental Institute, Baroda, Madhava has been referred to as Mādhavan vēṇvārōhādīnām karttā ... Mādhavan Ilaññippaḷḷi Emprān. It has been noted that the epithet 'Emprān' refers to the Emprāntiri community, to which Madhava might have belonged to.
The term "Ilaññippaḷḷi" has been identified as a reference to the residence of Mādhava. This is corroborated by Mādhava himself. In his short work on the moon's positions titled Veṇvāroha, Mādhava says that he was born in a house named bakuḷādhiṣṭhita . . . vihāra. This is clearly Sanskrit for Ilaññippaḷḷi. Ilaññi is the Malayalam name of the evergreen tree Mimusops elengi and the Sanskrit name for the same is Bakuḷa. Palli is a term for village. The Sanskrit house name bakuḷādhiṣṭhita . . . vihāra has also been interpreted as a reference to the Malayalam house name Iraññi ninna ppaḷḷi and some historians have tried to identify it with one of two currently existing houses with names Iriññanavaḷḷi and Iriññārapaḷḷi both of which are located near Irinjalakuda town in central Kerala. This identification is far fetched because both names have neither phonetic similarity nor semantic equivalence to the word "Ilaññippaḷḷi".
Most of the writers of astronomical and mathematical works who lived after Madhava's period have referred to Madhava as "Sangamagrama Madhava" and as such it is important that the real import of the word "Sangamagrama" be made clear. The general view among many scholars is that Sangamagrama is the town of Irinjalakuda some 70 kilometers south of the Nila river and about 70 kilometers south of Cochin. It seems that there is not much concrete ground for this belief except perhaps the fact that the presiding deity of an early medieval temple in the town, the Koodalmanikyam Temple, is worshiped as Sangameswara meaning the Lord of the Samgama and so Samgamagrama can be interpreted as the village of Samgameswara. But there are several places in Karnataka with samgama or its equivalent kūḍala in their names and with a temple dedicated to Samgamḗsvara, the lord of the confluence. (Kudalasangama in Bagalkot district is one such place with a celebrated temple de
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https://en.wikipedia.org/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva
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Jyeṣṭhadeva (Malayalam: ജ്യേഷ്ഠദേവൻ) () was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama (). He is best known as the author of Yuktibhāṣā, a commentary in Malayalam of Tantrasamgraha by Nilakantha Somayaji (1444–1544). In Yuktibhāṣā, Jyeṣṭhadeva had given complete proofs and rationale of the statements in Tantrasamgraha. This was unusual for traditional Indian mathematicians of the time. The Yuktibhāṣā is now believed to contain the essential elements of the calculus like Taylor and infinity series.
Jyeṣṭhadeva also authored Drk-karana a treatise on astronomical observations.
According to K. V. Sarma, the name "Jyeṣṭhadeva" is most probably the Sanskritised form of his personal name in the local language Malayalam.
Life period of Jyeṣṭhadeva
There are a few references to Jyeṣṭhadeva scattered across several old manuscripts. From these manuscripts, one can deduce a few bare facts about the life of Jyeṣṭhadeva. He was a Nambudiri belonging to the Parangngottu family (Sanskrtised as Parakroda) born about the year 1500 CE. He was a pupil of Damodara and a younger contemporary of Nilakantha Somayaji. Achyuta Pisharati was a pupil of Jyeṣṭhadeva. In the concluding verse of his work titled Uparagakriyakrama, completed in 1592, Achyuta Pisharati has referred to Jyeṣṭhadeva as his aged benign teacher. From a few references in Drkkarana, a work believed to be of Jyeṣṭhadeva, one may conclude that Jyeṣṭhadeva lived up to about 1610 CE.
Parangngottu, the family house of Jyeṣṭhadeva, still exists in the vicinity of Trikkandiyur and Alathiyur. There are also several legends connected with members of Parangngottu family.
Mathematical lineage
Little is known about the mathematical traditions in Kerala prior to Madhava of Sangamagrama.
Vatasseri Paramesvara was a direct disciple of Madhava. Damodara was a son of Paramesvara. Nilakantha Somayaji and Jyeshthadeva were pupils of Damodara. Jyeṣṭhadeva's pupil was Achyuta Pisharati and Melpathur Narayana Bhattathiri was Achyuta Pisharati's student.
Jyeshthadeva's works
Jyeṣṭhadeva is known to have composed only two works, namely, Yuktibhāṣā and Drkkarana.
The former is commentary with rationales of Tantrasamgraha of Nilakantha Somayaji and the latter is a treatise on astronomical computations.
Three factors make Yuktibhāṣā unique in the history of the development of mathematical thinking in the Indian subcontinent:
It is composed in the spoken language of the local people, namely, the Malayalam language. This is in contrast to the centuries-old Indian tradition of composing scholarly works in the Sanskrit language which was the language of the learned.
The work is in prose, again in contrast to the prevailing style of writing even technical manuals in verse. All the other notable works of the Kerala school are in verse.
Most importantly, Yuktibhāṣā was composed intentionally as a manual of proofs. The very purpose of writing the book was
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https://en.wikipedia.org/wiki/East%20Wales
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East Wales () is either a ITL 3 statistical region of Wales or generally a region encompassing the easternmost parts of the country.
Usage
The UK Office for National Statistics has as its highest level sub-division, East Wales, covering the whole east side of the country. It is defined as Powys, Flintshire and Wrexham, Monmouthshire and Newport, and Cardiff and Vale of Glamorgan. (The remainder of Wales is termed 'West Wales and The Valleys').
Sport
Welsh Athletics has four regions (East, West, North and South), with leagues for various disciplines having regional and inter-region competitions, particularly at school levels. The East Wales region covers Blaenau Gwent, Caerphilly (eastern half), Monmouthshire, Newport, South Powys and Torfaen.
The East Wales Bridge Association is one of four in Wales, the others being Mid, West and North. The East Wales Association has clubs in the historic counties of Glamorgan, Monmouthshire and Brecknockshire.
Rivers
Rivers in East Wales include:
River Usk
Ebbw River
River Llynfi
Sirhowy River
Some Notable People of East Wales
St. Tegfedd (Llandegveth)
Ian Gough
Taulupe Faletau
Aneurin Bevan
Ellis Shipp
Places of Interest
Cwmbran Stadium
Cwmbran Centre, The second largest shopping centre in Wales
Ysgyryd Fawr
Sugar Loaf, Monmouthshire
Offa's Dyke
Tintern Abbey
Pontypool Park
Blaenafon World Heritage Site
See also
Geography of Wales
Mid Wales
North Wales
South Wales
West Wales
List of Welsh principal areas by percentage Welsh language
Subdivisions of Wales
References
Regions of Wales
NUTS 2 statistical regions of the United Kingdom
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https://en.wikipedia.org/wiki/Circle%20bundle
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In mathematics, a circle bundle is a fiber bundle where the fiber is the circle .
Oriented circle bundles are also known as principal U(1)-bundles, or equivalently, as principal SO(2)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.
As 3-manifolds
Circle bundles over surfaces are an important example of 3-manifolds. A more general class of 3-manifolds is Seifert fiber spaces, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional orbifold.
Relationship to electrodynamics
The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with being cohomologous to zero, i.e. exact. In particular, there always exists a 1-form A, the electromagnetic four-potential, (equivalently, the affine connection) such that
Given a circle bundle P over M and its projection
one has the homomorphism
where is the pullback. Each homomorphism corresponds to a Dirac monopole; the integer cohomology groups correspond to the quantization of the electric charge. The Aharonov–Bohm effect can be understood as the holonomy of the connection on the associated line bundle describing the electron wave-function. In essence, the Aharonov–Bohm effect is not a quantum-mechanical effect (contrary to popular belief), as no quantization is involved or required in the construction of the fiber bundles or connections.
Examples
The Hopf fibration is an example of a non-trivial circle bundle.
The unit tangent bundle of a surface is another example of a circle bundle.
The unit tangent bundle of a non-orientable surface is a circle bundle that is not a principal bundle. Only orientable surfaces have principal unit tangent bundles.
Another method for constructing circle bundles is using a complex line bundle and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from we have that it is a principal -bundle. Moreover, the characteristic classes from Chern-Weil theory of the -bundle agree with the characteristic classes of .
For example, consider the analytification a complex plane curve . Since and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf has Chern class .
Classification
The isomorphism classes of principal -bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps , where is called the classifying space for U(1). Note that is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space Such bundles are classified by an element of the second integral cohomology group of M, since
.
This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because a circle is homotopically equivalent to , the complex plane with the origin remove
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https://en.wikipedia.org/wiki/Y-homeomorphism
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In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces.
It can be constructed by sliding a Möbius band included on the surface
around an essential 1-sided closed curve until the original position; thus it is necessary that the surfaces have genus greater than one. The projective plane has no y-homeomorphism.
See also
Lickorish-Wallace theorem
References
J. S. Birman, D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface, Trans. Amer. Math. Soc. 247 (1979), 87-124.
D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65 (1969), 409–430.
M. Korkmaz, Mapping class group of non-orientable surface, Geometriae Dedicata 89 (2002), 109–133.
W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307–317.
Geometric topology
Homeomorphisms
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https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan%20structure
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In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
History and the Kelvin problem
In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the hexagonal tiling, but although the first record of this honeycomb conjecture goes back to the ancient Roman scholar Marcus Terentius Varro, it was not proven until the work of Thomas C. Hales in 1999.
In 1887, Lord Kelvin asked the corresponding question for three-dimensional space: how can space be partitioned into cells of equal volume with the least area of surface between them? Or, in short, what was the most efficient soap bubble foam?
This problem has since been referred to as the Kelvin problem.
Kelvin proposed a foam called the Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex polyhedron with 6 square faces and 8 hexagonal faces. However, this honeycomb does not satisfy Plateau's laws, formulated by Joseph Plateau in the 19th century, according to which minimal foam surfaces meet at angles at their edges, with these edges meeting each other in sets of four with angles of . The angles of the polyhedral structure are different; for instance, its edges meet at angles of on square faces, or on hexagonal faces. Therefore, Kelvin's proposed structure uses curvilinear edges and slightly warped minimal surfaces for its faces, obeying Plateau's laws and reducing the area of the structure by 0.2% compared with the corresponding polyhedral structure.
Although Kelvin did not state it explicitly as a conjecture, the idea that the foam of the bitruncated cubic honeycomb is the most efficient foam, and solves Kelvin's problem, became known as the Kelvin conjecture. It was widely believed, and no counter-example was known for more than 100 years. Finally, in 1993, Trinity College Dublin physicist Denis Weaire and his student Robert Phelan discovered the Weaire–Phelan structure through computer simulations of foam, and showed that it was more efficient, disproving the Kelvin conjecture.
Since the discovery of the Weaire–Phelan structure, other counterexamples to the Kelvin conjecture have been found, but the Weaire–Phelan structure continues to have the smallest known surface area per cell of these counterexamples. Although numerical experiments suggest that the Weaire–Phelan structure is optimal, this remains unproven. In general, it has been very difficult to prove the optimality of structures involving minimal surfaces. The minimality of the sphere as a surface enclosing a single volume was not proven until
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https://en.wikipedia.org/wiki/Andr%C3%A9%20Hunebelle
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André Hunebelle (1 September 1896 – 27 November 1985) was a French maître verrier (master glassmaker) and film director.
Master Glass Artist
After attending polytechnic school for mathematics, he became a decorator, a designer, and then a master glass maker in the mid-1920s (first recorded exhibition PARIS 1927 included piece "Fruit & Foliage"). His work is known for its clean lines, which are elegant and singularly strong. He exhibited his own glass in a luxurious store located at 2 Avenue Victor-Emmanuel III, at the roundabout of the Champs Èlysées in Paris. Etienne Franckhauser, who also made molds for Lalique and Sabino, made the molds for Hunebelle's glass which was fabricated by the crystal factory in Choisy-le-Roi, France. Hunebelle's store ceased all activity in 1938 prior to World War II.
Hunebelle pieces are marked in several ways. The most common is A.HUNEBELLE-FRANCE in molded capitals either within the glass design or on the base. Other pieces are marked simply A.HUNEBELLE. There was also a paper label with A and H superimposed in a stylized manner. Since paper labels are frequently lost, many pieces may appear completely unmarked. In the author's collection there are pieces marked A.HUNEBELLE both with and without the word FRANCE, and a bowl marked MADE IN FRANCE that is identical to one shown in a Hunebelle catalogue. Hunebelle also used a more elaborate maker's mark imprinted on some glass pieces which had the word FRANCE encircled by the words MADE IN FRANCE MODELLE DEPOSE et R COGNEVILLE and with A. HUNEBELLE underneath (reflects mid 1930s partnership with COGNEVILLE).
In a short essay, he defined his stylistic aims as a glassmaker, explaining that he wanted to be "an adept of an abstract art where the geometric exactness, the poetry of line, and transparency are combined."
He also patented techniques for producing exact mouldings of items.
His glasswork displays a calculated modernism in contrast to influences derived from animals, plants and flowers which featured in the work of contemporaries such as René Lalique, Pierre D'Avesn and Marius-Ernest Sabino at the time. Hunebelle chose to focus on geometric forms, using technique and his scientific background to enhance light emission as much as possible. Surface contrasts, volume intersections, polished-non polished effects, geometry, light and poetry of line feature prominently in his work. Hunebelle employed both mold-blown and pressed-molded techniques in producing his pieces.
Filmmaker
Hunebelle was a publisher of a French newspaper called La Fleché. During World War II, he had no job until a friend Marcel Achard found him work in films for Production Artistique Cinématographique (P.A.C.) where he acted as an art director and later began producing films beginning with Leçon de conduite (1946). He directed his first film Métier de fous in 1948.
His next three films were a film series of French film noir featuring Raymond Rouleau as a journalist character mixing with c
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https://en.wikipedia.org/wiki/Brun%27s%20theorem
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In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.
Asymptotic bounds on twin primes
The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes.
Let denote the number of primes p ≤ x for which p + 2 is also prime (i.e. is the number of twin primes with the smaller at most x). Then, we have
That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor.
This bound gives the intuition that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms, the sum
either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant.
If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin primes. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes. Brun's constant could be an irrational number only if there are infinitely many twin primes.
Numerical estimates
The series converges extremely slowly. Thomas Nicely remarks that after summing the first billion (109) terms, the relative error is still more than 5%.
By calculating the twin primes up to 1014 (and discovering the Pentium FDIV bug along the way), Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6 as of 18 January 2010 but this is not the largest computation of its type.
In 2002, Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate that B2 ≈ 1.902160583104. Hence,
The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Dominic Klyve showed conditionally (in an unpublished thesis) that B2 < 2.1754 (assuming the extended Riemann hypothesis). It has been shown unconditionally that B2 < 2.347.
There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:
with value:
B4 = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.
This constant should not be confused with the Brun's constant for cousin primes, as prime pairs of the form (p, p + 4), which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n.
Further results
Let be t
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